Singular Perturbations I: Spaces and Singular Perturbations on Manifolds Without Boundary (Studies in Mathematics and Its Applications)

SINGULAR PERTURBATIONS I Spaces and Singular Perturbations on Manifolds without Boundary

STUDIES IN MATHEMATICS AND ITS APPLICATIONS

VOLUME 23 Editors: J.L. LIONS, Paris G. PAPANICOLAOU, New York H. FUJITA, Tokyo H.B. KELLER, Pasadena

NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD TOKYO

SINGULAR PERTURBATIONS I Spaces and Singular Perturbations on Manifolds without Boundary

LEONID S. FRANK Institute of Mathematics University of Nijmegen Nijmegen, The Netherlands

1990

AMSTERDAM

NORTH-HOLLAND NEW YORK OXFORD TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. SARA BURGERHARTSTRAAT 25 P.O. BOX 21 1,1000 AE AMSTERDAM, THE NETHERLANDS Sole distributors fo r the United States and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655, AVENUE OF THE AMERICAS NEW YORK, N.Y. 10010, U.S.A.

L i b r a r y o f Congress C a t a l o g i n g - i n - P u b l i c a t i o n

Data

F r a n k , L . S. ( L e o n i d S . ) . 1934Spaces and s i n g u l a r p e r t u r b a t i o n s on m a n i f o l d s w i t h o u t boundary / L e o n i d S. F r a n k . p. cn. (Singular perturbations ; 1) (Studies i n m a t h e m a t i c s and i t s a p p l i c a t i o n s ; v . 2 3 ) Includes bibliographical references (p. ISBN 0-444-88134-4 (U.S.) 1 . Global a n a l y s i s (Mathematics) 2. M a n i f o l d s (Mathematics) 3. S i n g u l a r p e r t u r b a t i o n s (Mathematics) 4 . Function spaces. I. T i t l e . 11. Series. 111. S e r i e s : F r a n k , L . S . ( L e o n i d S . ) , 1934S i n g u l a r p e r t u r b a t i o n s ; 1. OA372.FB v o l . 1 I PA6 141 5 1 5 ' . 3 5 2 s--dc20 [ 5 1 4 ' .741 90-7631

--

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Dddik 8.

J.-L. LIONS, L. SCHWARTZ, G.E. SHILOV

This Page Intentionally Left Blank

Vii

Contents

Introduction .............................................................

IX

Notation ..................................................................

1

Chapter 1. Manifolds. Functional Analysis. Distributions . . . . . . . . . . . . . . . . . . 5 1.1. Manifolds ......................................................... 5 1.2. Functional Analysis .............................................

10

1.2.1. Metric Spaces and Contractions ..........................

10

1.2.2. Topological Vector Spaces ................................

37

1.2.3. Banach Spaces ...........................................

42

1.2.4. Hilbert Spaces ...........................................

43

1.2.5. Classical Sobolev Spaces W,, and Holder Spaces CkiA. . . .45 1.2.6. Linear Functionals and Dual Spaces ...................... 51 1.2.7. Linear Operators .........................................

54

1.3. Distribution Theory .............................................

85

1.3.1. Test Function Space D ( V ) ................................

85

1.3.2. Distribution Space D'(V ) .................................

86

1.3.3. Some other Distribution Spaces ..........................

90

1.3.4. Parameter Dependent Distributions ......................

95

Notes ..............................................................

107

Chapter 2. Sobolev Spaces of Vectorial Order ...........................

109

2.1. Spaces on Rn ..................................................

109

2.2. Restriction to Hyperplane and Further Imbeddings Results . . . . . 114 2.3. Spaces of Distributions Supported by a Half-Space ............. 120 2.4. Spaces on R z ..................................................

122

2.5. Dual Spaces ...................................................

131

2.6. Sobolev Spaces of Vectorial Order on Manifolds . . . . . . . . . . . . . . . . 133 2.7. Spaces of One Parameter Families of Meshfunctions . . . . . . . . . . . . 138 2.8. Some Spaces of Two Parameter Families of Meshfunctions ...... 194 Notes ..............................................................

211

viii

Contents

Chapter 3 . Singular Perturbations on Smooth Manifolds without Boundary ..........................................................

213 3.1. Singular Perturbations with Constant Symbols ................. 213

3.2. Singular Perturbations with Homogeneous Symbols ............. 224 3.3. Singular Perturbations with Variable Symbols .................. 235 3.4. Continuity of Singular Perturbations in Sobolev Spaces of Vectorial Order ................................................ 3.5. Pseudolocality of Singular Perturbations ....................... 3.6. Asymptotic Expansions of Symbols ............................

239 241 243

3.7. Amplitudes, Adjoints and Products of Singular Perturbations ...250 3.8. The Stationary Phase, Laplace and Saddle Point Methods ...... 258

.................... 295 Diffeomorphisms and Singular Perturbations on Manifolds ...... 307 An Algebra of Difference Operators ............................ 328 Elliptic Singular Perturbations ................................. 366 3.13. Girding’s Inequality ........................................... 437 3.9. 3.10. 3.11. 3.12.

The Fourier Integral Singular Perturbations

3.14. Reduction of Elliptic Singular Perturbations t o Regular Perturbations .................................................. Notes ...............................................................

486 528

............................................................

533

Bibliography

ix

Introduction

Asymptotic analysis, which started as a mathematical tool for the treatment of special problems in mathematical physics affected by the presence of characteristic small or large parameters, has been rapidly developing during the last decennia, acquiring more and more global features and penetrating into different fields of mathematics and applied sciences. Although, originally, asymptotic analysis had a rather heuristic character, it was realized that, in order to guarantee the validity of formal asymptotic expansions, rigorous mathematical theories (especially uniform error estimates) were needed t o ensure further development and the applicability of existing formal techniques. The latter stimulated a vigorous growth of asymptotic analysis as an integral part of pure and applied mathematics. Singular perturbations being one of the central topics in the asymptotic analysis, they play also a special role as an adequate mathematical tool for describing several important physical phenomena, such as propagation of waves in media in the presence of small energy dissipations or dispersions, appearance of boundary or interior layers in fluid and gas dynamics, as well as in the elasticity theory, semi-classical asymptotic approximations in quantum mechanics, phenomena in the semi-conductor devices theory and so on. Elliptic and, more generally, coercive singular perturbations are of special interest for the asymptotic solution of problems, which are characterized by the boundary layer phenomena, as, for instance in the theory of thin buckling plates, elastic rods and beams. A perturbation is said to be singular since its structure and the nature of

X

Introduction

the phenomena which it describes is completely different from the ones which are proper t o the corresponding reduced problem. For instance, considering a gas flow around an obstacle in fluid dynamics in the situation when the di-

mensionless viscosity parameter (the inverse of the Reynolds number) is small, one has a mathematical model (the Navier-Stokes equations) which reflects the physical boundary layer phenomenon in a neighborhood of the obstacle, while setting the viscosity equal t o zero one gets a different mathematical model (the Euler-Lagrange equations), in which this phenomenon is completely lost. Considering a stochastic model which is a superposition of a deterministic process and of a “white noise” of a small level (described as a Wiener process with a small variance), one comes to the Kolmogorov-Chapman parabolic equation with a small diffusion term for the density of the stochastic process in question; it is a singular perturbation of the reduced hyperbolic equation, which describes the deterministic situation. Other examples come from the theory of elastic rods or beams. If an elastic beam at rest is subjected t o a strong pulling out longitudinal force described by a large dimensionless parameter, then using this parameter and setting it equal t o infinity, one can simplify the mathematical model, getting a reduced differential equation, which only partially refects the physical phenomenon. Indeed, for instance, in the case when the beam at rest is simply supported by its end points, the natural boundary conditions would tell that at the end points the displacements and the momenta of the forces applied must be zero. However for the reduced equation (which is of the second order) it is possible t o have only the displacements vanishing, while the momenta of the forces at the end points (not necessarily zero) are determined a posteriori. In fact, a boundary layer phenomenon in a neighborhood of the end points of the beam is present in this situation and should not be neglected. The linear singular perturbation theory and its possible applications is

Introduction

xi

the topic of this volume. Let A' be such a perturbation which is usually a differential (or integro-differential) operator affected by the presence of a small parameter

E

E (0, € 0 ) . One is interested in solving the equation

where f is a given second member., It is (impicitly or explicitly) assumed that the reduced equation (defined in a natural way and usually much simpler than (1)):

can be uniquely solved. Then one is interested in getting a convergent series

(3) for the solution u, of (l),and that is usually not possible, since, as a consequence of a singular nature of the perturbation A', the solutions of (1) do not depend analytically on

of

E

E

[-EO,EO]

E

even in the case when A' is a real analytic function

valued in some operator space.

Giving up the convergence, one asks for an asymptotic convergence of the series on the right hand side of (3), i.e. for each integer N

> 0 one would like

to have in a certain sense:

(4) Usually, formal asymptotic expansion techniques allow t o produce a relatively simple algorithm for computing recursively the coefficients uk, k 2 0, in the asymptotic expansion (3) or, even for more complicated forms of such an expansion, taking into account, for instance, the boundary layer phenomenon.

Introduction

Xii

A very important question, which arises afterwards, is a proof of the asymptotic convergence like (4) (or in a different form, appropriate to the situation considered). The only reasonable way to ensure the asymptotic convergence of approximate solutions to the solution of (1) is to have uniform a prion' estimates for u,, i.e. uniform upper bounds for the norm of the inverse

(d')-' (whose existence is, in fact, a part of the problem) as an operator from an appropriate data space V, into the solution space X,, operator

Such a question is not merely a matter of mathematical rigor, but it is crucial for the entire "raison d'Ctre" of the formal techniques which may, eventually, allow to determine uniquely uk, k

> 0, even in the situation, when the

solution to (1) does not exist or is not uniquely defined without additional restrictions.

For being specific, consider several examples. Example 1. Let q(z) (z E R3) be a real valued infinitely differentiable function and assume that q(z)

q ( o 0 ) for

1x1 2 r , r > 0 being sufficiently large.

Consider the following singular perturbation:

AEu:= u - E2div(q(z)grad u )

(6)

and the corresponding equatior

A'u, = f,

(7)

where f is a given infinitely differentiable function with compact support, i.e. f ( z ) vanishes outside of some ball in

R3,and

u,(z) is supposed t o vanish a t

infinity. The natural reduced operator A' for A' is the identity, so that uo = f, if uE in (7) admits an asymptotic expansion.

Introduction

Xiii

Furthermore, introducing the differential operator:

B ( z ,ax) := div(q(z)grad) = V . (q(z)V),

(8)

one can formally write an asymptotic expansion for u, in the form:

(9)

21,

-

= p k u 2 & ) ,

U2k(Z)

:= (B(.,ax))”(.),

k10

whose right hand side makes sense since f is smooth and has a compact support. Now the crucial question of an asymptotic convergence of the series on the right hand side of (9) to u, arises. Let us make the following basic additional assumption: inf q ( z ) = qo

(10)

> 0.

XER3

Under this assumption (10) (which is an ellipticity condition for the singular perturbation A , ) one can easily show the asymptotic convergence in (9). Indeed, integrating by part after multiplication of (7) by u,, using the Cauchy-Schwarz inequality and the basic assumption ( l o ) , one gets the following a priori estimate

(11)

(IIucII2.(P3)

+ &211vu&yR3J

1/2

I Y(Q)llfllLa(R3),

where

Introducing the norms of vectorial order s = ( ~ 1SZ, , sg) E R3,

where G(E) = Fx,Eu is the Fourier transform of u, one can rewrite (11) in the form:

5

l l ~ l l ( 0 , 0 , 1 ) , ~ Y(~)llfIl(O,O,O),c.

xiv

Introduction

Actually,using (13) and estimating more accurately by the Cauchy-Schwarz inequality, one gets the following sharp a priori estimate:

where y(q)

< 00

is defined by (12).

Differentiating (7) with respect to c and using the same argument, one gets for any integer s2 2 0,

s3

> 0 the following estimate:

where theconstant C ( S ~ , S Z , S ~ may , Eonly O , ~dependon ) s ~ , s ~ , s ~ , E o ,and Y(~) some derivatives of q ( x ) . An estimate like (15) can be established for any s = ( s I , s ~ , sE ~ )W3. Using (15) for each given s E

W3, one finds:

(16) llUe

-

E2kU2kIl(s),c

5 CE

2N

VN

Ilfll(sl,sa+2~,s3-2),,,

> 0,

V E E (O,Eo),

O
where the constant C

> 0 may only depend

on N , s,€ 0 , y ( q ) and some deriva-

tives of q ( x ) . Thus, (16) implies the asymptotic convergence in (9) and one may differentiate the asymptotic relation (9) with respect to

3:

any number of times

without loosing the asymptotic convergence, provided that f E C r ( R 3 ) , i.e. f is smooth and has compact support. Note that (9) is not satisfactory in the sense, that it provides the asymptotic approximations to ue, which all have their support coinciding with the support o f f . This is not the case for the solution u, of (6), (7) under the assumptoin

(lo), the support of u,

being the entire R3.

A more appropriate asymptotic formula for the solution u, of (6), ( 7 ) under the assumption (10) is provided by the following argument. First, assume that

xv

In trod u c tion q

q(z)

> 0 is a constant.

Then the solution of (6), (7) is given by the formula:

where FX+ and FCJx are the direct and inverse Fourier transform, respectively. Now, if q(z) is not a constant but still satisfies the conditions hereabove and, especially, the condition ( l o ) , one can still define the function: ti!')

(18)

: = ( 4 T ~ ' q ( z ) ) - ~ f ( y ) l ~ - y l - ~exp(-lz-yl/&(q(z))'/2))dy

=

J . 3

= (F;:.(l+

E 2 q ( z ) 1 ~ 1 2 ) - 1 F x - ~ f ):= ( z )(S'f)(z).

It turns out that

where

with a constant C > 0, which does not depend on

E,Q'

and f .

In other words, S' being the operator defined by (18) and introducing

) the functions u whose Fourier transforms are locally the spaces H (S),P (W3of integrable and have the norms (13) finite, VE E ( O , E ~ ) , one can rewrite ( 1 9 ) , (20) as follows:

A'S' = I - EQ',

(21)

I = identity,

where the family of linear mappings Q',

is equicontinuous, i.e. the norm of E

E

(O,EO), VEO

< 00.

&'

is uniformly bounded with respect t o

In troduction

xvi

I Thus, the norm of EQ' (0

(0 < E

< EO)

<

E

<

EO)

is strictly less than 1, i.e.

is an equicontraction, provided t h d

Hence, (21) yields for

EO

> 0 sufficiently

EO

EQ'

> 0 is sufficiently small.

small:

the series on the right hand side of (23) being convergent with respect to the operator norm in L(H(8-u),,(R3))uniformly with respect t o

E

E(0,~).

One can show that in fact one has also the following asymptotic relation:

SEA' = I - E Q ~ ,

E

E (O,EO),

where the family of linear mappings:

is again equicontinuous. Of course, for the zero approximation uLo) defined by (18) one has:

and moreover, both u, and u!)'

are supported by the entire W3.

Introducing

where the constant C > 0 does not depend on

E,

u, and f .

The basic difference between (16) and (26) is the fact, that the norm off on the right-hand side of (26) is the same, i.e. u i N - l ) defined by (25) converges

Introduction

xvii

at order O ( E ~ t o)u, in H(s1,,(R3),Vf E H ( s - v ) , ,(R3), thus, also for non-smooth second members f , while in (16) the larger is N

> 0 the smoother

the second

member must be in order t o have the asymptotic convergence of order O ( E ~ ) . Now, let us turn to the situation when q(x) does not satisfy condition

( l o ) , for instance, let us consider the case of q(x) z -1. Note that the formal asymptotic expansion (9) in this case takes the form: U,

-

C(-1)'E2'Akf, k10

where A is the Laplace operator, A = C a;,

aj

= 8/8xj.

However, equation (7), which ir. this case takes the form:

does not have a unique vanishing at infinity solution for f E CF(R3). One can consider two different classes of solutions satisfying at infinity the so-called Sommerfeld radiation conditions:

(29)

u,'(x) = O ( T - ' ) ,

(iaa,. f l)u,f(x)

= O ( T - ' ) , as 1x1 = T + CO,

where 8,. = d / d r is the derivative with respect t o

T

= 1x1.

Each solution u : is given by the formulae:

Thus, the right hand side of (27) does not converge asymptotically, since it does not 'know' t o which solution u,f or u; it should converge. In fact, different methods are needed, in order t o get convergent asymptotic approximations for the solutions u,f of (6), (7) when q(x)

5

qo

< 0,V x

E R3,

of course, u , f ( z ) being defined as solutions of ( 6 ) , (7) satisfying the respective Sommerfeld radiation conditions (29).

xviii

Introduction

Example 2. One of the efficient methods for solving approximately differential (and pseudodifferential) equations is the use of their finite difference approximations, which are, of course, perturbations of the approximated operators, the small parameter being the mesh-size of the uniform grid where the discretized difference equations are considered. Let us have a look at the boundary value problem:

c

-a,"u(c)

u = (0,l) = p ~ ( ~ ~ ) ,E au = {o,i}, = f (c),

cE

where 8, = d / d x , f is a given smooth function of

3:

E

r and p(x'), x' E d U ,

are given real or complex numbers. Since the solution u ( z ) of (31) is a smooth function of c E

8,too,

one

is templed t o use a higher order approximation of (31) on the grid v h =

{ c = k h , k=O, 1 , . . . , N }with integer N = h-' > 0. For instance, the following finite difference approximation of -8: has the

accuracy O ( h 4 ) on smooth functions:

where

a,,*

and a i , h are the forward and backward finite difference derivatives,

respectively, i.e.

Indeed, a straightforward computation shows that for any smooth function

u(x),c E

r,one has:

One is tempted to use (32) for solving numerically (31). However, one can not use (32) for all points on the grid the points

Xk

u h

but only for

= k h with 1 < k < N . Of course at the points 2 0 = 0 and

IN

=1

Introduction

XiX

one can use the boundary condition in (31). Still one extra boundary condition, say at the points x1 = h and XN-1 = 1-h, is missing and one has t o find this boundary condition in an appropriate way, since otherwise a discrete version of a boundary layer behavior in a neighborhood of the boundary dU = ( 0 , l ) will appear, i.e. solutions of the homogeneous equation

of the form C+q-"/h+C-q-(l-x)/h, with q = (7+@)

will emerge with non-

negligible coefficients C* . Thus, the approximation (32) along with appropriate boundary conditions is a singular perturbation of (31), h being the corresponding small parameter. Some perturbations by finite differences might destroy the basic structure and make disappear the fundamental properties of the operators which are being approximated. For instance, the basic property of the differentiation operator

8, = d/dx (x E R) is the fact that all the solutions of the homogeneous

equation &u(x) = 0 are smooth (in fact, constants). This property of 8, is preserved on the grid Rh = hZ = { x = t h , k E Z} by both the forward and the backward

finite difference approximations of 8,.

centered finite difference derivative

&,h

= (1/2)(&,h

+ a:,,)

& ,h

However, the does not enjoy

such a property anymore, since, besides the constants, also the non-smooth meshfunction (-l)"lh is a solution of the homogeneous equation &,hu(x)

=

0 on the grid Rh. The centered finite difference derivative is a non-elliptic approximation of

a,.

For the Hilbert transform

,

( H u ) ( z ):= (7ri)-1 v.p. the approximation

J. f(y)(z

- y)-ldy

Introduction

xx

is non-ellaptic (Hh is no longer invertible in (XhU)(2)

:= (Ti)-%

c

12(Rh)),

f(y)(z-y)--l(l-

while the approximation:

COS(7rh-1(z-y)),

2

E Rh

Y c R A \{s)

preserves on the grid Rh all the fundamental properties of the Hilbert transform. Example 3. Finite difference approximations of the heat equation provide a wealth of interesting situations from the point of view of singular perturbations. Let us consider the Cauchy problem:

(34) where cp(z), f ( z , t )are given smooth functions with compact support. Using the backward finite difference discretization as an approximation of

8, = a / d t on the grid W? = { t = k r , k>O}, one gets for the corresponding approximation ur(z, t ) , t E

W$

of u ( z , t ) the following singularly perturbed

recursion equations (implicit finite difference scheme):

For finding ur(x, t ) on each step of the recursion one has to invert the singular perturbation (1- &’A), r = E’

, which is precisely of the elliptic type discussed

hereabove. One might be tempted to use the centered finite difference proximating

at,,

for ap-

at since the accuracy in this case is O ( r 2 ) . This is the so-called

Richardson’s scheme. However,

at,, being a non-elliptic approximation of at,

one should not expect to have a finite difference scheme, reflecting the basic properties of the heat operator. First, there is a problem of imposing (or not) an extra initial condition, since the Richardson’s approximation is a three-step finite difference scheme. An easy Fourier analysis shows that actually one has to consider the corresponding discretized problem in this case on

W$ x RZ not as

xxi

Introduction

an initial, but as a boundary value problem with just one boundary condition at t = 0 and another one either a t t = co (the solution vanishes as t

< 03.

or at t = T

--+

+co)

In both cases the smooth part of the solution will be 'pol-

luted' by the non-smooth part generated by the solutions of the homogeneous equation having the form: v,(i,t)

= ( - l ) ( T - * ) /(T( ~ + T ~ A-~~)A' )/ -~( ~ - * ) l ~=$ ( z ) = (-i)(*-*)/~ (~;-,((1+~21t14)1/2

+T1t12)-(T-*)/r~z+E~)

(I,t),

> 0, the oscillatory non-smooth factor ( - l ) ( * - * ) / ,being always

for any finite T present.

A good scheme having the accuracy O ( T ~is )the following one: (36)

{

t

7

+

("

(T/2)A2 - A)@t,,)u,(z,t)

(

U(t,O)

= cp(I),

2

-+

= (l +

-A))f(i>t),

E R", t E R, = 7Z+,

where Z+ is the set of all non-negative integers and O t , , is the shift operator

on

-+ W, , i.e.

(Ot,,v)(t) = v(t+7).

Note that the corresponding singular perturbation t o be inverted on each step of solving the implicit finite difference problem (36), is again an elliptic singular perturbation having the form:

(37)

A' = 1 - c2A + ( l/2)c4A2, c2 = T ,

I

E R" .

Of course, hereabove one may use the usual finite difference approximation Ah of the Laplacian on the grid R i = hZ":

thus getting (by using the schemes hereabove) unconditionally stable timespace finite difference approximations of the heat equation with one condition at t = 0 (also for the Richardson's scheme), i.e. the approximations hereabove with A replaced by Ah given by (38), are stable (in the sense of nonaccumulation of the errors), whatever the mesh-sizes

T

and h are.

xxii

Introduction

One of the interesting aspects in the theory of singular perturbations is the question of the possibility to reduce such a perturbation to a regular one. This is what can be done for any elliptic singular perturbation. Namely, one can construct explicitly an elliptic singular perturbation SE such that the product

S'A'

will be a regular perturbation of the corresponding reduced operator

A'. Such an operator S' is given by (18) hereabove in the case of the elliptic singular perturbation A' given by ( 6 ) , (7) and satisfying (lo), the reduced operator for A' being the identity. The same kind of reducing operator can be also constructed in the case of elliptic finite difference operators with one

( h > 0) or two

(E

> 0, h > 0) small parameters.

This volume deals with linear singular perturbations (on smooth manifolds without boundary), considered as equicontinuous linear mappings between corresponding families of Sobolev-Slobodetski's type spaces H ( s ) , cof vectorial order s € R3. Chapter 1 provides the necessary (also for the next volume) functional analytic background aiming at the situations, characterized by the presence of a small parameter. Chapter 2 is devoted to the spaces H(,),' and their finite difference versions. Chapter 3 deals essentially with elliptic and hyperbolic singular perturbations and their finite difference counterparts. Here also the classical asymptotic stationary phase, Laplace and saddle point methods are presented and used later on for different purposes, as for instance, the local theory of singularly perturbed Fourier integral operators, diffeomorphisms and symbol transformations in the C*-algebra of the singular perturbations and so on. A special attention is given t o the sharp form of GQrding's inequalities and their applications. The next volume will be devoted to singular perturbations in the elasticity theory and, more generally, to coercive singular perturbations, as well as t o

In trod uction

xxiii

singular perturbations of dissipative and dispersive type. In a work of this kind it is impossible to provide even a first approximation to an adequate list of references, which would, at least remotely, reflect the wealth of publications in the field of the singular perturbation theory and its applications. The only way to produce a first approximation to the solution of the problem of providing a comprehensive bibliography, seemed to me t o restrict the references to publications, which are more or less connected with the topic of this book, thus, taking a chance not to mention some valuable contributions to the asymptotic theory. I would appreciate any suggestion indicating me either inadequate or missing (but still relevant t o the topic) references, so that they could be included in the next volume. I started working on the theory of difference operators in the late fiftiesearly sixties in MOSCOW, being strongly influenced by remarquable achievements in the general theory of partial differential equations and the initiated development towards the theory of pseudodifferential operators. This influence can be easily traced back in my work on the C*-algebra of one parameter families of difference operators. The work on coercive singular perturbations had as its starting point the year of my immigration to Israel (1972) and had been progressing during my staying at the Hebrew University of Jerusalem (1973-1976), my sabbatical leave at the University of Kentucky (1976-1977) and my tenure at the University of Nijmegen (1977-), where Wolfgang Wendt, Guido Sweers, Johannes Heijstek and Henk Norde have been working with me. Writing this book was not an easy task to me, because of a lack of communication and considerable amount of other professional commitments. Still it was enjoyable and challenging. I am sure, many deficiencies (including awkward

linguistics, so characteristic for people whose mother tongue is not English) can be found spread over the text of this and the next coming volume. I should be

xxiv

Introduction

mostly grateful for any suggestion or criticism aiming at the improvement of the book.

I am deeply indepted t o Alain Bensoussan, Bernard Helffer, Denise Huet, Jacques-Louis Lions and Harold Widom who agreed to read the first volume and a part of the second one of this book and helped me by their remarks and criticism t o improve the manuscript. This help has been invaluable to me.

I would like t o express my gratitude t o the Elsevier Science Publishers for encouraging me to carry out the work on this book. Writing it has yet taken much longer than both of us, the Publisher and myself, have ever expected.

Nijmegen, February 1990

Leonid Frank

1

NOTATION

IRn IRE

IRn

5

:

n-dimensional (real) Euclidean space

:

n-dimensional (real) Euclidean space of variables x = (xl,...,x

:

n-dimensional Euclidean space of dual variables 5 = ( 5 l , - - - , t n )

Bnr+ : half-space of x E R: Cn

> 0

n-dimensional complex space of variables

:

c

=

.

(cl,.. ,cn)

x 5 +. . .+x 5 : scalar product between x E IR and 5 E IRn 1 1 n n 5 2 2 i 2 2 3 : Euclidean norms in illRn and illn ( x l + ...+ xn) , 151 = (C1+ ...+ 5,)

<x,O

1x1

with x

)

=

=

5

respectively <x> = (1+/xI2)+for x E mn

<*

=

...,<:

(<;,

:

*

<

l
=

1 1

(lC1l

=

...+ 5 ' 5 n n

< I < * +

2

:

+...+l
the hermitian product in Cn :

Euclidean norm in Cn

U

:

open subset of Euclidean space Rn

uc

:

complement of a subset U in illn

U

<

the complex conjugate of

X

CC

V

means that the closure n

:

of

u is a compact subset

of an open set

" c RX z z+

:

set of all integers

:

set of non-negative integers

Zn =

ZX..

:

2:

.xZ

:

set of all the n-tuples m

set of n-tuples a

Znf+: set of n-tuples m

8

=

.

(a1,. .,a

(my,.-.,m

)

(ml,.. . ,m

7

1

=

...a n !

al!

:

length of a E Zn

)

with m . E 2 , 1 5 j 5 n I

with a , E Z + 3

)

with m , E Z, 1 5 j < n. mn E 1

5 a : means a . 5 B ~ I,< j i n for a , E~ zn

la/ = a +...+a

a!

=

=

z+

Notation

2

...xa

xa = x"1

sa

c1a~ ...E nan

=

h E (0,h0 1,

<

hZn

=

:

E

mn,a

for x E

for E, E I R ~

cL

5'

E : Z

E : Z

E (O,E 0 1 small positive parameters

one parameter family of grids {x =(xl,...,xn), x 1 5 j S n } with m . E Z, 1 5 j < n , h

7

Tn Erh

mn

:

dual of

a"

:

<,+

,...,

aEl. *-at;

'j

= Dal.. .Dan

X.

El...~cI a

For a u(c()

ern

1

with D

-ia/ax ., 1 5 j 5 n 1

-ia/aSj, l < j ~n

=

function u ( x ) :

aau

=

=

=

(sometimes denoted by D) with D

xn

Da = D"1

LI

3

, i.e. one parameter family of n-dimensional toruses

family of grids in IF?

x1

a

hm.,

(also denoted 1 , i.e. of all X,+ m . E Z, l S j < n , mn x = (x, x 1 , x . = hm '+ n 1 j' I a 15 j S n =.':2 . .a (sometimes denoted by a) with a X . = a/ax,, I 1 xn 1 al a with a = = a/aSj, l < j < n

5 :D

=

{ 5 = ( 5 , ,..., S n ) E I R n , ( h S 3 . ( S n ,1 5 j ~ n ~ , h E ( 0 , h o ~ , T ~ = T ~ , 1 5-

x,h,+

a"

%

j

E (O,hol

v

X u =

D u = (D

U,

x1

grad

u =

...,DX

U)

..,au/axn)

(au/ax,, . =

(-iau/axl,...,-iau/axn)

n m

Taylor series for u E C

:

Leibnitz'formula:

Q(x,D)

:

linear partial differential operator polynomial in D with coefficients in

m

C

(U), i.e.

3

Notation

If some q (x) with la1 = m does not vanish identically, then m is said to be the order of Q(x,D).

When q (x)

qa,

va,

la1 5 m, do not depend on x E U,

then we write Q(D). t Q(x,D)

:

transpose of Q(x,D), defined as follows

Q(x,D)*

:

adjoint of Q(x,D) defined by Q (x,D)*U (x)

C

=

Da (q: (x)u (x)1

la14m where qi(x) x E 2

A = (a/ax,)

supp u

:

is the complex conjugate of q, (x),

=

((11

5 m,

u.

+...+ (a/axn)’

:

the Laplace operator in variables x E Rn

the support of the function u(x), i.e., the closure of the set of points at which u(x) does not vanish

k -

C (U)

:

space of k times continuously differentiable complex-valued functions in the closure of U, where either k E Z or k =

Ckny(;)

:

subspace of C k ( t ) of functions whose derivatives of order k satisfy the

HBlder condition with 0 < y < 1, i.e.

~ x - ~ ~ - ~ I D ~ ( x ) - D<~ m, u ( V~a), Ila1 SUP Y X,YEU,X#Y m

=

k

m

C (U) 0

:

subspace of C ( U ) of functions with compact support in U; the m

elements of S(nn

+m

:

)

C

0

(U) are also called test-functions in U

the Schwartz space of complex-valued

m

C

functions u(x), x E En,

satisfying the conditions:

L (U)

P

:

Lebesgue space of (equivalence classes) of functions u such that lu(x)

m

L (U)

:

Ip

is integrable over U, 1 5 p 5

m,

i.e. the following norm is

Lebesgue space of (equivalence classes) of essentially bounded functions u in U, i.e. the following norm is finite:

Notation

4

where ess sup is the supremum of (u(x)1 over X E W E with E the set of Lebesgue measure zero where lu(x) I may be +-.

"

= f u(x)v*(x)dx (u,v) L2(W

:

the scalar or hermitian product in L2(U), also

denoted (U,V)~,where v*(x) is the complex-conjugate of v(x), x € U. 0

x,h

(Ox

=

,h' '. . "xn,h

)

:

the shift operator 3n the family of grids

4 , (Ox

hu) (x) = u(x+he ) . k k' -1 2 = h ((Ox h-l),...,(Ox - 1 ) ) = h (Ox,h-l) : forward finite x.h 1' nth difference derivative with respecc to x E IF(. (lattices)

-1

a*

-1 h-l((l-0-l ), ..., (1-0 h)) = h-'(l-O-' ) : backward finite x,h 3,h xn difference aerivative with respect to x E

=

x,h

-i2 x,hD* = ia* x.h x,h' D

x ,h

=

<(h,S) = (51(h,S1),..-,5n(h,Sn))ICk(h,Sk) = (ih)-'(exp(ihC,)-l)

:

Fourier

transform of D x,h' 5*(h,5)

(5;(h,S1) ,...,5~(h,Sn)),5;(h,Sk)

=

=

(ih)-'(l-exp(-ihE k) )

:

Fourier

transform of D* x,h'

~ ( 5 =) w*(s)

F x-ts -1 F

1 ( 51)

(0

,...,wn(Sn)),

(W~(C~),...,W~(C~)),

=

~ ~ ( 5 =, )-i(exp(iSk1-1). w;(ck)

=

-i(l-exp(-iEk)).

:

Fourier transform, Fx-tSu = ; ( 5 ) , ;(E)

:

F-l inverse Fourier transform, S+xu

=

=

f u(x)exp(-i<x,<>)dx. Rn

u(x),

5-tX

u(x) = (2n)-" f

;(<)exp(i<x,S>)dS.

an F x-tS,h

:

discrete Fourier transform on the lattice

Gh(<)

=

hn Z

X(

, Fx-t5,h~=

-h u (5),

u(x)exp(-i<x,C>).

X€<

-1 5 -1 a,h;h F : inverse discrete Fourier transform, F E-tx,h

u(x)

-h

=

u(x),

(21~)-~fu (5)exp(i<x,5>)d5. Tn 5 ,h The Poisson formula : relates the integral and discrete Fourier transforms =

as follows: iih(<) =

C

kEZn

;(5+21rh-'k).

5

CHAPTER 1

MANIFOLDS, FUNCTIONAL ANALYSIS, DISTRIBUTIONS

In this chapter basic definitions and results concerning smooth manifolds, classical functional analysis and distributions are presented. Since mostly this material belongs to the standard background, the proofs are given only

of the statements involving one or several small parameters, which are extensions of classical results to the situation characterized by the presence of these parameters and where the uniformity (equicontinuity) is needed for further applications to singular perturbations and their finite difference approximations.

1.1.

Manifolds

Let M be a set. Definition 1.1.1.

An a t l a s o f

CZUSs

Cp (p 2 1) on M is a cOZk?CtiOn

Of

p a i r s ( u , , + , ) ( j ranging in some indexing s e t ) s a t i s f y i n g t h e foZZowing 3

3

conditions: (i) Each

u . i s a s u b s e t of M and t h e { u . } cover 3

3

M.

(ii) Each 4 . is a b i j e c t i o n of U . o n t o an open s u b s e t +.(u.)of an 1

EucZidean space Bn and f o r any j (iii) The map

3 ,k

the set

+ 1 ( U1.n U,) ,

3

1

is open in

mn .

is a c P - b i j e c t i o n of t h e open s e t +,(uj n u,) i n nnonto t h e open s e t , ) i s c a l l e d a c h a r t of t h e a t l a s . (U fl u,) cnn. Each p a i r ( U . k j 3 1 I f a p o i n t x E M l i e s i n U . t h e n ( u , , + , )is s a i d t o be a c h a r t a t x.

+

,+

3

3'

3

It will be assumed that each t.wo points x,y E M have non-intersecting charts at x and y , respectively. Definition 1.1.2. Given an open s u b s e t

u

c M and a b i j e c t i o n

+

:

u

-P

u'

w i t h U' an open s u b s e t in m n , (u,+) is s a i d t o be compatible w i t h t h e

1. Manifolds, Functional Analysis, Distributions

6

a t l a s {(u.,$.)} if each map Oj$-' i s a c P - b i j e c t i o n o f t h e corresponding 3 3 d' ( a s d e f i n e d i n ( i i i ) ) .

open s u b s e t s in

Two a t l a s s e s a r e s a i d t o be compatible i f each c h r t of one a t l a s i s compatible w i t h t h e o t h e r a t l a s . An equivalence c l a s s o f a t l a s s e s o f c l a s s

s t r u c t u r e o f CP-manifold on

M,

on

Cp

M

is s a i d t o d e f i n e a

t h e dimension n of Euclidean space Rn i n

D e f i n i t i o n 1.1.1. being t h e dimension o f t h e cP-manifold A chart $ : U

x E

+

M.

. ..,on (x),

i s given by n coordinate f u n c t i o n s $ ( x ) , 1

Wn

u.

Examples 1.1.3. m

i s a C -manifold, whose a t l a s c o n t a i n s j u s t one

lo. Euclidean space Wn chart.

2O. Sphere

n-1

S

C

n IR

,n

Sn- 1

> 2,

= {x E

nn!

1x1 = 11 i s again a

m

c -

manifold, whose a t l a s c o n t a i n s a t l e a s t two c h a r t s ( U ,,$ , ) I , j = 1,2 t h e -1 1 1 maps $ . , j = 1 , 2 , being t h e s t e r e o g r a p h i c p r o j e c t i o n s onto two copies of n-1 B

.

m

Further c i r c l e S1 i s a C -manifold,whose a t l a s contains one c h a r t given by t h e angle coordinate 8 =

0-l

:

n1

-f

S

1

.

nn i s s a i d t o be a k-dimensional manifold n , i f in each c h a r t u a t x, t h e r e imbedded in Euclidean space Wn , k

D e f i n i t i o n 1.1.4. A s e t M c

e x i s t c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s f.: U 1

-f

W ,

1 5 j 5 n-k such

t h a t t h e i n t e r s e c t i o n u Cl M is d e f i n e d by t h e equations f . = 0 , 1 5 j 5 n-k 1 and, moreover, t h e v e c t o r s IVfi)lbj5n-k are l i n e a r l y independent.

If

f ,E 3

cP, p

2 1, t h e n M is a k-dimensional

cP-manifoZd imbedded in mn

.

In each c h a r t U 3 x one may introduce l o c a l coordinates by solving t h e equations f . ( xl,...,x

= 0 , 1 2 j 5 n-k

)

1

a s f u n c t i o n s of o t h e r

k

and by expressing n-k coordinates x . 1 c o o r d i n a t e s , t h e i m p l i c i t f u n c t i o n theorem being

a p p l i c a b l e h e r e , s i n c e v e c t o r s { V f . ( x ) ) l b j j n - k a r e l i n e a r l y independent, 1 V X E U .

Example 1.1.5. The r o t a t i o n group S O ( 3 ) of Euclidean space W3

m

i s a C -manifold 3 9 homeomorphic with ( r e a l ) p r o j e c t i v e space P ) imbedded i n t o W

Let M

mx

C

Wn

be a jc-dimensional CP-manifold

imbedded i n Wn

(which i s

.

. Tangent

a t each p o i n t x E M i s defined a s t h e orthogonal complement of n-k Euclidean subspace R spanned by v e c t o r s {Of, (x))lsj,
1.1. Manifolds

Vectors in TM

7

are also said to be tangent of M at point x.

Tangent space TM

can be also defined in a different way,

A map x = $(t), t E (t t ) 5 W , tl < 0 < t2, $(t) E M is said to be a 1' 2 differentiable curve on 14, if $(t) is a differentiable function of t.

are said to be equivalent, if $ . ( O ) = x I

Two curves $.(t), j = 1,2, 3

and lim t-'(Q1(t)-Q2(t)) = 0. t 4 Definition 1.1.6.

The equivalence c l a s s o f d i f f e r e n t i a b l e curves $(t), $ ( O )

=

x, i s s a i d t o

be a tangent v e c t o r t o M a t p o i n t x E M. The s e t of a l l tangent v e c t o r s a t x E M equipped w i t h t h e n a t u r a l s t r u c t u r e of a v e c t o r space is s a i d t o be

tangent space t o M a t p o i n t x E M. m

Let U

C

M be a chart on

C

-manifold M and let $(t)={$l(t),...,$n(t,)

be

acurveonM, $ ( O ) =x.ThenthenumbersF,.= $ . ( O ) ,

l<jZn with $.(t) = (d/dt)$.(t) 3 3 3 1 are said to be coordinates of the tangent vector 5 E TM at point x E M.

Union U TM of tangent vector spaces at each point x E M has a xEM x natural structure of a differentiable manifold of dimension 2n, where n is the dimension of M. Definition 1.1.7.

u xEMTMx i s c a l l e d tangent bundle o f cP-rnanifold M and i s denoted by

TM.

TM is again a CP-manifold whose dimention is twice the one of M.

A generic point in TM is the pair (x,S) where x E M and 5 E TM tangent vector of M at

X.

Local coordinates on TM may be introduced in the

following fashion. Let x l , ...,x Ux) and

El, ...,E

iS a

be local coordinates on M (in some chart

the components of a tangent vector 5 of M at x. Then 2n

...,xn,S1,...,Sn)

real numbers (xl,

are local coordinates of a point

(x,E) E TM. The map

is said to be the natural projection. -1 Inverse image p (x) of x E M is tangent space TM

X'

called also the

fiber at x of tangent bundle TM. If M is imbedded in Euclidean space, then the Euclidean metric naturally generates arc length and angle measurements, volume measure etc. on M. All these quantities can be expressed in terms of length of tangent

1 . Manifolds, Functional Analysis, Distributions

8

vectors, i.e. in terms of a positive definite quadratic form (in 5 ) given on each fiber TM

:

TMx 3 5

(1.1.2)

-f

<<,6> E IR+.

For instance the arc length l ( y ) of a curve y:

[tl,t21 3 t

+

x(t) E M on

M, is given by the formula: l(y)

(1.1.3)

t f J2 dt.

=

Definition 1.1.8.

A cP-manifold M with a given positive definite quadratic form <<,c> on each tangent space TM is said to be a Riemannian manifold, quadratic form < 5 , 5 > being the Riemannian metric on M. Remark 1.1.9.

...,x

Let U be a chart of an atlas on M with local coordinates xl,

in U.

Then Riemannian metric is given by the formula: 2 ds =

(1.1.4)

’’

Z a . (x)dx.dx lSj,ksn k’

...,dx

where (dxl,

I lajk(x) I I

)

a . (x) = a (x) kj

are the coordinates of a tangent vector x, matrix

is posqtive definite and of class CP-’.

Let f

:

M

+

N be a map from differentiable manifold M into differentiable

manifold N (bothof class $)

. Then

f is said to be differentiable (of class Cp) if

it is given in local coordinates on M and N by differentiable (of class

C)‘

functions. Definition 1.1.10.

Let f

M

+

N be differentiable (of class

and let Ji

:

iR

:

-f

cP) map. Let (x,c) E

M be a curve on M such that $(O) = x, $ ( O )

=

TM be given 6. Define

The map

is called the derivative of differentiable map f

: M

+

N

at point x E M.

It is readily seen that f ( 5 ) does not depend on the choice of a *x.

curve $(t) on M with $(O) = x,

Ji(0) =

5.

1.1. Manifolds

Let x =

...,yn) be local coordinates on M and N, ...,Sn) and ( q 1 , ...,q ) be the components of

(Xl,...,~n),

respectively and let ( E l ,

9

y = (yl,

respectively. tangent vector 5 E TMx and tangent vector TN f(x) ' Then one has:

,

Taking the union of all maps f,

x E M, one gets a single map from

X

tangent bundle T M into tangent bundle TN: f,

:

TM

f,(S)

TN,

-f

=

f, (6) for 5 E TMx. X

The map f, g

:

N

-+

:

TM

TN is again differentiable. Furthermore, if f : M

-f

K, h = g

f

o

:

M

+

K, then it holds: h,

= g,

o

-f

N.

f,.

Now a useful concept will be introduced. Let M be a $-manifold (p t 1). Definition 1.1.11. A p a r t i t i o n of u n i t y (of c l a s s cp) on a cp-manifold M c o n s i s t s of an open

covering { u . } of M and a system of functions $ .

1

3

:

M + R satisfying the

conditions: (i) For aLZ x E M one has:

I$.

1

(x) t 0 .

(ii) The support of $ , i s contained i n U.. 3

1

{u.} of

(iii) The covering

1

M i s finite.

(iv) For each point x E M one has:

z i If

$.(XI = 1.

1

{vk } i s another covering of

t o { u , } i f each v

M, then

i s contained i n some

it i s said t o be subordinated

u..

1 k 1 Each compact differentiable manifold admits a differentiable partition

of unity. One can consider CP-manifolds with boundary which are defined in the same way as previously with the only difference that the bijections $I. in 3

Definition 1.1.1

in this case are either between a chart U . and an open set

in Rn or between U .

...,x

If (xl,

1

)

n

3

M and some set in Rn in Euclidean space Wn

.

are local coordinates on a CP-manifold M in a neighbour-

hood of one of its boundary points, then the boundary aM of M in this neighbourhood may be given as the collection of the points (x~,...,x~-~,~).

10

1. Manifolds, Functional Analysis, Distributions

The boundary a M is a CP-manifold, whose dimension is n-1. A typical example of a CP-manifold with boundary is an open domain U

in Euclidean space Rn whose boundary aU is a CP-hypersurface of dimension n-1. 1.2. Functional Analysis In this section a (part of the) classical functional analytic background and its extension to the situation with a parameter dependance are presented, which will be useful later on in the following chapters. 1.2.1. Metric Spaces and Contractions Definition 1.2.1.

An a b s t r a c t s e t

x i s s a i d t o be a (complete) m e t r i c space, or simply, a p : x x x m+ i s w e l l - d e f i n e d , which X

m e t r i c space, i f a d i s t a n c e f u n c t i o n

-f

s a t i s f i e s the following conditions: (i) I d e n t i t y p r o p e r t y : px(x,y)

0 i f f x = y.

=

(ii) Symmetry: pX(x,y) = px(y,x). (iii) T r i a n g l e i n e q u a l i t y

p (x,y)+p (y,z) 2 p (x,z). X X X c x such t h a t p (x ,x ) + 0 (iv) Completeness: For each sequence {x v v21 X V l J f o r v , -t~ +m, t h e r e e x i s t s an element x E x such t h a t p (x,x ) -t 0

x

for v

v

+ +m.

Only complete metric space will be considered. If a set X ' equipped with a distance function p is not complete, then one

may also take the completion X of X' with respect to the convergence concept (topology) defined by

p

on X'.

Definition 1.2.2.

L e t x and Y be two m e t r i c spaces and l e t t h e map x 3 x + f ( x ) = y E Y be d e f i n e d on some s e t M c x and valued i n some s e t i n Y . The map f(x) i s s a i d t o be continuous a t some p o i n t x E M, i f f o r each E > 0 t h e r e e x i s t s a 0

6 > 0 such t h a t px(f(x),f(xo)) <

In other words, f : M sequence {x }

v v>l

for v

C

+

E

f o r each x E M w i t h px(x,x

0

Y , M C

M with px(xv,xo)

)

<

6.

X is continuous at xo E M iff for each -f

0 as v

+ m,

one has: p Y (f(xv),f(xo))-to

+ m.

Definition 1.2.3.

Two m e t r i c spaces

one-to-one

map f

x and :

x

+

Y Y

are s a i d t o be homeomorphic i f t h e r e e x i s t s a

o f x onto

Y

such t h a t f

:

x

+

Y and f-l

: Y

+

x

1.2. Functional Analysis

are c o n t i m o u s a t each p o i n z o f

X,

11

respectively of

Y.

Definition 1.2.4.

Let G. 5 I R ~be an open s e t . We s h a l l c o n s i d e r continuous f u n c t i o n s x(w), x

:

G.

+

x valued i n a m e t r i x space x w i t h t h e d i s t a n c e j i m c t i o n

p

Assume

X'

t h a t x is a d d i t i o n a l l y equipped w i t h a ,family of d i s t a n c e f u n c t i o n s

p

w'

which have t h e f o l l o w i n g p e r y e r t i e s : (i) For each w E G. t h e topology d e f i n e d by p

i s e q u i v a l e n t w i t h t h e one

induced by p x , i . e . t h e r e e x i s t s a c o n s t a n t y ( w ) such t h a t (y(w))

-1

PX(X,Y) 5 PU(X,Y) 5 y ( w ) p X (x,y),

(ii) The f u n c t i o n 51 3 w

+

p,(x(w),y(w))

v

(x,y) E

x

x

x

i s continuous f o r each couple o f

continuous f u n c t i o n s x ( w ) ,y(w), valued i n X. The m e t r i c space x equipped w i t h such a d i s t a n c e f u n c t i o n

p

i s said

t o be c o n t i n u o u s l y parametrized by t h e parameter s e t G. and i t is denoted by xw, w E G.. Example 1.2.5. kl o . Let X = C (U), U = (O,l), k 2 0 integer, be the set of all k-times continuously differentiable functions and let p (u,v) be the following X

distance function:

c max laJ(u(x)-v(x))I . O<j
=

with all the derivatives up to the order k. Let G. = (0,l)be the parameter set and let functions of

E

E

E (0,l). For continuous

valued in Ck(E) introduce the following distance functions

Pk,E (u,v):

Obviously, P ~ , ~ ( u , vsatisfies ) the conditions (i), (ii) in Definition 1.2.4.

The set of all continuous functions ( 0 , l ) 3

with distance function p ( 0 , k ) ,E

imbedded in Co to

E

u E Ck(6) equipped

k,E (0,k), E (U) has a structure of a vector space and it is -

Obviously, C

E +

hereabove, will be denoted by C

-

(U)

.

(e), the imbedding being continuous uniformly with respect

E (0,l).

2'. Let IRl C lR1 be a family of grids and let Ct(lR1 ) be the set of x,h x ,h all continuous functions of h E (0,l) defined on R 1 = hZ and valued in x,h

1. Manifolds, Functional Analysis, Distributions

12

there exists a constant C > 0 which does not depend on u E C

(0,O)

(ii) ,E

E

E (0,l) and

( u , O ) , the latter being

(AEu,O) 6 Cp

and such that p

O,E

OJE

valid with C = 1, since

I

(2~J-lexp(-(x-yj/~)dy < 1,

V

E (0,l).

E

u

-

Furthermore, it is readily seen that AE

: C

(0,O)

(U)

+

,E

c ( 0 , 2 ) ,E (U)

is an

ecpihomeomorphism (equiisomorphism in this linear case), i.e. that there exists a constant C > 0 which does not depend on U E C

-

(0,O)

E

E (0,l) and

(U) and such that ,E

-1

PO,€

(u,O) 6 p

2t E

V

(u,O),

(AEu,O) 5 Cpo

E

E (0,1),

,E

V u E C

2

. Let xh =

( 0 , O ) ,E

Ck(Rk,h) be as in Example 1.2.5.

the backward finite difference derivative, With a* x,h -1 1 (a:,hu) (x) = h (u(x)-u(x-h)), x E Rx,h , V

introduce the symmetric finite difference operator $,h,

Xx,h

=

(112)(ax,h+a:,h)

It is readily seen that V

ax,h

:

Ck+l 1 h (mx,h

+-

k 1 'h("x,h

' k

2 0, h E ( 0 , l )

is a family of equicontinuous mappings. h The family of mappings A , h is a family of equiisomorphisms, V k 2 0. Indeed, one has:

where

and it is easy to check that

h E (0,1)

(U)

I

1.2. Functional Analysis the metric space of the sequences u : Z

CL(mi,h

-f

13

C with the distance function p

X'

being equipped with the following family of distance functions

)

'k,h'

-1

(u(x+h)-u(x)) is the forward finite difference 1 nx,h Obviously, C;(ni,h) is a family of metric spaces with the natural

where (aXlhu)(x) = h

derivative on the grid

structure of a vector space, parametrized by a parameter set (0,l)3 h. Furthermore the space Ck (E) of all k-times continuously differentiable

?ii

is continuously imbedded in Ck(R1 ) by taking the h Fc_.h of each u E C ( R) , this imbedding being restriction TI u(x) onto x E lR1 h xth continuous uniformly with respect to h E ( 0 , l ) . functions on

Definition 1.2.6.

Let x

be two families of metric spaces parametrized by a c

,yw

with families of distance functions A family of continuous maps,

p,

mn, equipped

and uw, respectiueZy.

is said to be equicontinuous at a point x0 ( w ) , if f o r any E > 0 there exists a 6 > o such that p (A~(X),A~(X ) ) < E as soon as p (x,x ) < 6, uniformly 0 w o with respect to w E n, i.e. 6 = ~ ( E , x ) can be chosen independently of 0 w E a. x , Y ~ ,w E a, are said to be equihomeomorphic, if there exists an equicontinuous family of one-to-one maps A" : xu + yw, such that : yw + xw is equicontinuous, as well.

CAY-'

Example 1.2.7.

lo. With X

= E

C

(0,0),E

(u)defined

in Example 1.2.5

,

consider the following

family of maps A€: 1 (A'u)

(x) :=

I

-1 u(y) ( 2 ~ ) exp(-Ix-y//E)dy.

0

men C

(0,O)

: .E

c

( 0 , O ) ,E

6) c ( 0 , O ) +

(u)is a vector

-

,E

(U) is equicontinuous. Indeed, since

space and A E is linear, it suffices to show that

1 , Manifolds, Functional Analysis, Distributions

14

(Ah)-'

:

Ch(Rx,h) k 1

+

Ch k+2 (Rx,h), 1 h E (0.1)

is an equiisomorphism. Furthermore, one has: 2 (1/2)exp(-lxl)-G (x)=O(h ) , h

h

+

0,

uniformly with respect to x on each compact set in R

1 X

.

Definition 1.2.8.

Let

x be a metric space.

A map

T : X + X

such t h a t pX(T(x) ,T(y)) 2 BpX(x,y),

V (x,y) E X x X

w i t h some 8 E [o,I), i s said t o be a contraction. Let Xw be a family of metric spaces parametrized b y of maps: T

:

Xu

+

Xu,

w E

5

Rn

. A family

R

i s c a l l e d an equicontraetion i f t h e r e e x i s t s a nwnber e E not depend on w E n and such t h a t

[0,1)

which does

The following result is due to S. Banach and is known as the fixed point theorem. Theorem 1.2.9.

Let T

:

X

x be a contruction. Then there e x i s t s a unique element

-+

xo E X,

such t h a t xo = T(xo).

The following statement is a useful extension of the Banach contruction mapping theorem, which can be proved by exactly the same argument as Theorem 1.2.9. Theorem 1.2.10. I f TW

:

Xw

-f

(i.e. R 3 w

Xu is an equicontraction and T depends continuously on w E R,

+

p,(T,(x(w),y(w))

is continuous f o r each couple of p o i n t s

1.2. Functional Analysis (x(w),y(o)) E

x u ) , then the fixed point x

x

0

15 (w)

of T

o

:

x

w

+

x

w

depends

continuously on w E R as well. Example 1 . 2 . 1 1 . As an application of Banach's fixed point theorem (Theorem 1 . 2 . 9 1 ,

consider

the following initial value problem with a parameter E E ( 0 , l ) : 2

u (t)+u (t)-UE(t)-f(t) = 0, t > 0

Ea

{UE(0)

(1.2.1)

t E

where f E

C1

=

(E+)

0,

and

$J

E R are given.

Assume that f and 0 satisfy the conditions: (1.2.2)

2

f(t) 2 (1-y ) / 4 ,

$

c

(1-6+y-yo)/2 2

with some y > 0, 6 > 0, where yo = 1 - 4 f ( O ) . We are going to show that problem ( 1 . 2 . 1 )

u

:

R+

-f

has a uniq-ie solution u (t),

R , which can be represented in the form:

and for y

c1

:

IR+

+

W the following estimate is valid with some constant

=

E

= Cl(f,$):

provided that

E~

0

(f,$) is sufficiently small.

It is readily seen, that (1.2.8)

uo(t)

=

uL(t)+f(t) 0

and w(t) is the solution of the initial value problem:

1 , Manifolds, Functional Analysis, Distributions

16

(1.2.9)

a tw(t)+yow(t)-w 2 (t) =

0,

t E IR+,

= $

with y and $ defined by (1.2.6), (1.2.4). 0 Looking for u (t) in the form (1.2.3), one gets for y (t) the following initial value problem:

Notice that as a consequence of (1.2.2), one has: (1.2.12)

A(t)

t y,

v

t

E E+ a

-

Moreover, if $ 5 0 then w(t) 5 0, V t E R + , so that in this case on? has : A(t)-2w(t/E) 2 y, On the other hand, if $

v

t E

zt.

0, then $ < y o - 6 ,

as a consequence of (1.2.2),

and, moreover, one has in this case: 0 < w(t/e) < $. Thus, one finds for 0 < $ < y -6: 0

(1.2.13)

X(t)-2w(t/E) 2 y-2$ = y-2'$+1-yo

2 6,

again as a consequence of (1.2.2). Therefore, one has: (1.2.14)

p(E,t) := A(t)-2w(t/E) t 6,

v

t E

s+.

Now we may rewrite (1.2.10) as an equivalent integral equation in the following fashion:

17

1.2. Functional Analysis

Now consider in C(s+

where r = r(f,$) and

E~

=

)

a ball BrE of radius rE centered at 0 , i.e.

Eo(f,$) will be chosen later on.

One finds for each y E BrE :

where C = C(f,$) is such that

Thus, if r and tz0 are such that

then T defined by (1.2.161 maps BrE into itself. Furthermore, one easily finds for each pair (yl,yz) E BrE

(1.2.22)

X

BrE:

2rE0 < 6 ,

will be an equicontraction in BrE' v E E ( 0 , ~ ~ ) . It is readily seen that the following choice of E~ = s0(f,$) and

then T

r = r(f.9) is compatible with the conditions (1.2.20), (1.2.23)

E~

=

6'/(8C),

With such a choice of

r = 4C(2/2-1)/(2/26). E~

and r, there exists a unique y

(1.2.15), (1.2.16) or, equivalently, of (1.2.10). Thus, one has:

(1.2.24)

(1.2.22):

IIyEllc(~+,5 r(f,$)E,

V

E

E

( 0 , ~ ~ ) .

Now, (1.2.24) and (1.2.10) yield:

i.e. (1.2.7) with c,(f,$) = A(f,$)+c(f,$).

E BrE solution of

I . Manifolds, Functional Analysis, Distributions

18

The argument presented here can be easily extended to the initial value problem for the nxn systems of the form:

under the following assumptions: (i) The equation f(t,u (t)) = 0 has a solutions uo E C1($+). 0

(ii) The Jacobian-matrix f (t,u) has the eigenvalues h.(t,u), 1 5 j 2 n, 3

which in some neighbourhood of the curve (t,u (t)) E Wn+l satisfy the 0 condition: Re X.(t,u) 2 y > 0. 1 (iii) Vector ($-u ( 0 ) ) E Wn lies in a sufficiently small neighbourhood 0 of zero. Then again u (t) exists, is unique and can be represented in the form (1.2.3) with w(t) the solution of the following autonomous system:

a tw(t) +

(1.2.26)

f(O,uo(0)

+ w(t))

= 0,

t E W+ ,

As a consequence of (ii) and Liapunov's theorem, w(t) E 0 is an asymptotically stable stationary solution of system (1.2.26), so that if

0-u (0) lies in an attraction neighbourhood of zero, initial value 0 problem (1.2.26) has a unique solution w(t), which exists for all t Z 0 J, =

and is exponentially decreasing as t

-t

+a-

The same procedure as here above yields an integral equation for y (t) which again can be solved by the equicontraction argument in the family of metric spaces B which are balls of radius r& centred at zero in the space rc C(% ) of vector-functions y(t) with the norm l\ylL(%+) = sup ly(t)/ finite. t2O

I

Example 1.2.12. Consider the following boundary value problem: 2

u

+ f(u,E~ )

(1.2.27)

-E

(1.2.28)

u(E,x') = $(x'), 2

xx

=

0, x E U = (O,l), xt

E au

= {o,i},

where u = a u, u = a u , E E ( O , E ) is a small parameter, $ : a U + W is x x x x x 0 given and f (vl,v2) is a given function of variables v = (v v 1 E W2 with 1' 2

1.2. Functional Analysis real values: f

m2

:

+

w

19

.

We shall assume that f(v ,v satisfies the following conditions: 1 2 lo. f(v) is twice continuously differentiable. 20. f(0) = 0, fv (0,O) = y2 > 0. 1 We shall also assume that $ ( X I ) ,

x' E aU, is sufficiently small, i.e.

($(x')I 5 6 where 6 > 0 will be specified later on.

We are going to show that under the assumptions hereabove, the solution U(E,X) of (1.2.27), (1.2.28) exists, is unique and can be represented in the form:

z

(1.2.29) u(E,X) =

+ V(E,X),

W(X',(X-X'I/E)

x*Eau where w(x',x), x' E aU, are the (well-defined) solutions of the boundary value problems: (1.2.30)

X'

-wxx+f(w,(-l) w(x',o)

=

W )

=O,

X > O ,

$(x'), w(x',x)

+

0

for x

+

+-,

V

au

E

XI

and V(E,X) is at least twice continuously differentiable function of x E such that one has: (1.2.31)

I IvI Ic

I (€a ) kV(E,X)I

c max O5ks2 XE;

:=

(0,2), E )'(

5 CE

with some constant C > 0. Let us start by showing that each boundary value problem (1.2.30) for XI

=

as x

0 and x' = 1 has a unique solution w(x',x) which decreases exponentially +

+-,

provided that

Consider the case

XI

the same way. Denoting a

$ ( X I ) ,

= :=

f v2

rewrit.P ( 1 . 2 . 3 0 ) , fashion :

x' E aU, is sufficiently small.

0, the one of

(o),

w(x)

=

XI

:=

1 can be treated in exactly

w(O,x), $ = $(O), one may

(with x' = 0) esu'valently in the followinq

(1.2.32) w(x) = $ exp(X-x) + m

+

o

2 Law (y)+y w(y) - f (w(y) ,w (y))lG(x,y)dy := y Y

(T(w)1 (x),

where G(x,y) is the Green's function for the operator L(a L(a

)

:=

-a 2

+ aax + y

2

, x

E

X

)

m+,

with Dirichlet boundary condition at x = 0 and where h - is the negative

1. Manifolds, Functional Analysis, Distributions

20

zero of the characteristic equation A2-aA-y2 = 0, the positive one being denoted by A+. It is readily seen that G(x,y) is given by (1.2.33) G(x,y) = ( 2 ~ ) -f

( C2+iaF,+y2)-'exp

(-iyC)(exp(ixs)-exp (1-x) ) dC

R

so that with a = min{lA- , A + }

(1.2.34)

> 0, one has:

lG(x,y)I 5 Cx(

Introduce the notation: 11.2.35)

c

IIwIL;~=

sup exp(ax)Iaxw(x)I, 1

k = 0,1,2

,...

O<jO

and denote by Ck(%+

)

-

the set of all k-times continuously differentiable

functions on R+ for which expression (norm) (1.2.35) is finite. Further let

is a metric space with the distance function: Obviously, Mk r,a Pk;a(w1,w2) :=

IIW1 -w2 4 ; o

For operator T(w) defined by (1.2.32) one finds (1.2.36)

IIT(w1

th;a SItJl

+

2 C!~w~~l;av

where m

(1.2.37)

c

=

c(w) := sup I IG(x,y)lexp(ax-2ay)dy x>o 0

*

(1/2) max Ivl 41wlll;a

1

I a I =2

la:f(v)

-

I.

Furthermore, one easily checks that, in fact, one also has: (1.2-38) IIT(w) with C

=

1 (1.2.37).

I 1;a

5 C1I+I

2 +

CgJIWIIl;a.

max{l,(h-(], C2 = max{lA-l,A+lC(w), where C(w) is defined by

Thus, one finds:

(1.2.39)

IIT(W)l(i;a5 C1l$( + C2(r)r2,

v

1 w E Mria

.

21

1.2. Functional Analysis

Again, an easy computation shows that (1.2.40) IIT(w~)-T(w~)

5 rC3(r)/lW1-W21/1;o, v wj

1

E Mr;a, j

=

1,2,

with some constant C (r) > 0, which is bounded for r E [O,ro] with any 3 ro < m. We shall choose @ and r

=

r($) E (0,1/2] so that the following

inequalities hold:

(1.2.42) Ck =

sup Ck(r), k = 2,3. O
Now, choose

I$[

so small that it holds:

With r and @ chosen to satisfy (1.2.41) with Ck, k = 2,3 defined by (1.2.42), operator T defined by (1.2.32) is a contraction in metric space 1 so that (1.2.32) has a well-defined solution w(x) which decreases exponentially as x -f +-, along with its derivative, as a consequence of the 1 construction of M r;a' Furthermore, it is easily seen that, in fact, w(x) is twice continuously differentiable (in order to check it, one just has to differentiate twice the equation (1.2.32)) and, as a consequence of (1.2.30) and the fact that

.

f ( 0 ) = 0, one finds that w E C2 (F ) a + Thus, for I $(XI) I sufficiently small the boundary value problems (1.2.30) have well-defined solutions w(x',x), (x',x) E aU O(exp(-ax)), as x

x

z+,which are

+-.

+

Besides, as a consequence of (1.2.41), one has for w(x',x) the following estimate: (1.2.43)

Il k ;o

I~w(x',*)

< 2C1C21$lr

k = 0.1,

where the constants C1 and C2 are defined hereabove, so that one also has: (1.2.44)

Iwx(x',O)

I

< 2C1C21$1.

As a consequence of (1.2.30) and the fact that f(0) = 0, estimate

I . Manifolds, Functional Analysis, Distributions

22

(1.2.43) also holds for k = 2. It can be shown, but will not be done here, that, in fact, the boundary value problems (1.2.30) are uniquely solvable in the class Of exponentially decreasing functions for x

-f

+-,

v2 E R in El2, intersect the stable (for t

-f

iff the lines ($(x'),v2), +-)

invariant manifold (Curve)

X- of the first order autonomous system (1.2.45)

1

(t) = v (t), 2

2

(t) = f(v(t)),

t > 0,

the solutions v(t) belonging to X- being asymptotically stable for t Obviously v(t)

-f

+a.

0 is a saddle point type stationary solution of

(1.2.45), so that (1.2.45) has also one unstable invariant manifold X+ in a neighbourhood of the origin. Now, looking for U(E,X) in form (1.2.29) with w(x',x)

as above, one gets for (1.2.46)

:

au

x

z+defined

the following boundary value problem:

V(E,X)

L(E,x,~~) = vF(E,v,Ev~), x E

u , v(E,x')

=

0, x' E

au,

where we have introduced the notation: (1.2.47)

+ a(E,x)Eax

L(E,x,~,) := -E2a:

+

b(sA

with a(E,x) := f V

(1.2.48)

(w(E,x),EW

z

W(E,X) =

(€,XI)

2

t

b(E,x) := f (w(E,x)rEwX(ErX)) I v1

w(x',lx-x'I/E)

X'Eau and with (1.2.49)

F(EV,EV

)

=

f(w(E,x)+v,~w (E,X)+EV )-f(W(E,X),EWX(E,x))-

-EV

f

x v

2

(W(E,X),EWx(E,X))-Jf,J (W(E,X),EW (€,XI). 1

Obviously, one has (uniformly with respect to 2 2 F(Ev,Ev~) = O(V + ( E v ~ )1, i.e. for v E C

(0,1); E

El:

2 as v +(Ev~)

-f

0,

(a),with < r

one has:

s

Cr2

1.2. Functional Analysis

23

wirh some constant C = C(r) which is bounded for r E (O,ro] with V r0 < Furthermore, as a consequence of (1.2.48) and the exponential decreasing of W(E,X) for

E +

0 uniformly with respect to x E U ' c c U, one finds:

where we have denoted: (1.2.53)

=

$(XI)

lim w

(€,XI)

= w (x',O),

X'

E aU,

E"0

and the arrows mean the uniform converqence on (1.2.44),

Thus,

(1.2.52), (1.2.53) yield:

2 b(E,x) 2 yo > 0,

(1.2.54)

i'.

v x

E

u,

v

E

E (O,Eol,

(v) is continuous and fv (0) = y2 > 0. V1 2-1 Furthermore, partial integration yields for V v E C (U), v(x') = 0,

provided that E~ > 0, given that f

x' E

au,

-

v : U

(1.2.55)

+

R and L ( E , x , ~ ~ defined ) by (1.2.47):

2 2 Iv(x)L(E,x,~)v(x)dx = I[(Ev~)+ ( ~ ( E , x ) - ( E / ~ ) ~ ~ ( E , x ) )Idx, )v U

U

where a(E,x) and b(E,x) are defined by (1.2.48). The chain rule yields: (1.2.56)

Ea

(€,XI

f (W(E,X),EW (E,X))EW (E,x)+ v v 1 2 +f (W(E,X),EW (€,XI) E 2w (E,X). v 2v 2 xx

=

Therefore, using (1.2.43) with k = 1,2 and (1.2.56), one finds:

where the constant C > 0 does not depend on Choosing (1.2.58)

I $ (x') 1

6

E (O,Eo] and $(XI).

so small that

2 b(~,x)-(~/2)a(E,x) 2 (1/2)y0 > 0,

one gets:

Therefore, the solution of the boundary value problem: (1.2.60) is unique.

L(E,x,~~)v(x)= g(x),

x E

u,

v ( x ~ )= 0 ,

E

au

I. Manifolds, Functional Analysis, Distributions

24

-

We are going to show that for each g(x) continuous function onE there exists a solution V(E,X) with

SUP llvlic (6) 5 C < O<E<E (0,2),E

provided that

E~

is

0

sufficiently small. Let us rewrite (1.2.60) equivalentlv in the following fashion: -E

2 2 2 vxx+asvx+y v = g(x)+(a-a(c,x))sv + ( y -b(E,x))v,

x E U,

(1.2.61)

E au,

v(E,x') = 0 ,

(01, y2 = fv ( 0 ) are the same as hereabove. 2 1 Notice that the following estimates hold with some constant C > 0:

where a = f

V

as a consequence of the properties of w(x',x) constructed hereabove. Let GE(x,y) be the Green's function for the boundary value problem: (1.2.63)

L ~ ( E ~ ~ ):=V

2 -E2v +aEv +y v = h(x), xx x

x E

u , v(x')

= 0,

It is readily seen that GE(x,y) is given by the formula (1.2.64)

GE(x,y) =

E

-1 (G((X-y)E-I)-AE(y) exp(h-xE-l)-

-~~(y)exp(-h+(l-x)€-l)), (x,y) E

where (1.2.65)

G(x) := (2n)-'I

(C2+iaC+y2)-l exp(ixc)dc, R

AE(y),

B (y) satisfy the linear system AE

(1.2.66)

(y)+BE (y)exp

i,

= G

(-YE-')

(y)exp(X-E-') +BE (y) = G( (I-y)E-')

and h -, are the zeros of the characteristic equation (1.2.67)

2

A -aA-y2

= 0,

A- <

0 < A+,

introduced also hereabove. The residuum calculus gives immediately:

ux

U,

x' E

au.

1.2. Functional Analysis (1.2.68)

25

iG(x)

with a = min{ 1 A I , A + }

> 0, so that (1.2.64)-(1.2.68) yield:

where C > 0 is some constant, which may depend only on a and y. Moreover, it is easily seen that also the following estimates are valid (perhaps with a different constant C > 0 , which still depends only on a and y )

:

for (x,y) E ij x

U.

Thus, as a consequence of (1.2.62), (1.2.69), (1.2.70), one has the

following estimates with k = 0,l:

(1.2.71)

k SUE I ja-a(E,y) 1 j (€ax) G~(X,Y)/dy s CE, xEU U k {sup 1 IY -b(E,y) I I (€ax) GE(x,y) Idy 5 CE XEU u

and the same estimates (1.2.71) for ( € 3

) GE(x,y), as it is easy to check. Y Now, using the Green's function G (x,y) we may rewrite boundary value

problem (1.2.61) equivalently in the following fashion: (1.2.72)

V(E,X) = ( T

where we have denoted: (1.2.73)

(TE(V))

(x) ::

One has for each pair vl,v2 in the metric space C 1

distance p (v1'v2) =

I Iv

-v

II c(o,l),E

(0,1)

,E

(5)with

the

(if,:

where, of course, (1.2.71! have been used. Furthermore, for g E Co(<)

and each v E C

(0,1), E

(c),one also finds:

1. Manifolds, Functional Analysis, Distributions

26 Thus, if (1.2.76)

E~

is chosen in such a way that

C E < ~ 1,

where C is the same constant, as on the right hand side of ( 1 . 2 . 7 4 ) ,

then

the family of (linear) mappings

-

(1.2.77)

TE : C

(0,1)

rE

(U) -t c

(0,1)

,E

(U)

will be an equicontraction. Thus, ( 1 . 2 . 7 2 ) has a well-defined solution v E C Moreover, as a consequence of ( 1 . 2 . 7 5 ) ,

(G, v E

E

(O,E0).

the following estimate with some constant C > 0, which does not

of ( 1 . 2 . 7 2 ) depend on

(0,1) , E

one a l s o has for the solution V(E,X)

E

E ( 0 , ~ ~v)and ’ g:

provided that CEO < 1, where the constant C is the same as on the right hand side of ( 1 . 2 . 7 5 ) . one gets immediately the conclusion that for each 0(t)(= C ( U ) 1 , the well-define solution v of ( 1 . 2 , 6 1 ) (or,

Using ( 1 . 2 . 6 1 ) , 9

c(o,o)

,E

equivalently, of (1.2.60)

in fact, belongs to

C

(0.2) , E

(6)(the metric

space

of twice continuously differentiable functions with the distance function defined by ( 1 . 2 . 3 1 ) ) depend on

E

and, moreover, with some constant C > 0 which does not

E ( O , E ~ ) , v and g the following estimate is valid:

provided that

E~

is sufficiently small (i.e. satisfies the inequalities

specified hereabove). Denote by

( G ( 9 ) ) (x) =

V(E,X) the solution of ( 1 . 2 . 6 0 ) ,

may be regarded as a family of maps (linear operators) form C

(0.2) ,E

(u).In fact, if one takes instead of C0 -

(U)

of continuous functions g ( E , x ) (depending on

E

C

so that E 0- . ( U ) into

the matric space C

-t-

( 0 . 0 ) ,€

G

(5

E ( 0 , ~ ~with ) ) the same

distance function, as on C o 6 ) , then the (linear) maps

are a family of homeomorphisms (isomorphisms) which are Lipschitz equicontinuous, as a consequence of ( 1 . 2 . 7 9 ) .

i.e. one has for each pair g1 ’ 9 2

1.2. Functional Analysis

27

where the constant C > 0 is the same as on the right-hand side of (1.2.79)' 1 and g 0 > 0 is sufficiently small.

i.e. does not depend on that

E~

E

E (O,E

E

-

C (o,o),E(U),

k = 1.2, provided

Now, we come back to boundary value problem (1.2.46) with L ( E , x , ~ ) and F(E,V,EV ) defined by (1.2.471, (1.2.48) and (1.2.49), respectively. Using the operators G

constructed hereabove and having the property

(1.2.80), one may rewrite (1.2.46) equivalently in the following fashion:

where we have denoted:

Introduce the following family of metric spaces X rE (1.2.84) XrE

=

{v E C

5 rsl,

(0,2),E

where r > 0 will be chosen later on. One has:

as a consequence of (1.2.51) and (1.2.84). E XrE, k

Furthermore, one finds for each pair v

where C1 > 0 does not depend on

E

E (O,E

0

)

and v

=

1,2:

E XrE, k = 1,2.

Of course, again we have used (1.2.51). Now, choosing r > 0 so small that the following inequality is satisfied:

1. Manifolds, Functional Analysis, Distributions

28

(1.2.87) Crso < 1 , where C > 0 is the maximum of the constants C1, C 2 on the right-hand side of (1.2.85), (1.2.86), respectively, one gets the conclusion that the family of maps:

(1.2.88) Q, : MrE

+

MrE,

E

E

(o,Eo)

is a family of equicontractions (provided that

E

0

> 0 is chosen, as

specified hereabove), so that there exists a well-defined solution v E Mrs of (1.2.82). As a consequence of the construction, one has:

with any constant r

0 satisfying (1.2.87).

Using (1.2.461, (1.2.511, one also finds that, in fact, v E

C(o,z),E

(U)

and, moreover,

with some constant C > 0, which does not depend on

6 .

Finally, (1.2.891, (1.2.90) imply (12.31). One should emphasize that using exactly the same argument for the completely non-linear boundary value problem:

(1.2.91)

F(E

2UXX,EUX,U) = 0 , x E

u,

U(E,X') = $ ( x i ) ,

XI

E

au

one can prove the existence and uniqueness of the solution U(E,X) to

(1.2.91) for

6

E

( 0 , ~ and ~ ) j $ ( x ' ) I <,

6, x' E au with

E~

> 0, 6 > 0

sufficiently small, if the following conditions are satisfied: (i) F(v), v

=

(vl,v2,v3)is sufficiently smooth (at least twice continuously

differentiable in some neighbourhood of the origin in lR (ii) F(0) = 0, F

V

L(S)

(0)

# 0 and the polynomial

3

).

L(5),

1

:= -ac2+ibS+c, 5 E

R

does not have real zeros and has precisely one zero in the upper and the lower complex halfplanes, where it is depoted: (a,b,c) = (VvF)( 0 )

1.2. Functional Analysis

29

Moreover, under assumptions l o , 2' again U(E,X) may be represented in the form (1.2.29) with

V(E,X)

satisfying (1.2.31) and w(x',x) the solutions

of the following boundary-value problems: X'

F ( w ~ ~ , ( Wx,W) ~ ~ ) = 0,

X > 0,

W(X',O)

the latter being well-defined, provided that Indeed, since F V1

=

$(XI),

I $ (x')[

is sufficiently small.

(0) # 0 so it is also in some neighbourhood of zero 2 ( E uxx,~ux,u),V E E ( O , E ~ ) ,

in B 3 , to which belong the values of the vectors

provided that E~ is sufficiently small. Consequently, one may solve the 2 2 thus, reducing boundary equation F ( E u , E U , u ) = 0 with respect to E u xx x xx' value problem (1.2.91) to the form (1.2.27), (1.2.28). condition 2" being satisfied as a consequence of condition (ii). The essential requirement (ii) hereabove is the condition for the linearized (at zero) equation:

+ bEvX + cv = 0 aELv xx to be a proper elliptic singular perturbation (see vol. I1 of this book), which amounts in the situation considered to condition (ii).

I

Example 1.2.13. A

different nature of dependence upon a small parameter, namely a highly

oscillatory behaviour, is exhibited by the motion of a mass point in the presence of a strong force field in the vicinity of a smooth surface y in R3

,

the force field being directed towards y. Heuristically, it is obvious

that the mass point stays in a narrow neighbourhood of y and, as the force field becomes infinitely large, the motion occurs on y itself giving rise to a Lagrangian dynamical system on the smooth manifold y. One

says in this

case that the mechanical system is subjected to a holonomic constraint. Consider the motion of a mass point of mass 1 in B3

,

whose potential

energy in cylidrical coordinates (r,z,$) has the form: (1.2.92)

2 -1 2 UE(r) = ( 2 )~ (r-1) ,

the Lagrangian L

E

E (O,E~),

of this mechanical system being:

Here, as it is customary in the mechanics of mass points, the dot stands for the derivative with respect to the time variable t.

1 , Manifolds, Functional Analysis, Distributions

30

When

E

cylinder y

-f

=

0 the motion with initial data in a small vicinity of the {(l,z,$)

1

0 5 $ < 27,

-m

< z <

takes place on the cylinder

itself and i s characterized by the reduced Lagrangian Lo given by

which is a free motion (no potential energy) on y and, obviously, one has: $(t) = $0+C2t, z(t) = z +vt, where $ 0

0

=

$(O), C2

= i(O),

zo

=

~ ( 0 1 ,v

=

l(0)

are the initial conditions for $(t) and z(t). We shall analyze the perturbed motion described by the perturbed Lagrangian L Since

z

given by (1.2.93). and $ are cyclic coordinates ( L E does not depend on z and $ ) ,

the axial and angular momenta are preserved, the latter yielding the following first integrals for the mechanical system considered: (1.2.95) p (t) = z(t) E v,

2 . pm(t) = r (t)$(t) E p ,

so that for the radial component r(t) one finds the following Lagrangian

function

where we have denoted:

The Newton's equation for r = r(c,t) has the form: (1.2.98) 'Q(E,t) + (d/dr)VE(r) = 0 , where, as usual, the double dot stands for the second derivative with respect to t. From now on we shall assume that the motion starts from the cylinder y and has no initial radial velocity, i.e. that r(E,t) is subjected to the

following initial conditions: (1.2.99)

r(E,O) = 1,

r(E,O)

=

0.

Introducing the new unknown function (1.2.100) p(E,t)

=

E-2(r(c,t)-1),

p

(E,t),

1.2. Functional Anaiysis (1.2.98),

y i e l d f o r p ( ~ , t ) t h e following i n i t i a l value problem:

(1.2.99)

- n2 (1+E2 p ) - 3

p + p

E

31

=

0,

(1.2.101) P(E,O)

where S?

=

= 0

0, P(E,O)

=

$(O) i s t h e (given) i n i t i a l angular v e l o c i t y .

The c r i t i c a l p o i n t s (d/dr)VE( r ) , i . e .

S ( E2 ) of equation

a r e t h e zeros of

(1.2.101)

t h e s o l u t i o n s of t h e equation:

and only t h e p o s i t i v e one i s of i n t e r e s t . Obviously, i t has t h e following asymptotic behaviour: (1.2.103)

S ( E ~ )= G.

2

+ O(E

2

),

E

-f

0.

Furthermore, one has: (1.2.104)

2

2

( d /dr ) V E ( r ) > 0 ,

so t h a t a l l t h e motions i n t h e neighbourhood of t h e s t a t i o n a r y s o l u t i o n r(E ,t )

2 E S ( E ) of

a r e p e r i o d i c f u n c t i o n s of t with some period

(1.2.101)

21TE/W(E).

We a r e going t o apply t h e Poincar6-Lindstedt method i n o r d e r t o analyze t h e asymptotic behaviour of t h e s o l u t i o n of Since p ( ~ , t ) i s p e r i o d i c with period

(1.2.101).

~ T ~ E / w ( Ew ) e

s h a l l seek p i n t h e

form:

which y i e l d s t h e following i n i t i a l value problem f o r (1.2.106)

2 2 2 w (E)';+u-~ (1+~ u ) - ~= 0,

t h e frequency

W(E)

u(E,O)

Let

{

U(E,T)

W(E)

where

=

=

0,

being unknown, a s w e l l ; here t h e double d o t s t a n d s f o r

t h e second d e r i v a t i v e d2/d.r2.

(1.2.107)

= ;(E,O)

u(E,T):

2

= V ( T ) + E W(T)

1

+

E

2

v

+

E

4

*

U(€),

E4q!E,T!

32

1. Manifolds, Functional Analysis, Distributions

{u

= 0

V(T)

(1.2.108)

2

T),

(1-COS

W(T)

=

4 -3R (l-COS

T)

2 (3/2)Q .

=

we are going t o show t h a t with some c o n s t a n t C > 0 one has:

I lqll

SUP

2 C ([O,a])

O<E<E

:= sup c max l akT q ( E , . r ) / 5 C , O < E < E ~ OCkC2 OStLa

(1.2.109) sup O<E<E

I!.l(E)I

5 c,

0 provided t h a t a = a ( € ) i s such t h a t

a(€) = 0(E-4), as E -+ 0 . 2 Writing w ( c ) = 1+2E2u+E4K(E), s u b s t i t u t i o n of (1.2.107) i n t o (1.2.101)

(1.2.110)

y i e l d s f o r q(E,T) t h e following i n i t i a l value problem:

..

(1.2.111)

2 4 2 2 (1t2UE + E K ( E ) ) q + q = (K(&)t3n (2UtE K ( E ) ) ) V 2 2 2 4 6 -a (K(E)+3a (2U+E K ( E ) ) ) - F (2E V,E WIE q ) , = 0

q(E,o) = i ( E , O ) where

We choose

(1.2.113)

K(E)

K (E)

t o be:

2 2 -1

= - 6 ~ R ~ ( 1 + 3 c ~ R=~ -9R4(1+3~ )-~ R )

,

where JA(E.0)

(1.2.115)

2t3R 4 ( 2 v t2~K ( E ) )

:= K ( E ) R

2 4 L 2 ( E ) := 1t2uE + E K ( E )

a r e well-defined by (1.2.1081,

(1.2.113).

Notice t h a t f o r F(v,w,q) defined by (1.2.112) one has:

(1.2.116) F(v,w,q) = O ( V 2 + / W \ + / q l )a, s IVl+lW\+\ql

-f

0

1.2. Functional Analysis

33

and, moreover, one also has for F and F = (a/aq)F with some constant C > o : q 2 (1.2.117) IF (v,w,q)[ S C, IF(v,w,q)[ 5 C(v +Iwl+lql), 9

V(v,w,q) with IvI+[wl+lql 5 1/2. Let us rewrite (1.2.114)

equivalently in the following fashion:

where we have denoted: T

(TE(q))(T) :=

(1.2.119)

I'q0(E,T)

I

2

6

4

Sin((T-S)/W(E))F(E V(S),E W(S),E q(E,s))ds,

0 := A ( E , n ) (l-COS(T/W(E)).

Now introduce a family of metric spaces as follows: (1.2.120) xE := Cq E c"([o,aI) [ llq-qolI s 11 c0( o a]) wnere, as previously, Co([O,a]) is the set [v;?ctor space) of all continuous functions on [O,al and Ilgll = max Iq(c,T) I . c'([o,~I) O
Using (1.2.117),itisreadilyseenthatonehaswithsomeconstantC

1

> 0:

2

4

6

),E*w(s),E q(E,s) Ids 5 C , ~ E, V provided that

5

where A

O

E

E

E

( 0 , ~ ~ ) ~

> 0 is so small that

0

2 2 2 4 4 (20, +6s Q +E ( 1 + 2 A O ) ) 5 ( 1 / 2 ) , 0 0 0

A(E,n).

max

=

6

OSESE

0

Furthermore, again using ( 1 . 2 . 1 1 6 ) , constant C

2

(1.2.117),

one finds with another

> 0:

under the same condition ( 1 . 2 . 1 2 2 ) . Let a = a(€) be as in ( 1 . 2 . 1 1 0 ) , lim

E

4

so that one also has:

a(€) = 0.

€4

Thus, choosing

E

0

> 0 so small (if needed) that

1. Manifolds, Functional Analysis, Distributions

34 sup

E

4

-1

a(€) < C

O<E<E

with

c

=

0 max{C1,C2}, one gets the conclusion that T

E

:

X

-f

E

X

E

E'

E

(O,EO)

is a family of equicontractions, so that ( 1 . 2 . 1 1 8 )

E X

has a well-defined solution q(E,T), q

for

T

and also ( 1 . 2 . 1 1 4 )

E tO,a(E)1.

Thus, one also has:

with A.

defined as in ( 1 . 2 . 1 2 2 ) .

Using ( 1 . 2 . 1 1 4 ) ,

(1.2.113)

one gets immediately the first of the

(1.2.124),

the second one being an immediate consequence of

inequalities ( 1 . 2 . 1 0 9 ) ,

and the fact that

is well-defined by

V(E)

K ( E ) .

In fact, q(E,T) is infinitely differentiable function of with respect to E +

E),

i.e. all the derivatives of q(E,T) in

T

(uniformly

T

are bounded as

0 for T E [o,~(E)].

Finally, let us write the asymptotic formula for r(E,t) that we have established:

(1.2.125)

w

2

(E)

2 2 = 1 + 3 ~0 - 9 ~ ~ 0 ~ ,

provided that a(€) =

O(E

-3

)

and

0 is chosen small, as above.

>

E~

One also has the following asymptotic formula: r(s,t)

= I+E

2 2

4

0 COS(W(E)~/E)+O(E) ,

V t

E [~,a(~)l

(1.2.126) (,(E)

= 1 + ( 3 / 2 ) ~ ~ f i ~V,

provdided that a(E) = o ( E - ' )

and

E

E (0,~~)'

E~ >

0 i s sufficiently small as indicated

hereabove. One may differentiate the asymptotic relations ( 1 . 2 , 1 2 5 ) , with respect to t multiplying it after each differentiation by

(1.2.126) E.

Notice that the perturbed motion apporches the reduced one (the free motion on the cylinder y ) for t =

-3 O(E

)

performing high frequency

oscillations with periods of order O ( E ) and amplitudes of order E +

0.

2 O(E ),

as

1.2. Functiorial Analysis

35

This argument may be extended to mechanical systems effected by the presence of a strong force field directed towards some hypersurface

r

in

Rn of codimension one. Namely, with (r,$) the appropriate local coordinates

( 4 on the hypersurface and r the distance to the hypersurface), one may extend the argument hereabove to the Lagrangians of the form:

where a($) > 0 and L(r,$,;) L(r,$,;) = O(Ir1

3

is supposed to satisfy the condition: )

as r

+

0.

Furthermore, as usual, L($,$) is supposed to be a positive definite quadratic form with respect to the local velocity $ with coefficients smooth functions of the local coordinate $ and, moreover, the potential energy of the mechanical system (with the holonomic constraint forces) on the hypersurface

r

in Rn described by L($,+) is supposed to be a non-

negative smooth function of the local coordinates $ on

r.

Thus, these

conditions being fulfilled, the solution $(t) of Newton's equations, which correspond to L($,$), exists globally for all t E R . Introducing p

= E

-2

r, one finds the following Newton's equations for

the motion of the mechanical system:

where the subindices $ and

6 stand

for the gradients of the corresponding

scalar functions with respect to $ and

6, respectively. 2

Again in the layer of thickness O(E

)

around the hypersurface the

dynamics of the mechanical system (with the initial condition on the hypersurface) is mainly a superposition of a slow motion on the hyperssurface, described by the local coordinates $ ( ~ , t )with the main term $(t) solution of the system

and of a fast oscillatory movement in the normal direction to the hypersurface described by the coordinate r(E,t) = E 2 p(E,t) and subjected to the

1. Manifolds, Functional Analysis, Distributions

36

2

initial conditions r(E,O) = ;(E,O)

,i

,+) ,

= O(E ),

whose main term has the form:

,i

B(g ), C(g ) and W ( E ) are defined by the initial 0 0 0 . 0 0 conditions g ( 0 ) = 4 and g ( 0 ) = go; besides, one also has: 0

where A ( g

0

W(E) =

2 a(gO) ( ~ + O ( E) ) ,

E -t

0,

where a($) > 0 is the same as on the left-hand side of (1.2.127). To estimate the remainders in the asymptotic formulae for g(E,t) and 2 r(E,t) = E p(c,t) one uses again the argument based on Banach's

theorem concerning the equicontractions.

x w i t h the distance function p i s said c x such t h a t f o r each t o be separable i f there e x i s t s a sequence {x 1 k k2O 6 > o and f o r each x E x one can find an element x of t h e sequence

Definition 1.2.14. A metric space

kg

Ixk}k>O such t h a t P(X.Xk0) < 6 .

Any sybset of

x having t h e property hereabove i s said t o be dense i n

X.

Examples 1.2.15.

lo. The set

C

0

(b) of functions which are continuous on

= [0,11 with the

distance p ,

is separable. Indeed, the countable set of all polynomials with rational coefficients may be chosen as the sequence {x k k>O in Definition 1.2.14. The set 1 of all sequences of the form x = (ao,al, 2 an,.

.

...,

P

.)

with a k

real numbers equipped with the distance p ,

is separable.

...,rn,O,...)

Indeed, the sequences of the form (ro,rl,

with n 2 0 any

integer rk, 0 5 k I n, any rational numbers, form a countable subset of 1

P

and, thus, may be chosen as the sequence { x } in Definition 1.2.14. k k2O 3 O . The set L (U), u = (0,l) with p E [ O p ) , of all functions x(t) such that /x(t)1'

P

is integrable over

U,

is separable, the distance p

P

on L ( U )

P

1.2. Functional Analysis

37

being defined as follows:

Indeed, the countable subset of all polynomials with rational coefficients is dense in L (U).

P

4'.

of all almost everywhere on U = ( 0 , l ) bounded functions

The set L,(U)

with the distance p,,

is not separable. Here vrai max stands for the maximum of lu(t)-v(t) I over V \ E with E the set of Lebeargue's measure zero where u(t)-v(t) may be infinite. A

subset K in a metric space M is called compact, if each sequence

{ x ~ } c~K>, contains ~ a subsequence {x nk~k>O,whichconveqesto some elementxEM. Compactness criterium. A s e t K

5

M i s compact i f f f o r each 6 > 0 t h e r e

e x i s t s a f i n i t e + n e t i n M, i . e . a f i n t i e s e t o f elements {xk}lsksn C M such t h a t f o r each x E M t h e r e i s an element xk o f t h e &net whose d i s t a n c e t o x i s l e s s than 6 : p(x,xk) 1.2.2.

< 6.

Topological vector spaces

Definition 1.2.16. A set

x i s s a i d t o be a v e c t o r space o v e r t h e f i e l d o f r e a l o r complex

nwnbers, i f f o r i t s e l e m e n t s , c a l l e d a l s o v e c t o r s , two a l g e b r a i c o p e r a t i o n s , a d d i t i o n and s c a l a r m u l t i p l i c a t i o n , a r e d e f i n e d , w i t h t h e f o l l o w i n g usual algebraic properties: (i) To every p a i r (x,y) E x a way, t h a t x+y = y+x

and

x

x corresponds a v e c t o r

(x+y) E

x,

i n such

x+(y+z) = (x+y)+z;

b e s i d e s x c o n t a i n s a unique v e c t o r 0 ( t h e zero v e c t o r or o r i g i n of X) such t h a t x+O

=

x, V x E

-x such t h a t x+(-x)

x

and t o each x corresponds a w e l l - d e f i n e d v e c t o r

= 0.

(ii) To every p a i r (a,x) w i t h x E X and a r e a l or complex number

corresponds a v e c t o r ax E Ix = x,

X,

i n such a m y t h a t

a(f3x) = (aB)x,

1. Manifolds, Functional Analysis, Distributions

38

and such t h a t the two d i s t r i b u t i v e laws hold:

The symbol 0 i s a l s o used f o r t h e zero element of t h e s c a l a r f i e l d . A v e c t o r space over R

( r e a l numbers) i s s a i d t o be a r e a l v e c t o r space,

and t h e one over C ( t h e complex numbers) i s c a l l e d a complex v e c t o r space. D e f i n i t i o n 1.2.17.

A set

x

i s said t o be a topoZogical vector space, i f t h e following four

conditions are f u Zfi Z led: ( i )A

colZection

Y

of subsets of x i s singled out w i t h the following

properties:

0 and X beZong t o the system

lo. The empty s e t

2O. The union of any number 3'.

of subsets i n

Y

Y;

i s again a subset i n Y ;

The i n t e r s e c t i o n of any f i n i t e number of subsets i n

set i n

Y

i s again a sub-

Y.

The elements of t h e c o l l e c t i o n

Y

are c a l l e d open s e t s and t h e

co2lection Y i t s e l f i s said t o be a topoZogy on X. Euch open s e t containing a point x E

of

x i s said t o be a neighbourhood

X.

A point x E M c x i s called an i n t e r i o r point of the s e t M, i f M contains a neighbourhood u(x) of X.

Obviously, a set M i s open i f f every x (ii)x

i s a Hausdorff space and

of p o i n t s x

1

and x

2

Y

E

M i s an i n t e r i o r p o i n t of M.

i s a Hausdorff topclogy, i . e . each p a i r

in x have d i s j o i n t neighbourhoods u(xl) and u(x,)

:

u(xl) n u(x2) = 0. A point x E x i s said t o be a Z i m i t point of a s e t M c X, i f each

neighbourhood of x contains a t Zeast one point of M other than A set M c

x i s said t o be closed,

Every p o i n t x E

X

X.

i f i t s complement fiM i s open.

is a cZosed s e t . The closure k of a s e t M c

-

i n t e r s e c t i o n of a l l closed s e t s t h a t contain M; M

=

X

i s the

M U M' where M' i s the

c o l l e c t i o n of a l l l i m i t p o i n t s of M. 0 The i n t e r i o r M o f a s e t M i s the union of a l l open s e t s t h a t are

subsets o f M. A sequence Ix 1 i n a Kausdorff space (write: l i m x

= x)

i f every neighbourhood

nf i n i t e l y many of the p o i n t s xn.

x E x of x contains a l l but

x converges t o a p o i n t U(X)

1.2. Functional Analysis

39

It is readily seen that x = lim x is well-defined. n(iii) x is a r e a l or complex v e c t o r space. (iv) The v e c t o r space o p e r a t i o n s a r e continuous w i t h r e s p e c t t o t h e

topology

i . e . by d e f i n i t i o n t h e mapping

Y,

x

x

x 3

(x1,x2)

-f

is continuous (if x 1. E X, j

(Xl+X2) E

x

1,2 and if V is a neighbourhood of x +x 1 2' then there should exist neighbourhoods V.(x.) such that V1(x1)+V2(x2) c V) I 1 =

and sirni Zarly t h e mapping x

ip

x

3 (a,x) + ax E

i s continuous, where x E X, a E

0

0

x

i s t h e f i e l d of t h e r e a l or complex numbers (if

and V is a neighbourhood of ax E X, then for some r > 0 and

some neighbourhood W(x) 3 x one should have 6W c V whenever 15-a A set A

C

I

<

r)

.

X is said to be symmetric if x E A implies that (-x) E A,

as well. If U c X is a neighbourhood of zero in X, then (-U) n U is a symmetric neighbourhood of zero in X. A collection Y' c Y is said to be a base for the topology Y if every

member of Y (that is, every open set) is a union of members of YO.

A collection Z of neighbourhoods of a point x E X is a local base at x if every neighbourhood of x contains a member of Z. If E c X and if

2

is the collection of all intersections E

V with

V E Y, then Z is again a topology on E inherited from Y.

If a topology Y is induced by a metric p , then

p

and Y are said to be

-

compatible with each other.

Two topologies Y and Y on X are said to be equivalent, if for each neighbourhood U E Y there exists a neighbourhood and, conversely, for each

c

?

E

?

such that

c U

there exists V c Y such that V c V.

Example 1.2.18. Let X be the set of all infinitely differentiable functions in an interval U

=

(a,b) C R , which have a compact support in U, i.e. for each function

x(t) in X there exists an interval U' cc U such that x(t) vanishes outside U'. Obviously, X is a vector space. Let the neighbourhoods of zero be

defined as follows: for each 6 > 0 and each n 2 0 integer a neighbourhood ) I < 6, for U(n,6) is the set of all functions x(t) E X such that / x ( ~(t)

1. Manifolds, Functional Analysis, Distributions

40

k = O,l,.. .,n. With the neighbourhoods of each element xo(t) E X defined as the sets of functions x(t) such that x(t)-xo(t) form a neighbourhood of zero,

x

becomes a topological vector space, which is usually denoted either

m

by D(U) or by c;(U). Example 1.2.19.

Let Xh, h E (O,l), be the family of vector spaces of sequences u(h,x), x E F$l,= hZ with real or complex values, which depend continuously on + -, i.e. for any integer k one has: <x> lu(h,x) I + 0 as

h E (0,l) and are rapidly decreasing as 1x1

k 2 0 and for each sequence u 1x1 + -, where < x > ~= l+lxl 2

.

:

R1 h

+ C

Introduce the neighbourhoods of zero in Xh as follows: for each 6 > 0 and every integer k > 0, n 2 0 , a neighbourhood of zero U (6,h,n) is the set h of all sequences u(h,x) such that

..,

p ~~ < x > ~ u(h,x) / a ~I < 6 , for m = 0,1,. n (1.2.128) / ~ l ~ :=, s ~~ ; x,h xERh and for h E (0,1), With the topologies defined in such a way, h

+

Xh becomes a family

of topological vector spaces. Considering each u(h,x) E Xh as the Fourier coefficients of the periodic function C(h,S), ;(h,c)

=

h X u(h,x) exp (-ix<), xERh

one gets another family of vector spaces Xi of all periodic functions ;(h,5) with period 2n/h, which are infinitely differentiable in 5 . For each 6 > 0 and every integer k 2 0 , n 2 0 define a neighbourhood U (6,k,n) of h zero in Xi as the set of all periodic functions <(h,S) such that

where we have denoted: -1 2 (1.2.130) sh(c) := (ih) (exp(ihS)-l), * = l+lSh(C)

I .

It is easy to show that the (dual) topologies defined by (1.2.128) and (1.2.129), (1.2.130), respectively, are compatible. I

1.2. Functional Analysis

41

Tile following types of topological vector spaces are useful in applications: 1

.X x

2 ' . 3

.

is locally convex if there is a local base whose members are convex: is metrizable if its topology Y is compatible with some metric

and the corresponding metric respect to in

p.

X is said to be a Frechet space if X is locally convex, metrizalbe

p

is complete (i.e. each Cauchy sequence with

p

has a limit in X) and invariant with respect to the translation

x: p(x+z,y+z) = p(x,y), V x,y,z

in

X.

The following example of a Frdchet space plays an important role in the distribution theory. Let U

m

C

IRn be an open set and denote by C (U) the vector space of

all complex valued infinitely differentiable functions on U. For each m n the Schwartz space of all u E C (IR ) , whose C IRn denote by D K support lies in K.

compact K

m

Introduce a topology on C (U) which makes it into a Frechet space such that

DK

each bounded

is a closed subspace of Cm(U) whenever K

To do this, let us choose compact sets KN, N = 1,2, in the interior of KN+l and U = N = l,Z,

C

U and, moreover,

m

set in C (U) equipped with this toopology is compact.

..., by

U

...,

such that KN lies m

KN. Define seminorms PN on C (U),

Nd 1

setting:

P (u) = max max /D:u(x) N lal
I. m

They define a metrizable locally convex topology on C (U), a compatible translation invariant metric p(u,v)

:=

c Nt 1

p

being the following one:

2-NP (u-V)(l+P (u-v))-l. N N

Furthermore, for each compact K

C

U,

DK

m

is a closed subspace of C ( U ) , and

a local base is given by the sets:

0

N

m

:=

{u E Cm(U)

1

P ( u ) < N-'}, N

N = 1,2,

...

Each I) C C (U) is again a Frechet space with the same local compactness K m property as C ( U ) itself.

1. Manifolds, Functional Analysis, Distributions

42

1.2.3.

Banach spaces

Definition 12. 20 .

A vector space B is said t o be a Banach space if t o every x E B a non-

negative nwnber I ( x (1 i s associated, caZZed the norm of

X,

i n such a way

that ( i ) IIx+yli 5

I

( i i ) lax1 (iii) 11x1

1

1

=

>

I I x t l + l l y l I, l a / 11x1 1 , v

o if

x

+ 0,

V (x,Y) E B x 8, x E

B , a r e a l (or complex) n u d e r ,

( i v ) every Cauchy sequence I x } c n n>O

B has a l i m i t x E B, i . e . B is

complete in the metric defined .Ey the norm on 8 . Example 1.2.2 1.

lo. For each given i n t e g e r k t 0 t h e space C k ( E ) of a l l k-times continuously d i f f e r e n t i a b l e f u n c t i o n s u on some closed i n t e r v a l

5=

[a,b],

equipped with t h e norm: (1.2.131)

IuI

=

Ck(6)

z ma5 OSj
Ia:u(x)

1,

i s a Banach space, t h e topology defined by t h e norm (1.2.131)

equivalent with t h e uniform convergence on

being

with t h e d e r i v a t i v e up t o t h e

o r d e r k. 2 ' . The spaces C (0,k) ,E t h e family of norms:

(u)introduced

i n Example 1.2.5 and equipped with

(1.2

i s a family of Banach spaces. A l s o t h e spaces C k ( R 1

h x,k with t h e family of norms: (1.2.133)

:=

) introduced i n Example 1.2.5 and provided

Z s u p 1 Ia:,hu(h,x)I, O<j
i s a family of Banach spaces.

h E (O,l),

I

The following r e s u l t i s known a s Kolmogorov's theorem: A t o p o l o g i c a l v e c t o r space X i s normable ( i . e . a norm compatible with topology can be introduced on X) i f f X possesses a bounded convex neighbourhood of zero. Two Banach spaces B1 and B2 are s a i d t o be isomorphic i f t h e r e e x i s t s a one-to-one

continuous map f : B1 -+ B 2 , which p r e s e r v e s t h e a l g e b r a i c

o p e r a t i o n s on B 1 and B2, i . e .

f (ax+%y)= af ( x ) + %(fy ) , V (x,y) E B1

and V a , % r e a l o r complex numbers.

X

B2

1.2. Functional Analysis Let B be a Banach space and let Bo Two elements xk E B, k = 1,2, class X, if (x -x 1

2

)

E

Bo.

43

B be a closed subspace of B.

C

are said belonging to the same equivalence

The linear structure and the norm on the set of

the equivalence classes are induced by the linear structure and the norm on B and the Banach space B/BO, thus obtained is called the quotient space of B modulo B

0'

1.2.4.

Hilbert spaces

Definition 1.2.22.

A vector space H i s c a l l e d a (complex) Hilbert space (or unitary space) i f t o each ordered p a i r of vectors x and y i n H is associated a complex number (x,y), called the inner product or scalar product of x and y, such t h a t

t h e following properties are v a l i d : (i) (x,y) = (y,x)* (The upper star denotes complex conjugation.)

, .

(ii) (x+y,z) = (x,z)+(y z ) (iii) (Ax,y)

=

A(x,y),

(iv) (x,x) 2 0 ,

V x E

H

V (x,y,A) E

H,

(x,x)

(v) Every Cauchy sequence {x 1

=

x

H

x

0 iff x

t

C. = 0.

ti converges t o some element x E H , i.e.

H is comp2ete with respect t o t h e norm:

a complex vector space w i t h an inner product over t h e f i e l d

If H is

o f the r e a l x w n b e r s , then H is c a l l e d a r e a l Hilbert space. The following basic properties of the Hilbert spaces can be easily established: (1.2.134) (1.2.1353 (1.2.136)

1 (x,y)1 S (1x1 ( y (I H (Cauchy-Schwarz inequality), 1 /x+y(I H 6 ( ( x (IH+( / y (I H (The triangle inequality), I ly[I H 5 I [Ax+y(I H for every A E C iff (x,y) = 0.

Every nonempty closed convex set V c H contains a unique element x of minimal norm. An

element x is said to be orthogonal to a set E c H if (x,y) = 0, V y E E. If M is a closed subspace of H and M I is the set of all elements

orthogonal to M, then (1.2.137)

H

=

M B MI,

h is

a closed subspace of H and, moreover, one has:

1. Manifolds, Functional Analysis, Distributions

44

i.e. each element x E ff can be uniquely represented in the form: x = x M If M

C

+M' f xM

EM,

xiEd.

ff is a closed subspace, then (Ml)'

=

M.

If {xnInzO is a sequence of pairwise orthogonal vectors in

ff, then

each of the following three statements implies the other two: (i) E x nzO

converges, in the norm topology of

ff.

(iii) X (xn,y) converges, for every y E H. ntO A collection of vectors {e 1 C ff is said to be orthonormal system 1, k = j n if (e e.) = dkj = {o, k z j . k' I

Let L

5

be the closed subspace of ff spanned by an orthonormal

system {e 1 and let x E L. The numbers n n2l

,...

(1.2.138) f. = (x,e.), j = 1,2 7

3

are called the Fourier-coefficients of x E L with respect to the orthonormal basis {e 1 in L. n n2l One has Parceval's identity for every x E L:

Moreover, for every x E ff one has the Bessel's inequality:

If ff does not contain a non-zero element which is orthogonal to an

ff, then {en>n2l is an orthonormal basis in ff and in that case Parceval's identity (1.2.139) holds for every x E ff with

orthogonal system {en}n2O

C

f. given by (1.2.136).

I

A

Hilbert space

ff which has an orthonormal basis {enlnZ1 is called

separable. Every separable complex (real) Hilbert space ff is isometric and isomorphic to the complex (real) Hilbert space l2 of the sequences u

=

{u 1 n ntO

equipped with the inner product (u,v) =

z nZO

u v* n n'

where again the upper star stands for the complex conjugation.

1.2. Functional Analysis

45

In particular, the space L2(U), U = (0,l) of all square integrable complex valued functions u(x) on U with the inner product := (1.2.140) (u,v) L*(U)

J u(x)v(x)*dx

and l2 defined hereabove are isomorphic and isometric (Riesz-Fischer). 1.2.5. Classical Sobolev spaces W and HBlder spaces Ck" Plr Let U

Rn be a bounded open set. For p E

C

[l,m),

any r-times continuously differentiable function u

:

U

integer r 2 0, and + C

define the

Sobolev norm:

where

Taking the closure of the set of all r-times continuously differentiable functions in

with respect to the norm defined by (1.2.141), (1.2.142),

one gets the Sobolev space W (U). P.r The elements of W (U) which are not r-times continuously differentiable Pfr functions on are said to have generalized derivatives up to the order

u,

r t 0 which are in L ( U ) .

P

One can give a different definition of a generalized derivative a"u E L (U) of a function u E L (U), la1 5 r. Namely, let $ E Cr(?J) and X P P let J, have a compact support in U (i.e. $ vanishes identically in some neighbourhood of the boundary aU of U). A function v E L (U) is said to be the generalized derivative aau of

P

u

E

L

P

( u ) if for any function

J, E Cr(u) with compact support in

u one has:

(U) as the vector space of the Thus, one can define the Sobolev space W PPt functions u : U + C which have all generalized derivatives up to the order

r >= 0 in L (U). P Notice, that two functions which are elements of W (U) and coincide a.e. PCr have also all their generalized derivatives up to the order r, which (U). Thus, coincide a.e. and, therefore, are indistinguishable in W Ptr (U) are (equivalence) classes of functions, prr

actually the elements of W

1 . Manifolds, Functional Analysis, Distributions

46

which coincide a.e. and have all their generalized derivatives up to the order r 2 0 in L (U).

P

It can be shown but will not be done here, that both definitions of the Sobolev spaces W (U) are equivalent. Of course, W (U) is a Banach P,r Prr space. Define the Sobolev space W ( l R n ) as the closure of the vector space p,r of all r-times continuously differentiable functions with compact support in X i R n , with respect to the norm

Sobolev spaces are very useful in applications. Aslo very useful are k,X Hdlder spaces C (U). Again let U u

:

U

+

C Xin be a bounded open set. A continuous function 0,X C is said to belong the Hdlder space C (U) with A E ( 0 , l ) if the

following norm is finite: (1.2.144) IIull

o,x c

(U)

. .=

SUP-

-

Ix-Yl-Alu(x)-u(Y)

I.

(x,y)Euxu,x#y

k,X Hdlder space C ( u ) with integer k 20 and X E (0,l) consists of all ktimes continuously differentiable functions u(x) such that

or equivalently,

where, of course,

I Iul I

=

COG)

max lu(x) 1 . XEU

The following classical result is due to S.L. Sobolev and Morrey. Theorem 1.2.23.

L e t u c =“be bounded and l e t

au b e a smooth rnanijold.

u has a l l g e n e r a l i z e d d e r i v a t i v e s of order l e s s

If u E

w

PrZ

(U) t h e n

than r and, moreover, i f

rp > n , 0 2 k < r-(n/p), t h e n u E ck(,) and t h e r e e x i s t s a constand which does not depend on u E w (u) and such t h a t

C > 0

P#r

More p r e c i s e l y , t h e r e e x i s t s a f u n c t i o n v E u a.e.

and such t h a t ( 1 . 2 . 1 4 7 )

which c o i n c i d e w i t h ck,A (u)

holds w i t h a c o n s t a n t

C >

0 , which a l s o

47

1.2. Functional Analysis coes n o t depend on u and v.

Very useful for the numerical ixeatment of partial differential equations are finite difference versions of the classical Sobolev spaces k,X W (U) and Hdlder spaces C (U). P,r Again, let U C lRn be a bounded open set with a smooth boundary aU, p E [ 1 ,m) and r LO an integer. Denote

n

Uh = U where n\

=

<,

h E (O,ho)

n . hZ LS a one parameter family of grids in lRn, and introduce

the discrete version of L -norms for sequences (meshfunctions) u(h,x)

:

X$

P

+

C on Uh in the following fashion:

lu(h,x) Iphn)'",

h E (O,ho)

P

h Furthermore, a family of discrete norms similar to (1 2.141), are defined in the same way:

Remark 1.2.24. It can be shown but will not be done here that the following norms are equivalent with (1.2.149) uniformly with respect to h E (0,h 1 , V ho > 0. 0 Let v(h,x) : + C be any sequence (meshfunction) such that

<

u(h,x) : u(h,x), V x E U Introduce for u(h,x)

:

h

X$

<

is finite. and (1.2.149) for v with U = h C and with Uh as defined hereabove the -f

following norm:

where the infimum is taken over all meshfunctions v with

1 \ v /1 P,r;uh

<

m

and such that v(h,x) E u(h,x), V x E Uh. Then, there exists a constant C > 0, which does not depend on h and u(h,x) and such that

<

(U ) , h E (0.5 ) , consist of all meshfunctions The spaces W p,r h 0 u(h,x) : + C, which depend continuously on h E (0,h ) in the sense of 0 the topology defined by the norms (1.2.149) and, moreover, such that

1. Manifolds, Functional Analysis, Distributions

48

Obviously, W ( U ) is a family of Banach spaces and the vector p,r h space E (U ) for which semi-norm (U,) of all meshfunctions u E W P,r p,r h (1.2.150) vanishes, is a closed subspace of W (Uh), i.e. again a family P*r of Banach spaces. Two meshfunctions u and v in W (U,) are said being equivalent or Per belonging to the same equivalence class, if /u-vI = 0 where 1 - 1 p,r;u p,r;u is the norm defined by ( 1 . 2 . 1 5 0 ) . In other words, u and v are said to

be equivalent, if they belong to the same equivalence class in the family (U ) modulo of quotient spaces Wp,r(Uh)/Ep,r(Uh), h E (O,ho), of W p,r h

2 2 2 w (x;y) := afi-lh-nexp(lx-yl /(]x-yl -h ) ) ,

h

w (x,y) E 0 ,

for Ix-yl < h

for (x-yl L h,

h

where

1 a =

I

2

exp(p2(p - 1 )

-1

)dp

0 is chosen in such a way that

1

w

V y

(x;y)dx E 1,

E

lRn,

V h > 0.

Rn For every function u E W (V) define its averaged u (x) with the PPr h averaging kernel w (x;y) hereabove as follows: h

where 1 u(x) is the extension of u 0

outside the open set V

C

Rn

:

V

-f

C

by settint it identically zero

.

Let U C C V. It can be shown but will not be done here that for each u E W

p,r

(V) one has for its averages uh as defined hereabove:

I 1Uh-J

Ip,r;u - t O

ash+O.

Obviously, u (x) is infinitely differentiable function with respect h , V h > 0.

to x E lRn

With each u E W (V) associate the meshfunction u(h,x) : n\ -t C, Per which is the trace of the averaged u (x) of u ( x ) on the family of grids h Using HBlder's inequality one can show that u(h,x) belongs to W (Uh) P.r

$.

1.2. Functional Analysis

49

and, moreover, with some constant C > 0, which does not belong on

u E W (V) and h E (O,l), it holds: P,r lblp,r;u

I juI \p,r;Uh5

CI / u /

1 p.r;V'

1.1

is the norm defined by (1.2.148)-(1.2.150). p,r;U Thus, the traces of the averaged of all u E W (V) is a linear - p,r subset in w (U 1, provided that U C C V, i.e. U C V. p,r h The finite difference analogy of Hdlder spaces C k n X are defined in

where

(u)

<

an obvious way by introducing the families of norms for the meshfunctions u(h,x)

:

-f

C as follows:

-

-

Obviously, traces of functions u E CkpX(v) on Uh = U I UCC

V, form a linear subset in the spaces

C

k,X -

<

with

(Uh), h E (O,l), of the

meshfunctions u(h,x), whose norms (1.2.151) are uniformly bounded with respect to h E ( 0 , l ) (i.e. are bounded by a constant which does not depend on h E (0,l)). The following result is a finite difference version of the Sobolev'sMorrey result stated in Theorem 1.2.23. Theorem 1.2.25.

Let u c R~ be an open bounded s e t w i t h a smooth boundary k i s the l a s t i n t e g e r such t h a t 0 5 k < r-(n/p), X

e x i s t s a constant c

> 0

=

au.

I f rp > n,

r-k-n(p), then there

which does not depend on h E (O,ho)and u E W

P,r

and such t h a t

Proof. __ First consider the case when Uh = RE and r

A

=

=

1, so that p > n, k = 0 ,

1-(n/p). We shall show that

where the constant c > 0 depends only on p and n; here, of course,

(u,)

1. Manifolds, Functionai Analysis, Distributions

50

a x , h ~=

(ax,huf

Obviously, for each x E Rn\{O} h (1.2.154) u(h,x)-u(h,O) = h

C

one has:


YET where

r

Y.

,>u(h,y),

c Rf: is the set of points whose distance to the line tx,

0 5 t 5 1, in Rn is minimal, and ~ ( y )is the tangent vector to the broken

line passing through the points of Tx whose each piece is parallel to one of the coordinate axes x .

I'

< j 5n.

Let a > 0 and denote (1.2.155) Q, := {x = (xl,...,xn) E

~ f :I

jxjI 5 a, I

c

j 5 n}.

Let u.(h) be the mean value of u(h,x) over Qa: 7

(1.2.156) u (h) 0

:=

a-n

C u(h,x)hn. xEQa

Then (1.2.154), (1.2.155) yield:

C u(h,y)hn+l. (1.2.157) u (h)-u(h,O) = a-n X 0 XEQ, YETx Yrh It is readily seen (one has to use the induction argument, that

e

(1.2.158) c

< ~ ( y ) , a ~ , ~ > ~ ( h =, y )c c O5t6a xEQt

T(X)

,ax,h u(h,x),

XEQ, YETx

t = kh, 0 < k < a/h. Therefore, (1.2.157), (1.2.158) and HBlder's inequality (for sums) yield:

C a

-n O
where p ' is such that l/p + l/p' = 1. BY translation and by triangle inequality one also has:

for each pair of points (x,y) E Rf:

X

RE

such that a

=

2 Ix-yl , so that

1.2. Functional Analysis

51

(1.2.160) y i e l d s (1.2.153) with c = 22-n/P(n+l-n/p). Using t h e induction argument, one g e t s (1.2.152) f o r Uh

=

and

RE

using Remark 1.2.24, one f i n a l l y e s t a b l i s h e s (1.2.152) f o r each U

h'

Remark 1.2.26. In t h e proof of Theorem 1.2.25 we have used an a d a p t a t i o n t o t h e d i s c r e t e s i t u a t i o n of t h e method of prving t h e Morrey's r e s u l t f o r c l a s s i c a l Sobolev space, which is t o be found i n [ B r e z . l ] 1.2.6.

Linear f u n c t i o n a l s and dual spaces

8 be a r e a l ( o r complex) Banach space.

Let from

B

A l i n e a r mapping f ( x )

i n t o t h e f i e l d of r e a l ( o r complex) number i s c a l l e d a l i n e a r

functional. A l i n e a r f u n c t i o n a l f ( x ) defined on t h e whole Banach space

t o be continuous a t a p o i n t xo E {xnlnZl c 8 such t h a t

1 Ix

-xoI

1

B i f f(xn)

+

0 for n

+

+

f ( x ) f o r any sequence

E B i f it

0

m.

A l i n e a r f u n c t i o n a l f ( x ) on a Banach space

point x

8 is said

8 i s continuous a t each

i s continuous a t j u s t one p o i n t x

0

E 8.

A l i n e a r f u n c t i o n a l f ( x ) i s s a i d t o be bounded on

8 i f there exists

number c 2 0 such t h a t (1.2.161)

If(x)I 5 c / I x I / , V x

E 8.

The l e a s t number c such t h a t (1.2.161) s t i l l holds i s c a l l e d t h e norm of t h e f u n c t i o n a l f ( x ) and i s denoted by

\

If

I I.

Thus

A l i n e a r f u n c t i o n a l f ( x ) on a Banach space

8

s continuous i f f f ( x )

i s bounded.

D e f i n i t i o n 1.2.27.

A f a m i l y of l i n e a r f u n c t i o n a l s f ( x ) on a f a m i l y o f Banach spaces 8

w'

depending upon a parameter w i n some parameter s e t fi 5 Rn i s s a i d t o b e equicontinuous ( o r equiboundedl i f t h e r e e x i s t s a c o n s t a n t c 2 0 , which does n o t depend on w E 0 and such t h a t

a

1. Manifolds, Functional Analysis, Distributions

52

The least constant c such t h a t (1.2.163)

s t i l l holds i s c a l l e d t h e

of l i n e a r f u n c t i o n a l s f norm of t h e family {f ( x ) loEn by sup on

wcn B .

1 I f w /[

where

1 I f w (1

( x ) and i s denoted

i s t h e norm of f ( x ) a s a l i n e a r f u n c t i o n a l

Example 1.2.28. U = ( 0 , l ) be t h e family of Banach spaces introduced i n

Let C (o,o) Example 1.2.21.

Let y E U be given. Consider t h e following gamily of l i n e a r functionals f

E

( v ) on C E

( O , O , ) ,E

6 ):

-'I vE( y )exp

(1.2.164) f E ( v E ) := ( 2 ~ )

( - Ix-y

1 /E) dy.

U

Then t h e family (1.2.164) i s equiuontinuous on C

( 0 , O ) ,E

(5). Indeed,

one

has :

One of t h e fundamental r e s u l t s i s t h e following theorem of HahnBanach. Theorem 1.2.29.

Let B be a Banach space equiped w i t h the norm 1 I 1 I and l e t M c B be a subspace of 8. Let f ( x ) be a l i n e a r functional on M such t h a t (1.2.165)

/ f ( x )1 5

Ci

1x1

IB,

v

x

E

M.

Then there e x i s t s an extension of f ( x ) onto the whoZe Banach space 8 such that /f(x)1 5

where c 2

0

C ( 1x1

18,

V x E

8

i s t h e same constant as on t h e r i g h t hand-side of (1.2.165).

Obviously, The Hahn-Banach theorem i s s t i l l t r u e f o r t h e f a m i l i e s of equicontinuous f u n c t i o n a l s . L e t 8 be a Banach space. The s e t

f u n c t i o n a l s on

8' of a l l continuous l i n e a r

equipped with t h e norm (1.2.1621,

c a l l e d t h e dual space of

i s given a Banach space,

8.

Existence of a convex neighbourhood of zero i n a Banach space

B

is

t h e necessary and s u f f i c i e n t condition f o r t h e n o n - t r i v i a l i t y of t h e dual

1.2. Functional Analysis

space 8 ' . Thus, t h e dual space of L . ( U ) ,

U

Rn

C

53

,p

< 1 i s t r i v i a l . The

Using t h e induction argument, one g e t s (1.2.152) f o r Uh

=

and

R:

using Remark 1.2.24, one f i n a l l y e s t a b l i s h e s (1.2.152) f o r each U

I

h'

Remark 1.2.26. In t h e proof of Theorem 1.2.25 we have used an a d a p t a t i o n t o t h e d i s c r e t e s i t u a t i o n of t h e method of prving t h e Morrey's r e s u l t f o r c l a s s i c a l Sobolev space, which i s t o be found i n [Brez. 1 1.2.6.

1

Linear f u n c t i o n a l s and dual spaces

8 be a r e a l ( o r complex) Banach space.

Let

A l i n e a r mapping f ( x )

from 8 i n t o t h e f i e l d of r e a l ( o r complex) number i s c a l l e d a l i n e a r functional. A l i n e a r f u n c t i o n a l f ( x ) defined on t h e whole Banach space

E 8 i f f(x

t o be continuous a t a p o i n t x {xnlnZl C 8 such t h a t

1 Ix

0 -xoI

I

0 for n

+

)

+

+

-.

A l i n e a r f u n c t i o n a l f ( x ) on a Banach space

point x

E 8 i f it

8 is said

f ( x ) f o r any sequence 0

8 i s continuous a t each

i s continuous a t j u s t one p o i n t xo E

A l i n e a r f u n c t i o n a l f ( x ) i s s a i d t o be bounded on

8. 8 i f there exists a

number c 2 0 such t h a t (1.2.161)

If(x)I 5 c / I x I / , V x

E 8.

The l e a s t number c such t h a t (1.2.161) s t i l l holds i s c a l l e d t h e norm of t h e f u n c t i o n a l f ( x ) and i s denoted by (1.2.162)

I If1 I

= sup I f ( x )

xE8

I/j

1x1

1

=

1 If I 1 . SUP

Thus If(x)

I.

11x1 I=1

A l i n e a r f u n c t i o n a l f ( x ) on a Banach space

8 i s continuous i f f f ( x )

i s bounded.

D e f i n i t i o n 1.2.27.

A f a m i l y of l i n e a r f u n c t i o n a l s f ( x ) on a family o f Banach spaces 8

w'

depending upon a parameter w i n some parameter s e t fi 5 R~ i s said t o be equicontinuous (or equiboundedl if there e x i s t s a constant c 2 0 , which

F i f a one-to-one continuous map j of 8 onto a l i n e a r s u b s e t of F i s given, which preserves t h e a l g e b r a i c o p e r a t i o n s . Thus, one has:

space

\ 1 j (x) I I

1

5 C 1x1

IB

i s called natural.

with some c o n s t a n t C > 0 . I f C

=

1 then t h e imbedding

1. Manifolds, Functional Analysis, Distributions

54

I f t h e imeage of

8 i n F i s dense, then t h e imbedding i s c a l l e d dense,

Linear Operators

1.2.7.

Let

x1

and X 2 be two t o p o l o g i c a l vector spaces equipped with t h e

Hausdorff copology. A map A : X,

+

x2 i s said t o be a continuous linear

operator from x1 i n t o x

i f the following conditions are s a t i s f i e d : 2 i s defined on the whole of X1;

lo. A 2O.

A i s l i n e a r , i . e . f o r each couple x and y i n

( r e a l o r complex numbers) A(ax+By)

=

c1

x, and f o r each two scalars

and 5 one has:

aAx + 5Ay.

3O.A i s continuous a t each point x F: x , i n t h e sense of the topologies on x1 and x2, i . e . f o r each neighbourhood u 2 ( h ) of AX i n x2 t h e r e e x i s t s a neighbourhood ul(x) o f x i n x1 such t h a t f o r every x' E ul(x) one has:

Ax' E U2(AX).

A familyog l i n e a r mappings

A

:x1+x2,

respect t o t h e p a r a m e t e r s e t n ) i f f o r :

KCX

1

o E n , i s s a i d t o b e equicontinuous ( w i t h

each given u 0 E G the l i n e a r operator

x 1 + x 2 i s continuousand, moreover, theimage of each bounded s e t under the map A

I w E n ) i s uniformly bounded with respect t o w E n ,

i . e . the union u w E n A w ~belongs t o a bounded s e t i n x2.

81 and B 2 be two Banach spaces. A linear map A : 8, + 8, i s said t o be bounded i f there e x i s t s a constant c > 0 such t h a t 1 / A X ( ( s c ( l x ( 1 , V x E 8,. The l e a s t constant c > 0 Let

x2

x1

w i t h such a property i s called the norm of

I IAl

18 1

2

A :

B,

+

8, and i s denoted by

.

A l i n e a r map A :

8, + 8, i s continuous i f f it i s bounded.

Obviously, one has:

Let {Au}u>O be a sequence of bounded l i n e a r mappings form 8, i n t o 82' The sequence t A v } v > O , A : 8, + 8,, i s s a i d t o be convergent i n t h e norm topology t o some bounded l i n e a r o p e r a t o r . A

if

1 IA 0 -A v 1 I B,+BB,

I t i s s a i d t o be s t r o n g e l y convergent i f f o r each x

-f

E 8,

0 for v

one has:

+-.

1.2. Functional Analysis

55

I / A O ~ - A v1 ~ I

-Z 0 for v -Z a . The sequence {Av}v>O is weakly convergent to 82 some linear operator A , if for each x E 8 the sequence {Av}v,>O is weakly 1 convergent to A x . 0 The norm convergence implies the strong convergence and the latter implies

the weak convergence. The following result known as the Banach-Steinhaus theorem is important for applications. Theorem 1.2.30.

Let

{ A ~ I ~be> a~

sequence o f continuous Zinear o p e r a t o r s

from t h e Banach space B , i n t o t h e Banach space B,.

:

B,

+

8,

Assume t h a t f o r each

x E 8 , one has:

Then one a l s o has:

i . e . there e x i s t s a constant c IlAVxl I

1

5 C//x/

0

,

such t h a t

V x E B,,

V v > 0.

B2

If A map A-l

:

:

B,

8,

+

+

B2 is a one-to-one linear mapping onto B,,

then the linear

8 , is well defined and is called the inverse of A.

The following result, known as the open mapping theorem, plays an important role in applications. Theorem 1.2.31.

Let

A :

8,

+

a constant r

B , be a continuous Zinear mapping onto B,. 0 such t h a t t h e b a l l Biz) = {y E B , I I

i s contained in t h e image of t h e baZl ~ ( l = ) Ix E B , 1

I

Then t h e r e e x i s t s

/YI I

<

r} c 8 ,

B2 11x1

I

< 1) c

B,

Bl

under t h e Zinear mapping A , i . e .

Particularly important for applications is the following consequence of the open mapping theorem: Corollary 1.2.32.

Let A

:

8,

+

8 , be a continuous Zinear one-to-one

mapping o n t o €3.,

Then i t s

1. Manifolds, Functional Analysis, Distributions

56

inverse

A

-1

:

B2

+

B1 is again a continuous l i n e a r mapping onto B

Definition 1.2.33. A family of linear-mappings Aw

: EW +

Fw, w

1'

E 0, i s

said t o be equibounded lor equicontinuous) i f one has:

The following extension of Banach's theorem is useful in applications. Theorem 1.2.34. L e t

: E~

A

F ~ ,w

+

E 0, be linear,one-to-one and equi-

continuous. Assume t h a t one has:

Then -1 AW

: F w + E

W'

w E R

i s equicontinuous. Proof. One -

only has to show that

If the last inequality does not hold, then there exists a sequence

{wk}k21 c R such that

E Fw such that

y,

so that there exists a sequence

k

k

Introduce

-1 x = A , yW , zw Wk k k k

=

xu /IIxw k k

\IE

,

k = 1,2,.,.

w k

so that the last inequality may be rewritten as follows: -1 )lAwkZwkllFw 6 k

,

for

k Let u

E EW,

1 luW/I E

= W

1 and define:

I/zW

k

/IE

=

w k

1,

k = 1,2,...

1.2. Functional Analysis

zw

= u

w

for

w

# wk

and

z

w

=

z

w

for

w

k

=

w k'

Obviously, one has:

the latter being a contradiction with ( 1 . 2 . 1 6 9 ) . Remark 1 . 2 . 3 5 . Aw

:

Ew

+

I

The set L(Ew;Fw) of all linear equicontinuous mappings

Fw, w E R , is a Banach space equipped with the norm:

then L(E ;F ) = L(E ) is a Banach algebra, i.e. besides = F o w the natural linear structure it has a l s o multiplication as a natural

Besides, if Ew

algebraic operation having the property:

Of course, it is readily seen that the set of invertible operators is open in L (Ew;Fw). Let A

o

:

E

w

+

Fw, w E R be a family of bounded linear mappings from

the family of Banach spaces E

into the family of Banach spaces F

.

If gw E FA, i.e. gw is a continuous linear functional on Fw, then the family of functionals fw(Xw) := gw(Awxw), xw E Ew, w E R , is a family of continuous linear functionals on Ew, i.e. f E E ' and, morew w over, one has:

Thus, the correspondence g

w

+

f

w

which is called the adjoint of A Obviously, one has:

so that

defines a linear operator A

.

* 0'

1. Manifolds, Functional Analysis, Disfributions

58

A;

:

F;

E;

+

is again continuous.

It is equicontinuous iff A

: Eu

-f

Fw, w E

R , is equicontinuous.

Let E and F be two Banach spaces and let D(A)

5E

be a vector sub-

space where a linear operator A is defined and takes D(A) into some vector subspace R(A)

5 F. D(A) and R(A) are called respectively, domain

and image (or range) of A . The set N(A)

C E

whose image under the action

of A is zero, is called the kernel of A and sometimes denoted by ker A. Obviously, N(A) is a vector subspace in E. The map A

D(A)

:

R ( A ) is one-

-f

to-one iff N(A) = { O } . In that case the equation Ax = y is uniquely resolvable, V y E R(A). This equation is said correctly resolvable if there exists a constant C > 0 such that one has:

Equation

y is said to be resolvable everywhere if R(A)

Ax =

=

F,

densely resolvable if the closure R(n) in F coincides with F and normally resolvable if R(A) is closed. The correct resolvability of the equation y is equivalent with the existence of the bounded inverse operator

Ax =

A

-1

:

R(A)

-f

D(A).

A linear operator A

:

D(A)

if for each sequence {x 1 k k>l one has: xo E D(A) and yo = D(A) = E, is bounded iff A If A

:

D(A)

+

Axo. :

F, with D(A)

C D(A)

E

+

5 E,

such that xk

-f

is said to be closed

xo and AXk

A linear operator A

:

E

+

+

yo (k

+

-),

F, i.e. with

F is closed.

F is closed, then its kernel N(A) is a closed subspace

+

of E. Equation

Ax =

y with a closed linear operator A is correctly

resolvable iff it is uniquely and normally resolvable. Let A

:

D(A)

+

F

and assume that D(A) is dense in E. Let g E F', i.e.

g is a continuous linear functional on F. Then the linear functional g(Ax)

is defined on D(A). If

g(Ax)

is bounded, i.e. Ig(Ax) I 5

CI

1x1

IE

I

V x E D(A) with some constant C > 0, then g E D(A*) where the linear

operator A*

:

D(A*)

+

E' is defined by the equality (A*g)(x)

V x E D(A). Functional A*g defined by this equality for

v

=

g(Ax),

x E D(A) may be

extended by continuity onto E, since D(A) is dense in E and g is bounded on D(A). Thus, one may assume that A* maps D(A*) space of continuous linear functionals on E). A*

C :

F' into E '

D(A*)

+

(the Banach

E' is called the

adjoint of A; A* is always closed. If F is reflexive, i.e. (F')'=F"=F,

1.2. Functional Analysis

59

then D(A*) is dense in F'. Along with the equation (1.2.172)

AX = y ,

x E D(A) E E ,

y E F

consider the adjoint equation: (1.2.173)

A*g = f,

For ( 1 . 2 . 1 7 3 )

g

E D(A*)

f E E'.

_C F ' ,

one has the same concepts of unique and correct resolvability,

as well as these of the resolvability everywhere and on a dense linear subset in E ' . The equations (1.2.1721,

(1.2.173)

are connected in the following

sense :

Equation ( 1 . 2 . 1 7 2 )

i s densely resolvable i f f equation ( 1 . 2 . 1 7 3 )

is

uniqueZy resolvable; Equation (1.2.172)

i s uniquely resolvable i f ( 1 . 2 . 1 7 3 )

is densely

r e s o lvab l e . The kernel N(A*) of A* is the orthogonal complement to the range R(A) of A.

Thus ( 1 . 2 . 1 7 2 )

i s normally resolvable i f f i t can be solved for each second

member y which i s orthogonal t o N(A*). The linear sets N(A) and R(A*) are orthogonal, as well; however, they are not necessarily orthogonal complements of each other. Equation ( 1 . 2 . 1 7 3 ) is said to be c l o s e l y resolvable if R(A*) is closed in E ' and it is said

normally r e s o l v a b l e , if ( 1 . 2 . 1 7 3 )

can be solved for each second member f,

which is orthogonal to N(A).

If

(1.2.173)

i s normally r e s o l v a b l e , then it i s c l o s e l y resolvable, as

well. The converse of the latter statement is generally speaking false. If A is closed then the closed and normal solvability of ( 1 . 2 . 1 7 3 ) are equivalent. The close solvability of ( 1 . 2 . 1 7 3 ) property of ( 1 . 2 . 1 7 2 ) : M in

there exists a constant C > 0 and a dense set

R (A), such that for each y1 E

(1.2.172),

If ( 1 . 2 . 1 7 2 ) on R(A*).

is equivalent with the following

M

there is a solution x 1 E D(A) of

which satisfies the inequality:

I lxll I E

i s everywhere solvable, then (1.2.173)

Equation (1.2.173) solvable on R(A).

IC I

IyIl

IF.

i s c o r r e c t l y solvable

i s everywehere solvable i f f ( 1 . 2 . 1 7 2 )

i s correctly

1 . Manifolds, Functional Analysis, Disfributions

60

Normally solvable equation (1.2.172) with a closed operator A is called n-normal if N(A) = ker A is finite dimensional: dim ker A <

m.

Normally

solvable equation (1.2.172) with a closed operator A is called d-normal, if the orthogonal complement coker A to the range R ( A ) of A is finite dimensional: dim coker A

m.

As it has been said hereabove, for a closed operator A the correct

solvability of (1.2.172), (1.2.1731, i.e. the validity of the a p r i o r i

estimates:

with some constant

C >

0, is equivalent withtheunique solvabilityevery-

where of the both equations (1.2.172) and (1.2.173). However, in many cases it is possible to establish a priori estimates, which are weaker than (1.2.174). Let E be compactly imbedded in another Banach space Eo and let A

:

Q(A)

+

F, P(A)

5 E,

be closed.

The validity of the following a priori estimate is equivalent with the n-normality of equation (1.2.172):

The d-normality of ( 1 . 2 . 1 7 2 )

is equivalent with the validity of an analogous

a priori estimate for the adjoint operator. Let A be closed, D(A) dense in E and let F' be compactly imbedded in

a Banach space

where

C > 0

G.

Then (1.2.172) is d-normal iff one has:

is some constant.

Equation (1.2.172) (and, respectively, the operator A) is said to be noetherian if (1.2.172) is at the same time n-normal and d-normal. The integer: (1.2.177) K(A)

:=

dim ker A

-

dim coker A

is called the index of equation (1.2.172) and of the operator A. If A and B are both noetherian and D(B) is dense, then BA is again

1.2. Functional Analysis

61

noetherian and, moreover, one has: K(BA)

=

K(A) + K(B)

If D(A) is dense in E and A in (1.2.172)

K

Id A is noetherian, p

(A) :

-K(A).

0.

=

E

=

and the corresponding operator A are called

Noetherian equation (1.2.172) of Fredholm type if

is noetherian, then A* in the

is noetherian, as well, and K(A*)

adjoint equation (1.2.173)

+

F is bounded and has its norm sufficiently

small, then A+Q is noetherian too, and

If A is noetherian and K

E

:

+

F is compact, then again

K(A+K) = K(A).

L e t Ew ' Fw ' w E 0 be two f a m i l i e s of Banach spaces and Fa, w E R, be l i n e a r and equicontinuous. Assume t h a t t h e r e

Corollary 1.2.36.

l e t Aw

:

Ew

+

are two Banach spaces Eo, F ~ ,two f a m i l i e s Jw

s

:

:

xo Yw

+

xw,

+

YO'

-1

Jw

-1

sw

: Xu

+

X

: Yo

+

Yw

and a continuous l i n e a r operator (1.2.178)

1rJw E

"

J

W'

sw,

w E

0,ofequiisomorphisms:

0'

:

E~

+

F~ such t h a t t h e diagramme

Aw

swfs;l A

FO 0

i s c o m t a t i v e , V w E R. i s noetherian i f f A Then

0

i s noetherian and, moreover,

K

( A ~ )=

VwEfl.

Indeed, one has, as a consequence of the commutativity of diagramme (1.2.178):

A O = J A S

w w w'

so that

V w E f l

K

( A ~,)

1 . Manifolds, Funcfional Analysis,

62

since both

Jw ,

Sw

Distributions

are equiisomorphisms of the corresponding families of

Banach spaces.

If diagrame vanishes a s

Iw-oo I

that

is c o m t a t i v e modulo operators whose norm

(1.2.178)

w + oo

a , then t h e same conclusion is s t i l l t r u e , provided

E

i s s u f f i c i e n t l y smaZl.

Indeed, in that case one has: A.

= J

A S + w w w

Q

w’

where

I IQwl

IE +F

0

so that one has for

+

0,

as

w

+

w

0’

0

< 6 with 6 > 0 sufficiently small:

lw-wol

= K ( JA S ) = “ ( A , ) . w w w

K(A )

0

1 (1), E ( S ) E ( 0 . ~ ~ 1be , the family of Hilbert spaces of

Example 1.2.37. Let S1 be the circle of length 1. Let H 1 = (11,12), 11 2 0 integer, E functions u : S1 + C equipped with the inner products:

where

(k) is the Fourier transform on

S

1

,

1

u (k)

:= .f

exp(-2nkix)u (x)dx, k E

0

and ;*(k) is the complex conjugate of

(k).

Let A€

where a E

m

C

:= E

2 2 2 D,a(x)Dx

D + D2 x’ x

=

-id/dx

1 ( S ) , a(x) > 0, V x E S1.

Obviously,

*E

’

1

H(2,2) , E ( ’

-t

is equicontinuous, since u E

H(o,o) , E

E H

(2,2), E

(s’) iff

Z,

1.2. Functional Anulysis

Furthermore, one has: A.

=

D

2

and

is a continuous linear mapping, where H2(S 1 ) Sobolev space of order 2. = coker A.

Obviously, ker A. the index

=

'

1

H(2,0) , E

{c} with c E

C

(S )

is the usual

any constant, so that

0.

K(A )

=

RE : =

E 2Dxa(x) 2

0 Introduce

63

+ 1

Obviously,

is equicontinuous. We are going to show that for

EO

> 0 sufficiently small R

is

invertible and

is equicontinuous. Introduce S E as follows

1

(sEuE)(X)

sE (x,y)uE(y)dy,

=

0 where sE(x,y)

:=

Z (1+4n2E 2k2a(x))-lexp(Znki(x-y)), kEZ

(x,y) E S'xS 1 .

One has: (1.2.179) R s (x,y) = 6(x-y) + EqE(x,y), (x,y) E slxS1 € 6

where 6(x-y) is the Dirac 6-function and (1.2.180) qE(x,y)

1 q(x,E,kE)exp(Znki(x-y)), kEZ (1.2.181) q(x,e,ks) := 4~k€Dx(a(x)/r(x,2nk€))+ ~D~(a(x)/r(x,2i~ks)), 2 :=

r(x,r))

=

2 l+a(x)r) .

I. Manifolds, Functional Analysis, Distributions

64

It is readily seen that the following inequalities hold for q(x,E,kE):

I 6 (1.2.182) (D;~(X,E,~E) where C

P

cP (l+(kEl)-l,V

x E sl, V

E

E

V k E Z, v p E z+,

(O,co),

may depend only on its subscript.

Furthermore, since x

+

q(x,s,kE) is a Cm-function on S1 and (1.2.182)

hold for any integer p 2 0 (uniformly with respect to

E

E ( 0 , ~ ~ )one ) . has

m

for the Fourier transform q(m,s,kE) of q(x,E,ks) as a C -function of x E (1.2.183) Iq(m,e,ks)I

5 CN (l+m2)-N(l+\ksl)-1, V m E Z, V k

E 2, V N

v where the constant C

N

E

E

> 0, (O,E0),

depends only on the integer N > 0.

Now, (1.2.189) yields:

where Id is the identity operator and where 1

(1.2.184) vE(x)

:=

QEuE(x) =

I qE(x,yluE(y)dy. 0

Using (1.2.180), one finds for the Fourier transform jE(m) of VE(X) on

1

S :

(1.2.185) where

<

(m) =

E q(m-k,E,kc)iE(k), kEZ

( k ) is, of course, the Fourier transform of uE(x) on S

1

.

Now, if B = (bmk)-m<m,k<m is a linear operator in the space l2 of I I = 1lwkl2, then one has the 2 ' k following discrete analogue of Schur's lemma(see Lemma 3.12.8):

~ < ~I Iw the sequences w = { w ~ ] - ~ <with

Indeed, for each w E l2 one has:

Thus, (1.2.184). (1.2.185) yield:

1

S :

1.2. Functioiul Analysis

where C > 0 is a constand and B )=:b(

65

with

(1.2.187) b ' mk = (l+m2 ) (l+k2)-l q(m-k,E,kc).

Now, taking N = 3 in (1.2.183), using (1.2.183), (1.2.186), (1.2.187) and the inequality: (l+m2) (l+k2)-l 5 4(l+(m-k)2 ) , one finds:

where C 1 does not depend on

E

E

(O,E

0

).

Therefore, the linear map

defined by (1.2.184) is equicontinuous, V constand C(E

)

Now,choosing

E~

0

1

, so

<

-,

i.e. one has with some

so small that

one gets the conclusion that €QE, E E ( O , E H2(S )

E~

> 0:

0

)

is an equicontruction in

that (I+EQ€)-'

:

H ~ ( 1S)

-f

H ~ ( 1S) ,

E

E

(o,E0),

is equicontinuous. Thus R

-1

:=

SE(Id

+

EQE)-'

(1.2. 188)

is equicontinuous, since the same argument as above using (1.2. 1%)

(the discrete version of Schur's lemma) shows that

is equicontinuous V

E~

<

-.

1.

66

Manifolds, Functional Analysis, Distributions

Therefore, one gets the conclusion that the diagramme

is commutative, V

E

Therefore, A E index

K(A )

ker A E

E

=

= K(A )

0

coker A

E (O,E : H(2,2)

=

),

O

= 0, V

provided that

1 ,E(S ) E

+

H(o,O)

,E

E

> 0 is sufficiently small.

(O1 s)

is noetherian and its

E ( 0 , ~ ~ Besides, ) . one also has:

Ic(E)}, where C(E) is any complex valued function of

0

(O,E0).

Example 1.2.38. Let a(<), 5 f [-a,n] be continuous ~II-periodicfunction: a(-n)

=

a(n), and let b(h,<): [O,hol

x

[-n,n]

variables (h,S) and 2n-periodic in 5 : b(h,-s) Further, let c(x,h,c)

: R

X

[O,ho]

X

+ C

be continuous in

! b(h,n),

V h f [O,ho].

[-n,nI be continuous in variables

(x,h,<), twice continuously differentiable with respect to x f R and 2n periodic in 6: c(x,h,-nj

c(x,h,n), V ix,h) E R

Z

x

[O,hol.

Furthermore, assume that one has with some constant K > 0 :

where, as previously, <x>' Let

E$,

=

=

2

l+x

.

hZ be a lattice in IR with meshsize h and 1

(

2 %

discrete analogue of L (IR) on the lattice 2

IR,,,

)

be the

equipped with the family

of inner products and norms:

where, as previously, v*(x) stands for the complex conjugate of v(x). Let F be the Fourier transform on the lattice h

and let Fh' be its inverse:

?-It

67

2.2. F u n c t i u d Analysis For A(x,h,hS) = a(hS) + hb(h,hS) + hc(x,h,hS)

with a(E), b(h,S), c(x,h,S) as defined hereabove, and for each meshfunction u E 12(%

)

%,

which vanishes for x E

1x1 2 r

r > 0 sufficiently

large (depending on u), introduce the family of finite difference operators

4, on the lattice

I€$,by

First, we show that

A,,

the formula:

may be extended as a linear equicontinuous mapping

from 1 2 ( %) into itself:

3, :

12(\h)

Indeed, one has for v =

h E (O,hol.

12(%),

-t

A,," :

Gh(C) = (a(hc) + hb(h,hc))

+ h(2?r)-'

;h(C)

+

&h (S-q,h,hn) ih(q)dri,

J

lhrl\
where

is the Fourier transform of x

+

c(x,h,hrl) on the lattice

%-

Let S(E;,h,hrl) be the integral Fourier transform of c: 6(S,h,hq) = Fc( .,h,hrl)

=

i c(x,h.h~)exp(-ixE)dx. IR

It is readily seen that the following Poisson formula relates (1.2.190) ch(S,h,hn)

=

ch and 6:

&(5+2nkh-',h,hn) ,

C

kEZ

the series on the right hand side of (1.2.190) being convergent uniformly with respect to h E [O,h0 1, hS E [--71,711,hrl E [--TI,TT]. Indeed, for x E

one finds:

and the uniqueness of the Fourier transform on the lattice

%

yields

I . Manifolds, Functional Analysis, Distributions

68

(1.2.190), provided t h a t t h e series on i t s r i g h t hand s i d e converges One f i n d s i n t e g r a t i n g by p a r t : c(S,h,ri) =

J

c(x,h,q)exp(-ixS)dx =

IR =

<5>-2

2

1 c(x,h,n) (l-ax)exp(-ix5)dx

=

IF? =

-2

<<>

J

2 (l-ax)c(x,h,n)exp(-ixS)dx,

IR s o t h a t (1.2.189) y i e l d s : (1.2.191) \;(5,h,rl)

1

<x>-2dx

5 K<5>-2

= ITKic>-*.

IR I n e q u a l i t y (1.2.191) implies convergence of s e r i e s (1.2.190) and, moreover, i t i s r e a d i l y seen t h a t one has with some o t h e r constant K1 > 0:

Hence, one has:

and t h e same argument y i e l d s :

Thus, Lemma 3.12.8

(Schur) i m p l i e s t h a t f o r t h e n o r m o f t h e f a m i l y o f i n t e g r a l

o p e r a t o r s Ch with t h e k e r n e l s ch(S-rl,h,hn),

t h e following i n e q u a l i t y holds:

sup Olh
"hi

'L2(-n/h,n/h)'L2(-n/h,r/h)

0

so t h a t t h e Parceval i d e n t i t y y i e l d s f o r

i.e.

'

'TI

1'

9, introduced

hereabove:

1.2. Functional Analysis

69

is equicontinuous. In the sequel we shall assume that a(S) is continuously differentiable as a 2n-periodic function and satisfies the condition:

Introduce the family of subspaces 12(%,+) functions u € 12(%),

u

:

%

+

Analogously, we denote by 12(lF$,-) c 12(%) functions u € 12(\

),

C

12(\)

of all mesh-

\.

C, which vanish for x < 0, x €

the subspaces of the mesh-

which vanish for x 2 0. Also denote by

multiplication by the characteristic function of the set

x

the

F+ = {x b

0).

Consider the following family of finite difference operators in 12(\,+

) :

(1.2.193) Thu

:=

Since a(n),

X \u,

n

Th

:

12(\,+)

-f

12(%,+).

€ [-n,n] is continuous, 2n-periodic and does not

vanish, the curve [-n,nI 3

n

+

a(q) in the complex plane is closed and

does not pass through the origine. Therefore the so-called winding number m,

is a well-defined integer for a(n) and shows how many times any generic point on the curve [ - n , n ]

3

n

+

a(n) rotates arround the origin when

11

varies from -n to IT. We are going to show that the family of finite difference operators T h defined by (1.2.193), is noetherian, that its index K ( T ) does not depend h on h € (O,hol and that, moreover, (1.2.194)

K

=

m.

First, consider in 12(%,+) difference operators

where

the following family of finite

1 , Manifolds, Functional Analysis, Distributions

70

with a(5) as introduced above. Notice that the winding number for

is zero, so that

is a continuously differentiable periodic function of 5 E [ - n , n I . Introduce 6

J q ( n ) (l-exp(i(5-n)?6))-'dn, 6 > 0. Inl<TI 6 Notice, that q ( 5 ) can be extended as an analytic function in the

(1.2.195) q -+ ( 5 ) = ( 2 n ) - l

[Re 5 [ 5

half-strip

TI,

Im 5 > 0, V 6 > 0 .

6

Analogously, q - ( S ) can be extended as an analytic function in the halfstrip

IRC

<j

s v ,

Im 5 < 0 ,

v

6 > 0 . Moreover, the limits:

exist, are analytic functions in their respective half-strips and are continuous on the closure of these respective half-strips. Indeed, obviously, q , ( S )

are analytic in their respective half-strips

Im 5 > 0 and Im 5 < 0, since the corresponding integrals are uniformly convergent along with their first derivatives with respect to 5 , when Im

f,

t p > 0 and Im 5 S - 0 < 0 , respectively. Now let f, E [--71,n1.

Then, one has

1.2. Functional Analysis

71

The integral on the right hand of the last formula being uniformly convergent (since q ( 5 ) is continuously differentiable), one draws the conclusion that

6

limit (1.2.196) for q+(5) exists and is a Continuous function of 5 E

[-T,T].

6

The same argument yields the same conclusion for q-(5) as 6 4 0. Notice that if u E 12(\,+

then its Fourier transform uh(S) on the lattice

),

be extended analytically to the half-strip ively, if u € 12(%,-),

%

can

IRe 51 5 n , Im 5 > 0. Respect-

then its Fourier transform on the lattice

can be extended analytically to the half-strip

IRe 51 5

71,

IF$,

Im 5 < 0.

Introduce

Then, obviously, a + ( < ) are cotninuous 2r-periodic functions of

-

5 E

[--11,711

and a+(5) are analytic in their respective half-strip -

Im 5 > 0

and Im 5 < 0. Furthermore, one has the following factorization for a

0

(1.2.197) ao(S)

(6):

a+(5)a-(5).

=

Let us rewrite the equation (1.2.198) X+aOu

f,

=

f f 12(\,+)

in the following fashion: (1.2.199) a u = f - v, 0 where v E 1 ( R ) is also unknown. 2 h,After the Fourier transform on Rh equation (1.2.199) becomes: (1.2.200) a0(hS)ih(S) = f ( 5 ) - jh(S), where Gh, ? are analytic for Im 5 > 0 (and 2~rh-l-periodicin Re h analytic for Im 5 < 0 (and 2Th-l- periodic in Re 5 ) .

c),

vh(<) is

Using factorization (1.2.197), one may rewrite (1.2.200) equivalently in the following fashion: (1.2.201) a+(hS)Gh(S) = (a-(hS))-'Fh(5) Since (a-)-'Gh(5)

-

(a-(hS))-l;h(S).

is analytic for Im 6 < 0, and a u is analytic for + h

Im 5 > 0, one has: -1T[+(a-) vh = 0, ~

-1-1 Il+(a+) uh = (a+) u

h'

1 , Manifolds, Functional Analysis, Distributions

12

where iI+ are defined by (1.2.196).

-

Therefore, applying to both sides of (1.2.201) the projection operator

Il+ defined by (1.2.196), one finds:

Obviously, one has:

where

Indeed, (1.2.203), (1.2.204) follows immediately from (1.2.2021, since IT is an orthogonal projection in L (-IT/h,iT/h) and iI+ + iI- = Id. -1

Using the inverse Fourier transform Fh

, and

the Parcevals identity, one

finds for the inverse operator of X+ao:

Now (1.2.206)

we may rewrite

x+%

x+\

+

=

x+(aOOm

in the following fashion:

hQh),

where 0 is the shift operator, (Bu) (x) = u(x+h),

(Ornu)(x)

=

u(x+mh),

and Qh is the finite difference operator defined by b(h,C) andc(x,h,E),i.e. -1

Qh = Fh (b(h,hC) + c(x,h,hS))Fh. Since -1

(x+ao)

:

12(%,+)

+

12(%,+

),

h C (O,hol,

is equicontinuous, we may rewrite (1.2.206) in the following fashion: (1.2.207)

x+%

=

x +ao ( O m + h(x+ao)-1 X+Qh).

Obviously, one has for each u E 12 ( %,+)

:

1.2. Functional Analysis

13

where

Y2

=

SUP

OShSh

0

a s it has been shown hereabove. Furthermore, obviously one has: X+aoBm = m x+aoem = x+B X+ao f o r m 2 0.

x +a o x + O m

f o r m 5 0 and

Since

-1

h(X+ao)

X+Qht

h

E (OthO1

has i t s norm s u f f i c i e n t l y small, V h E (O,hol when ho i s s u f f i c i e n t l y small, one g e t s t h e following conclusion f o r t h e index K ( % ) ,

using t h e

p r o p e r t i e s of t h e index mentioned hereabove: (1.2.198)

K(\)

m

= K ( X a )+K(x+8 ) .=

+ o

m,

given t h a t h,+) x+ao : 12(R

'2(%,+)

i s an equiisomorphism ( a s it was y e t shown) and s i n c e , obviously, m K ( x + G ) = m. Indeed, f o r m 2 0 one has: ker x+Om

=

{6(x-kh) }05k<m

, Coker

x+Om = { O } ,

and f o r m < 0 i t i s v i c e versa; here 6 ( 0 ) = 1 and 6 ( X I x E

..{OI.

5

=

0 for

The same conclusion (1.2.198) is s t i l l t r u e when a depends on x E R , t o o and s a t i s f i e s t h e conditions:

1

. a(x,5)

1 is in C (R )

is 2 ~ - p e r i o d i c i n 5 EX=

Rx

x

2'.

a(x,c)

a s a f u n c t i o n of x with 1 s u f f i c i e n t l y l a r g e , it 1 2 and i s i n C ( R ) as a f u n c t i o n of (x,S) E R2 =

IR

5; =

am(c)+ a'(x,<),

1. Manifolds, Functional Analysis, Distributions

14 where x

a ' ( x , < ) s a t i s f i e s the inequalities:

+

Notice t h a t t h e winding number m i n t h i s case does not depend on x

E

R ,

s i n c e it i s a continuous f u n c t i o n of x E IR, whose values a r e i n t e g e r . Remark 1.2.39.

One may t h i n k of using f i n i t e d i f f e r e n c e approximations f o r

solving numerically t h e Wiener-Hopf i n t e g r a l equations of t h e form:

X+(X) ( u ( x ) +

J

k(X-y)u(y)dy) = f ( x ) ,

x E R+

,

R+ where t h e k e r n e l k ( x ) i s f o r i n s t a n c e , a continuous function of x E R k

E

and

LI(R). Let u s r e c a l l t h a t t h i s equation i s uniquely s o l v a b l e i n L 2 ( R + ) i f f

G ( 5 ) of k ( x ) s a t i s f i e s t h e condition:

t h e Fourier transform

and t h e winding number

K

for 1

+ G ( 5 ),

i s zero: c = 0 . If

K

> 0, then

K

boundary c o n d i t i o n s on u should be imposed i n

a d d i t i o n t o t h e i n t e g r a l equation i t s e l f i n o r d e r t o have t h e unique s o l v a b i l i t y , while f o r

K

< 0 one may introduce

-K

p o t e n t i a l type o p e r a t o r s

with unknown d e n s i t i e s ( i n t h e c a s e considered, unknown r e a l o r complex numbers a s d e n s i t i e s ) f o r t h e same purpose ( s e e [Goh-Kr,

11, [Goh-Feld,

and [Esk, 11 f o r t h e multidimensional c a s e ) . While considering a f i n i t e d i f f e r e n c e approximation of t h e WienerHopf equation h e r e above of t h e form

where P ( 8 ) i s a polynomial of t h e s h i f t o p e r a t o r 8 on %,+, with h 1 c o e f f i c i e n t s C -functions of h E [O,h 1 , P o ( l ) = 1 , one has t o t a k e c a r e 0

11

1.2. Functional Analysis

75

of the index of the family of finite difference (discrete Wiener-Hopf or Toeplitz) operators on the left hand side of (1.2.199), which should coincide with

K.

This may not always be the case, as shows an example of the WienerHopf integral equation with k(x) = exp(-(x[) and its following finite difference approximation: x+(x) (uh(x+h) +

huh(y)exp(-jx+h-yl 1 ) = fh(x),

Z YE%,+

Indeed, it is readily seen that the index equation with k(x) = exp(-lxl) ,sirice 1+;(5)

=

K =

2

0 for the integral 2 -1

( 5 + 3 ) ( 5 +I)

.

For the approximating finite difference equation one finds after the % of kh (x) := exp(-)x-h)):

discrete Fourier transform on ah(0)

:= =

8 + kh(e) -h =

e(B2-2e(Chh+hShh)+1) ( 0 2 -20Chh+l)-l,

where 0 = exp(ih5) is the discrete Fourier transform of the shift operator

,

which is the winding number for ah ('8) when 8 goes around the unit circle r , Thus, the index

Kh

Kh

:=

-1

(217)

[arg ah(e)lr = 1, t/h > 0 ,

while for the integral equation itself

K =

0.

Of course, this kind of difficulty does not appear, if one uses, for instance, the following apprcpriate finite difference approximation:

or any other approximation P (8) of the identity such that P ( 0 ) # 0 for h 0 18 1 = 1 and the winding number of P ( 0 ) along the unit circle 10 I = 1 is 0

zero (see also [Goh-Feld, 11 as far as projection methods for WienerHopf operators are concerned). -1

For k(x) = X+(x)exp(-x), k(S) = (l+iS) the index K = 0. For its -1 h. -h finite difference approximation k (x) such that k (0) = (l+i<) , h

1 . Manifolds, Functional Analysis, Disfributions

76

-I ic = h (0-l), the index

of the corresponding discretized equation 1s h zero for h E (0,lj and -1 for h E ( 1 , 2 ) . Of course, using k ( x ) = k(x), h h * -1 , i<* = h-l(l-e-'), one gets or k (x) such that kh(B) = (l+ic ) Vx E K~

finite difference approximations of the corresponding integral equation whose index is zero for all h > 0 . h Indeed, for k (x) = k(x), x E IR,,

I +

, one

finds:

-1 = ((l+h)e-exp(-h))(e-exp(-h))

ih(e) h

and for the backforward approximation a * of a one has: x,h x 1

+

-h kh(6) = (he)-'

so that in both cases K~ = 0 , '+h > 0 .

I

Now several definitions and results will be given, which are concerned with the linear equations, containing a spectral parameter. Let

A

:

€)(A)

+

R(A)

with domain D(A) and image R(A) in a Banach space E, be linear and closed. is said to be a regular p o i n t of A if the equation:

A complex number (1.2.210)

Ax x

:=

(A-XId)x

=

y, x E D ( A ) ,

y E E,

is correctly and densely resolvable. The set of all regular points is called the r e s o h a n t s e t of A, and the complement of the resolvant set in the complex plane is said to be the spectrum of t h e operator A . The operator A being closed, its spectrum is a closed set in the complex plane, its resolvant set being open. The linear operator RA(A), (1.2.211)

RX(A) = (A-iI)-'

defined for each A in the resolvant set of A, is called the resolvant of A; sometimes it is also denoted by

%.

The resolvant R x is an analytic function of h in the resolvant set, valued in the Banach space L ( E ; E )

of all bounded linear operators in E. Each

connected component of the resolvant set of A is a natural holomorphy domain for its resolvant Rx

and RX can not be extended analytically through

the boundary of this component. For each couple

x , of ~

regular points of A the following Hilbert's (or

1.2. Functional Analysis resolvent) identity is true: RA

-

R

ll

= (A-u)R

A

R

P’

as well as the following consequence of the latter identity: k (d/dA) RA If A

:

E

-f

E

=

k+1 k!RA , V k E

Z+.

is bounded, then its spectrum lies in the disk

1 I A l lE+}.

The l e a s t rA 2 0 such that 1x1 4 rA } still contains the spectrum of A, is called the s p e c t r a l radius of A. The

t

1x1

5

following Gelfand’s formula is valid for rA:

and the limit on the right hand side exists for each linear bounded operator A:E+E. If A.

is an isolated singular point of the resolvent R A , then RA

admits a Laurent serie’s expansion:

where the linear bounded operators An

: E

-t

E commute which each other

and satisfy the relations: €or k >= 0 , m 5 -1,

AkAm = 0 , An

=

A ~ + ’ , for n t I, 0

A-p-q+l

=

for p , q 2 1.

-A A -p -ql

The last relation with p = q = 1 implies that the residue A-1, A-1 : = Res RAIA=A 0

has the following projection property:

Also the following formulae are valied: (A-AOId)A

If A

0

n

=

A n-1‘

v n # 0 and (A-AOId)AO = A-l+Id.

is an m-pole (a pole of order m) of the resolvent RA, then A.

is an

1 , Manifolds, Functional Analysis, Distributions

78

eigenvalue of the operator A and, moreover, one has: = R( (A-hoId)n), V n 2 m. I~/(A-~)

R(A-~)= N ( (A-AOId)"),

Besides, in that case E can be represented as a direct sum: E =

N ( (A-AOId)") fB R ( (A-AOId)"),

V n 2 m,

and for each element RAx, x E E, one has the expansion:

= A.x, ej = Aj-kx, Ae = eo, Ae. = Ae. ,-1 +ej-2, 1 5 j S k 5 m, j i 1-1 the element eo E E being an eigenvector (associated with A 0) and the elements eo,...,ek-l being called the Jourdan chain associated with A

where f

0'

If A

1 A 1-l;

:

E

-+

E is bounded, then its resolvent Rh vanishes at 1

=

as

if A is unbounded, then its resolvent RA has a singularity at

infinity. In the latter case if, moreover, A = point of R A , then A =

m

m

is an isolated singular

is an essential singularity.

The set consisting of the spectrum of an unbounded operator A

: E + E

and of the point A

=

is called the extended spectrum of A.

m,

The extended spectrum is always non-empty. A non-empty part A l of the spectrum A(A) A is said to be a

SpeCtPffl

of a linear closed operator

S e t if A l is at the same time closed and open

in the spectral set A(A) equipped with the natural Hausdorff topology. For each bounded spectral set A1 there exists a closed Jourdan contour r l , which lies in the resolvent set of A, encloses the spectral set A 1 and does not enclose any point of the complement spectral set A 2 = A \ A l . One may always choose the orientation on to the left when one goes along

rl

rl

in such a way, that A

1

stays

arround the spectral set A1.

The operator

P1

:=

-1

-(2i~i)

I

RA(A)dA

rl is the so-called Riesz' projector. The Banach space E can be split in a direct sum: E = E El

=

PIE and E2

=

(Id- P1:E.

8 E2, where

The subspace E l belongs to the domain D(A) of

A and is invariant under the action of A. The spectrum of the restriction A

1

of A to E l coincides with A l .

The restriction A2 of A to the linear Set

(Id- P1)D(A) is a closed linear operator in E 2 , whose spectrum coincides

1.2. Functional Analysis

79

with A2. If Al,...,A are mutually disjoint bounded spectral sets for a linear k closed operator A, then A = A @ ... B 1 $ @ $+1, E = E 1 B... Ek Ek+l' Ej, 1 6 j 6 k, are bounded and Ak+l : where A . : E . Ek+l + Ek+l is I 3 closed.

'

@

-f

The corresponding projectors

P.P.= 1 3

P ,

1 6 j 5 k+l, are mutually orthogonal:

0, for i # j, and the spectrum of A. coincides with Aj, l < j
An important class of operators A

:

E

+

E is the one, for which

the resolvent RX (A) is a compact operator at least for one X

=

X o in the

0

resolvent set. In this case, Hilbert's identity implies that RX is compact for all X in the resolvent set. An operator A

:

E

+

E whose resolvent R A

is a compact operator, is necessarily closed and has a spectrum A(A) which consists of isolated points. Each point in A(A) is a pole of RX(A) of a finite order and, at the same time, an eigenvalue of A. The kernels N( (A-XOId)k, , k = 1.2,. . . with Xo E A (A) have finite dimension and, moreover, there exists no such that N( (A-hoId)no)

=

N( (A-XOId)k), V k 5 n0'

while for k 5 no one has:

* k If the domain D(A) of A is dense in E, then the kernels N(( (A-hoId) ) ) k . have the same dimension as the kernels N((A-X Id) ) , k = 1,2,

...

0

Let X E A(A), i.e. (1.2.210) is not uniquely resolvable. The complex number X is said to belong to the discrete spectrum of A if (1.2.210) has more than one solution for y = 0; X is said to belong to the continuous spectrum if (A-XId)-l is defined on a dense set in E and is unbounded. If A

:

E

-f

E is compact, then its spectrum A(A) consists of at most

m ) , which are poles (Ak -r 0 for k countable set of eigenvalues {X 1 k kZO %, k t 0, of the resolvent RX(A). The kernel -f

of finite multiplicities

N((A-AkId)) is the eigensubspace associated with Xk and the dimension of this kernel is the multiplicity of the eigenvalue X k' The subspace

Nk

:=

N( (A-hkId)mk) is called the root subspace, associated

with the eigenvalue Xk, and its dimension is called the algebraic multiplicity of X k' One may choose in

Nk

restriction of A to

a basis consisting of Jourdan chains so that the

N k will have as its matrix (with respect to the chosen

basis) the normal Jourdan form. We shall end this section by presenting some definitions and results

1. Manifolds, Functional Analysis, Distributions

80

concerning the classical regular perturbation theory. Let 5

+

A(<) be a family of continuous linear operators in a Banach

space, which depends analytically on 5 E C, 5

A(<) =

k

%, 9, E L ( E ; E ) ,

15

1

<

r, i .e.

151 < r,

k>O the series being convergent in the operator norm in L ( E ; E ) , V 5 with 151 < r. Let A

0

E h(A(0)) be an isolated point in the spectrum of A(0) and

assume that the corresponding invariant subspace (associated with A 0 1 (A(0)) has dimension m < m . Let U C C be an open set, such that EAO

U

n

A(A(O)) = {A 1. Then there exists a 6 E (O,r), an integer k 5 m and an 0

6 the set U

integer n such that for 151

n

A(A(5))

A1(5), ...,A ( 5 ) ; moreover, each function A , ( < ) ,

k 1 analitically on the main branch of the function z = = Xo,

condition: A . ( O )

P.(c), 1

1

consists of the points

1 6 j 5 k, depends

c1ln

and satisfies the

1 5 j 6 k; besides, the corresponding projectors

1 5 j 5 k, admit Laurent series' expansions with respect to z = 5

P.(5) 1

=

C

A. 5

s2-N

Is

s/n

1/n

,

where A . E L ( E ; E ) are continuous linear operators (F. Rellich). 1s

Of special interest is the following simplified version of Rellcih's

result. Let A(5) = A0+5A1, 5 E

C,

151 < r, be a family of self-adjoint

A operators in a Hilbert space H. Further, assume that A(0) = . complete set of eigenvectors {e(O)

has a

associated with the eigenvalues

(O) be an being an o:thonormal basis in H. Let A") k ih:o)jn>co' {en lnbO isolated simple eigenvalue of . A (i.e. of multiplicity one). Then the

eigenvalue A ( 5 ) and the associated eigenvector ek(5) of A(<) are analytic k functions of 5 for 151 < 6 5 r, and, moreover, one has:

where

and an analoguous formula may be obtained for eL2). Neither the Rellich result, nor the perturbation formulae in the specific case hereabove are

,

1.2. Functional Analysis

81

true when the perturbation becomes singular (i.e., for instance, when and A1 are unbounded self-adjoint operators in strictly stronger than A

0

H and A1 is in some sense

).

Example 1.2.40. Let us consider here an example in order to illustrate the complexity of spectral problems when the perturbation is singular. Let ( 0 , l ) 3

E

-+

A€ := a-E

AE be the following family of differential operators:

2 3

a , a

=

d/dx

considered as unbounded operators in L (U), U = (0,l). 2 Such a singular perturbation appears in the linear theory of dispersive singular perturbations. The natural domain of A

is the Sobolev space H 3 ( U ) , equipped with

the family of norms:

for each given Sobolev norm

with respect to

1

I . I I (It21,E being equivalent to the with constants which depend on E (i.e. non uniformly

E (0.1) the norms

E

I 1.1 1 E

E (0,l)).

It is readily seen that the family of linear mappings

is equicontinuous. It is also readily seen, that the spectrum of A

with H (U) as its 3

domain, is the whole complex plane. We shall further restrict the domain of A D(A

)

=

{u E H

(1,2), E

(u)

I

by defining it as follows:

u(o) = au(o)

=

~ ( i =)

01

and shall be concerned with the investigation of the spectrum of the family:

Obviously, (1.2.212) is equicontinuous. First, we show that (1.2.213) Ail

:

L2(U)

-+

D(AE)

is equicontinuous too. Indeed, AE1 is a family of integral operators,

I . Manifolds, Functional Analysis, Distributions

82

(Ailf)(x) =

f GE(x,y)f(y)dy, U

where the kernel G (x,y) can be represented in the form: (1.2.214) GE(x,y) = EE(X-y)+EE(Y)+Ea E (y)(exp(-x/-E)+exp(-(l-x)/E)-l)Y E -EE ( 1-y)exp ( - ( 1-x)/ E ) +exp (-l/~) g, (x,y), the kernel EE(x), (1.2.215) EE(x) := (1/2)(l-exp(-lxl/E))sgn x being a fundamental solution of the operator (1.2.216)

( a x--E 2a 3X)E€ (x) =

6(x),

ax-€'a3

x

on R

, i.e.

E IR m - -

with 6(x) the Dirac mass, and gE(x,y) being in C (UxU) and such that (1.2.217)

sup max J E O<E51 (x,y)EUxU

m m-k k ax aYg,(x,y)

I

<

m,

o

5 k

c

m,

v m

2

o

One has: (1.2.218)

k /(€a) EE(x)/ S 1/2, k = 0,1,

k f Ek-1 /axEE(x)/dx 5 1/2, k = 1 , 2 , IR

so that it is readily seen that the following inequalities hold for

GE(x,y) defined by (1.2.214), (1.2.215):

where, of course, (1.2.217) has been used, too. Using Schur's Lemma (Lemma 3.12.8), one gets the conclusion that for u

E'

U

(x)

:=

I U

one has:

2

GE(x,y)f(y)dy = (As'f) (x), f E L (U),

1.2. Functional Analysis where C 1 = max {C,1/2] does not depend on f and

( a x-E 2 d x3 E

F.

(1.2.216) imply:

NOW (1.2.214),

Thus

83

=

f w .

2 3

a u(x) belongs to a bounded set in Lz(U) and, moreover, one has: X

Now, (1.2.219), (1.2.220) yield:

the latter being precisely the claim (1.2.213). Furthermore, the family of integral operators

is compact (uniformly with respect to

E

E

(O,E~]),

2 for each f E L (U) the

(0 < E 5 E 0 ) belongs to a bounded set in H1(U), the latter family A-'f 2 being compactly imbedded into L (U)

.

For each f E H (U), which satisfies the conditions: 3

(1.2.221)

1 f(x)dx

= 0, f(0) = 0,

U

one has the following estimate:

where the constant

c

>

0 does not depend on f and

natural reduced operator for A

(Ai'f) (x)

=

and where Ail is the

E '

1 (1.2.223)

E

-1

1 Go(x,y)f(y)dy

=

0

70 f(y)dy,

the kernel GO(x,y) = H(x-y) = 1 for x > y and H(x-y) = 0 f o r x < y being the point-wise limit of G (x,y) as

E

*

0:

I . Manifolds, Functional Analysis, Distributions

84

G (x,y) = lim G~(x,Y), 0

v

(x,y) E

UxU,

E+O

as a consequence of (1.2.214), (1.2.215), (1.2.218). -1 -1 Indeed, let uE = A f, uo = A f and introduce 0

(1.2.224)

vE (x) = uE (x)-uo (x)-aEexp(-x/E)-bEexp( - (1-x)/ E ) -cE '

where the constants aE,bE,cEare chosen in such a way, that vE E D(AE), i.e. v ( 0 ) =

a XvE ( 0 )

=

~ ~ ( =1 0. )

It is readily seen that one has:

f(y)dy + O(exp(-l/E)), as 6 -t +0, U the latter asymptotic formulae for f satisfying (1.2.221) taking the form: bE = Ef(0) -

(1.2.226)

a

= bE = c

= O(exp(-l/E)),

as

E -t

+O.

For vE(x) defined by (1.2.224), vE E D(AE), one also has:

(a-E 2a 3)vE(X)

=

E

2 3 aXfw,

so that (1.2.213) implies:

Now, (1.2.224) , (1.2.226), (1.2.227) yield (I .2.222)

. If f (x) does not

satisfy (1.2.221), one still has the following estimate:

where again the constant C > 0 does not depend on f and

E.

Indeed, one gets (1.2.228) by using (1.2.2241, (1.2.2251, (1.2.227) 2 and the fact that the L (U)-norms of the functions eXp(-X/E), exp(-(l-x)/E) are O ( E 1/2) , as E * +O. Notice that Ao, the inverse of (1.2.223), has its spectrum only at infinity: A ( A ~ )= compact operator A

{ - I , while the spectrum of each AE, the inverse of the , has for each E > 0 only the discrete spectrum:

-1

It will be shown in Chapter 5 that a l l eigenvalues X E of A

are simple,

strictly positive and for each given n > 0 che following asymptotic formula holds for :A:

1.3. Distribution Theory

X E = ( 2 / 4 3 ) ~ - l + v2n2~/J3+ 0 ( s 2 ) ,

85

E +

0.

I Distribution theory

1.3.

In this section some basic facts from classical distribution theory are briefly sketched and several aspects of a possible extension of the distribution theory to a parameter dependent situation are presented, which will turn out to be useful in the following chapters. hlile presenting the

1

classical distribution theory, we follow essentially [Sch, 1 Sh, 1

1

1.3.1.

Test function space D ( U )

Let U

and [Gel

-

where the complete proofs of all the statements can be found.

5 lRn

be a nonempty open set. For each compact K

C

U, the Frechet

space DK was yet described hereabove. The union of the spaces DK when K ranges over all compact subsets in U, is Schwariz's test function space Obviously, D ( U ) is a vector space and $ E D(U) iff @ E

D(U).

m

C

( U ) and

the support of $ (the closure of the set where $(x) # 0 ) is a compact set in U. For @ E D(U) and each integer N L 0 let us introduce the norms: N = 0,1,

...

The restrictions of these norms to any fixed

DK

c D(U) induce the same

topology as do the norms p N hereabove. The same norms (1.3.1) can be used to define a locally convex metrizable topology on D(U). However, D(U) equipped with this topology is not a complete topological vector space, since one can indicate Cauchy sequences in D ( U ) which converge (in the sense of the topology defined by (1.3.1))

m

to C -functions whose support is not compact in U.

Introduce the topology on D ( U ) by saying that a sequence { $ v } v 2 0 C D ( U ) is convergent to zero if for each N = O,l,

norms (1.3.1) uf 4" vanish as v

K

C

+

5 K.

the

and, moreover, there exists a compact

U such that thesupportsof all @v, v = O , l ,

supp $, $v

+ +m

... given

... belong

to K:

This topology being translation invariant, we say that

$ (in D(u)) if $,

= $-$v + 0

(in D ( u ) ) . D(u) is a locally convex

complete topological vector space, where each closed bounded set is compact. Let L be a liiiear mapping of D ( U ) into a locally convex topological vector space X. Then each of the following three properties implies the other two:

1. Manifolds, Functional Analysis, Distributions

86 (i) L

:

U(u) + x

(ii) L

:

D(U)

+

is continuous.

X is bounded.

(iii) The restrictions of L to each DK As

C

D(U) are continuous.

a corollary, one has: every differential operator P(x,ax),

with coefficients a

E C m ( U ) , la1 L m, is a continuous linear mapping of

U ( U ) into itself. Distribution space D'(U)

1.3.2.

A linear functional L

:

D(U)

+

C, which is continuous with respect to the

topology on D(U) (described hereabove) is called a d i s t r i b u t i o n in U. The space of all distributions in U is denoted by D'(U).

The action of

u E D'(U) on a test function $ E D(U) is denoted by a,$>. If L

:

D(U)

+

C is a linear functional, then the following two conditions

are equivalent: (i) L E

D'(u).

(ii) For each compact K C <

m

C

U there exist an integer N 5 0 and a constant

such that the inequality

holds for every 4 E DK. Every locally integrable function f in U with complex values is an element in u ' ( U ) , whose action on each $ E D(U) is given by the formula: = J f(x)$(x)dx, U

where dx is the Lebesgue measure on U.

-

Similarly, if p is a complex valued Bore1 measure on U, or if p is a positive measure on U with p ( K ) <

u

for every compact K c U, then

E D'(U) and

If

is a differential operator with coefficients aa E Cm(U),

/(TI <

m, then

1.3. Distribution T h e o y

87

is a continuous linear mapping defined in the following fashion:

< P ( x , a )u,Q>

P*(x,a

where

If f f'D

E

)

=

,

is the formally adjoint of P ( x , a

):

D'(U) is locally intrgrable in U, then its distributional derivatives

are defined as follows:

or, more explicitly:

m

The multiplication by a function @ E C (U) obviously is a continuous linear mapping of A

D'(u)

into itself.

natural (weak*-) topology on

D' (U),

induced by the one on D(U) is

defined as follows: A sequence of distributions

i L v 1v>o

t

D'(U) is said to be convegent

to a distribution L if one has: lim
,$> = < L , $ > ,

V $ E

D(u).

V+-

Furthei, { L v j v Z O c D ' ( U ) is a Cauchy sequence if {
C, V

Each Cauchy sequence { L v } V 2 0 v

-+

-.

C

C

is a

Q E D(U). C

D'(U) has a unique limit L E

D' (U),

as

Moreover, one also has:

and, more generally,

P(x,ax)LV

+

P(x,ax)L,

for each differential operator

as v

P(x,a ) D' (U)

Besides, Cm(U) is dense in on

D' (u)) .

-f

-,

with coefficients in Cm(U). (with respect to the weak*-topology

1. Manifolds, Functional Analysis, Distributions

88

-

Let U' c c u (i.e. U' c

u ) . Two

are said to coincide in U' if

distributions L , and 1, in D'(U) =


v 6E

D(U').

Let {Uk}k20 be a collection of open sets in lRn whose union is U. Then there exists a sequence { $ j ) j 2 0

C

D ( U ) , with j i . 2 0, such that

1 (i) each $ . has its support in some member of the collection {Ukjkz0, 1 (ii) Z J , . ( x ) = 1, V x E U, j 20 1

(iii) to every compact K c U correspond an integer N 5 0 and an open set V

3

K such that

z

v

$.(x) = 1 ,

OSjSN

'

x E

v

Such a collection { J , . ] is called locally finite partition of unity in 3

U , subordinate to the open cover {Uk } ktO Let {Uk}k,o be an open cover of an open set U

5 Rn

and assume that

to each U corresponds a distribution 1, E D'(Uk) such that 1, k

U

k

=

L . in 3

I7 U . whenever this intersection is non-empty. Then there exists a unique l

distribution L E D ' ( U ) such that L = L in Uk for every k = O , l , ... . k Using a locally finite partition of unity { $ . I . in U subordinate to the ] 120 cover {Uk}kzo, one defines L as follows:



=

j20

1

is a member of the collection {U 1 such that U k , contains the k k>O 1 support of J , , , j = O,l, ... .

where

U

k. 3

1

Let V be the union of all open sets where a given distribution

L E D'(U) vanishes. The complement W V is said to be the support of L . Let L E D'(U) and denote supp L its support in U. If $ E Cm(U) and J, E 1 in some open subset V c U containing supp L , then $ L = L . If supp

is a compact subset of U, then there exist

constant

C

an integer N 2 0 and a

such that

m

Furthermore, in that case L extends uniquely to a continuous linear functional on p

N

m

C

(U) (equipped with the topology, defined by the seminonns

hereabove). The least integer N 2 0 such that ( 1 . 3 . 2 )

still holds is

called the order of the distribution L . Let L E D(U) and assume that supp L = {xo3, where xo is some point in U; further, assume that the order of L (which is finite in this case) is

1.3. Distribution Theovy

89

N 2 0. Then one has:

where car In1 5 N, are constants,

<6x ,@

=

$(x,),

0

6x0 = ~ ( x -0x) is the Dirac distribution, V 4 E D(U), and the derivatives a"6 are interpreted in x x 0

the distribution sense. Let L E D ' ( U ) . There exists a collection of continuous functions {fa(x) l a E z ; in U, such that each compact K only finitely many fa and one has

C

U intersects the support of

in the distribution sense:

Moreover, if L has finite order, then the collection {f (x)IaEZ+, can be chosen in such a way, that only finitely many of fa(x) are different from zero. Let U

5

IRn

5

V

x' direct product f ( x )

x

IR" be open sets and let f E D ' ( U ) ,g E D'(U). The Y g(y) is the distribution in D'(UxV) whose action on

each test function x ( x , y ) ,

x

E 2)'(UXV) is given by the formula:

The direct product of distributions is commutative: f x g associative: f X (gx h) = (f x g )

X

h

=

=

g x f, and

f x g x h.

If a distribution f(x,y), f E D'(UxV), is translation invariant in variable x, i.e.

then f!x,y) = lx x g(y), where g E D'(V) and l X is the (regular) distribution in D ' (Rn) defined by the formula

Let fk E D ' ( R n ) , k

=

1,2,

and assume that the support of at least

one of these distributions is compact. Then their convolution f *f 1 2 is defined as follows:

1 . Manifolds, Functional Analysis, Distributions

90

Of course, the convolution is commutative and associative. Moreover, one has:

Each of the following two conditions guarantees the contrinuity of the convolution f*g (with respect to the weak *-topology in ( D ' ( R n ) ) as a

D' (Rn)

function of f E

f v * g + f + g

(i) either supp fv

C

D' ( R n

valued in for

fv+f

if

..., where

K, V v = 1.2.

(ii) or if supp g is compact in

:

K is a compact in E n ,

xn.

The following result (due to L. Schwartz and often called the kernel theorem) is important for applications. Let B ( $ , $ ) be a bilinear functional x D(V), where U 5 m," , V 5 Rn. Y Assume that 8 ( + , $ ) is continuous with respect to each argument 4 and Q.

on D ( U )

Then there exists a distribution f E D'(UxV) =

such that B ( $ J , $ ) =

.

1.3.3. Some other distribution spaces Again let U

5 Rn

m

C -functions in

U

m

be an open set and C

(U)

the vector space of the

with real or complex values. Introduce the following

Fr6chet topology on C m ( U ) : a neighbourhood O ( m , 6 , K ) the set of all functions 4 E

Cm(U)

such that

for given 6 > 0, integer m 2 0 and compact set K sets { O ( m , & , K ) ) ,

m E Z+, 6 > 0, K

m

of zero in C (U) is

C

U. The collection of m

C U

defines the topology on C (U). m

The dual space E ' ( U ) of continuous linear functionals on C (U) is the subspace of all distributions in D'(U), which have compact support in U. Obviously, D'6

(x-x,)

,V

c1

E

Zy,

v

xo E Rn , is an element in E' (Pi"

)

.

Important for applications is also the Schwartz space S'(Rn) of tempered distributions which is a subspace of 0 ' ( E n ) m

. The

space

S(Rn ) of

rapidly decreasing C -functions $(x) is introduced as a Frechet space equipped with the topology defined by the norms:

1.3. Distribution Theory where, of course, < x > ~= lilxl

2

91

.

Any differential operator p(a

is a continuous linear mapping from

)

s ( R n ) into itself, as well as the multiplication by any polynomial p(x), p

:

S(Rn )

-f

S(Rn

.

)

Thus, also a differential operator p(x,a

)

whose

coefficients and all their derivatives have polynomial growth of some fixed degree N 2 0 for 1x1

S(W"

is a continuous linear mapping from S(Rn

+ m,

into itself. Obviously, S(IRn)

I>

D(Rn

).

)

Moreover, Z)(Rn) is dense in

).

The space of tempered distributions linear functionals on

s ' (Rn)

is the one of all continuous

s ( R~ ) .

The Fourier transform

is a continuous linear mapping from ~ ( I R " )onto itself, its inverse being:

This is a consequence of the formulae:

9 E s ( R n ) and i(c',CN) can be

The following result is useful in applications. Let assume that supp $I

5

=

{x

(x',xn), x

=

extended as an analytic function of 5 moreover, the following norms of

c$

Ry is denoted by supp Q

5 %!

=

{x

For each u E

so(zy) : =

< 0, V

5 ' E Rn-' , and,

s(lRn )

, whose

support belongs to

similarly, the one of all $ E s(Rn

(x',xn), x

s ' ( W n)

fornIm 5

are finite:

The subspace of all testfunctions @

-

b 0 1 . Then

5 Ol, is denoted by S (Tii") 0 -

),

.

define its Fourier transform by the duality

between S'(Rn) and s(IRn), i.e.

By duality, the Fourier transform is an algebraic and topological isomorphism of S'(Rn )i.e. F S ' ( R n ) linear mapping.

-f

S'(IRn) is a continuous surjective

1. Manifolds, Functional Analysis, Distributions

92

Also by duality, any differential operator p(x,a

)

with coefficients

which(with all their derivatives)grow as a polynomial of some fixed degree N L

o

as 1x1

is a continuous linear mapping from S ' ( R ~into Atself.

+ m,

If the convolution f*g is well defined for f and g in S'(Rn), then one has :

Also, one has for each f Dx =-ia

E S ' ( R n ) and each differential operator p(D

)

,

with constant coefficients: F(p(Dx)f) = p(S)f(5).

For distributions u E S'(Rn) whose Fourier transforms

u(c)

are

locally integrable functions such that <<>' u ( E ) E L2 ( a n ) with some s

E R , one can introduce the

The set of all u

norms:

E S ' ( W n ) , whose Fourier transforms are locally integrable

and have the norm

II 1I *

finite is called the Sobolev-Slobodetsky space

of order s E R and is denoted by H (Rn1 . For s = m Z 0 integer Hm (Rn) coincide with the classical Sobolev spaces introduced hereabove. Each Hs(Rn

)

is provided with the Hilbert space structure, the inner product on

Hs(Rn

)

being defined as follows:

-*

where v

=

J

<02' ;(S)<(E)*dS,

stands for the complex conjugate of

6.

The following Paley-Wiener-Schwartz result gives the characterization of distributions u E D ' ( R n ) with compact support. (i) If a distribution f E 0 ' ( R n ) has its support in the ball { 1x1 5 r} (hence, in fact, f E S ' ( n n ) ) and if f has order N, then its Fourier transform

f ( 5 ) can be extended as an entire function of 5 E Cn and, moreover, there exists a constant C > 0 such that

(ii) Conversely, if g ( 5 ) is an entire function of 5 E Cn which satisfies the inequality hereabove for some N and C, then there exists f

E st(Rn)

with its support in the ball { 1x1 5 r} such that F ( 6 ) = Fx+Ef, Nbeing its order.

1.3. Distribution Theoy

A

continuous li'near functional on

distribution in

;

which is denoted by

s 3 ( Z n ) is called a

all such functionals form a subspace of

one can also define the partial Fourier transform f(S',x

s'(?ii") by

=

)

s' (Rn)

The restriction of f E an element in S ' ( m n

to R :

is a distribution in

belongs to

s ' (Fy )

s' (xy)

to

.

s(%: )

We shall denote by

s'0 (Zy )

F ,f of x '+S

the formula:

and, using the Hahn-Banach theorem, one can extend each f E

and by

D' (Ry) ,

' being well-defined for each

so(xy) in the usual way,

each f E

tempered

s' (R: ) .

The partial Fourier transform FX 5'

$ E

93

of all $ E

the restrictions to R :

the subspace of distributions in

. The spaces s'o ( g+n 1

and

s(F:

s'

s(Rn) ,

(Rn)r whose support

are dual. Moreover, the

)

is an algebraic and topological partial Fourier transform F X'+E' isomorphism of s ) onto itself.

(x:

Let Uk, k

= 1,2,

be two open sets in IRn and let Ji

:

U1

+

U2

be a

m

C -diffeomorphism. Denote by J $

$-l

: U2

+

the Jacobian of the inverse diffeomorphism

-1

U1. For any f E D'(U ) define the composition foJI E D'(Ul) by 2

the formula: =


-1

I>,

)/J

V $ E D(Ul).

$-I The mapping D'(U2) 3 f

m

f,$ E u'(U

) is continuous. Since C (U) is 1 dense in D'(U), one can extend the chain rule to the distributions:

ax

(fo$)

k

c

=

l5j5n

+

(ax $ ) ((ax f ) q ) , k k

v

f E

u'(u2).

v f

E

Similarly, one has:

C'$f). Also

has :

if $l

=

: U1

+

($09)(f.$),

U2 and Ji,

v

'$ E C r n ( U 2 ) ,

U' ( U 2 ) .

m

:

U2

+

U3 are two C -diffeomorphisms, then one

1. Manifolds, Functional Analysis, Distributions

94

Now, one can define distributions on a Cm-manifold M. Namely, let be an atlas on M. A distribution f E D'(M) is the collection

(U.,@,), 3 3 JEJ

{fjjjEJ, f . E D ' ( @ j ( U j ) ) , such that

I

If M is an open set in IRn then the new definition of distributions in

D'(M) coincides with the previous one, given hereabove. Of special interest is the case when M is the n-dimensional torus Tn, Tn = { z = ( z

,...,zn), zk =

exp(iCk),

[Ck/ 5

n, 1 5 k 5 nl.

Functions $ on Tn can be identified with functions J, on B i n , which are 2n-periodic in each variable: Denote by

D(Tn)

$ ( t l , ...,En)

=

+(exp(iS,),. . .,exp(iCn)).

the space of all functions @ on Tn such that $ E Cm(Rn).

For each @ E D(Tn) introduce its discrete Fourier transform (Fourier coefficients $(k) of $ ( E ) ) by $(k) =

I @(5)

exp(-i
Tn One has: sup Nli(k)I kEZn

N = 0,1,

m,

... .

These norms define a Frechet space topology on D(Tn), which coincides with the one given by the norms:

where, of course,

$(C1

,...,Cn)

=

...,exp(iCn)).

$(exp(iC1),

One introduces D'(Tn) as the space of continuous lineair functionals on D(Tn). The Fourier coefficients of any f E

'Toeach f E D'(Tn) correspond

D' (Tn) are

defined by

an integer N b 0 and a constant C <

m

such

that

[ i(k) I

5 CN, V k

E Zn.

Conversely, if f ( k ) , k E Zn, satisfies the last inequalities with some

1.3. Distribution Theory N 2 0 and C <

95

then there exists f E D'(Tn), whose discrete Fourier

m,

transform is f(k). The convolution f*g for f and g in

D' (Tn) is

in 0'(Tn), whose Fourier coefficients are f(k)g(k)

defined as the distribution

. Again,

the convolution

is commutative and associative. It is also a continuous function (in each

D' (Tn).

factor) valued in If J 1

...

J2 C

C

are finite sets whose union is Zn, and if f E D'(Tn),

then the partial sums f ,( 5 )

Z f(k)exp(-i)

:=

7

kEJ . 3

converge to f(<) as j

in the weak*-topology on D'(Tn).

+ m

We shall also introduce the space s(Zn) of rapidly decreasing sequences $(k), k E Zn, equipped with the Frechet topology defined by the norms:

The dual Fr6chet space of all continuous linear functionas is called the space of tempered sequences and is denoted by 5" (Z") . Each element of s'(Zn) is a sequence u(k) whose growth for [ k l N most as with some integer N . 1.3.4.

-f

m

is at

Parameter dependent distributions

Now some classes of parameter dependent test functions and corresponding distribution

spaces are introduced, which will be useful in the

following chapters. Let B$ = hZn, h E (0,h0 1, be a one parameter family of grids (lattices) in

mn. A

map u(h,x), u

For meshfunctions $

:

<

: IF$

(1.3.4)

+

Cp is called a (vector valued) meshfunction.

-f

Cp introduce the semi-norms: 1

where, as previously,

h,:a

=

/a:,h$(h,x)I,

k,j = 0,1,

...,

axl,h...axn,h, "1 an axk,h being the forward finite

difference derivative in variable xk for x E Denote (1.3.5)

''

]$Ik

,

:=

sup I$[k,j;h, k,j O
= 0,1,

...

0

Similarly to the Schwartz space S(lRn), introduce the family

sh = S(-nhn)

of test function spaces of meshfunctions u(h,x), whose semi-

1. Manifolds, Functional Analysis, Disfribufions

96

norms (1.3.4) are finite, V k,j

E

Z+, V h E (O,ho]. Denote by s(Zn)the

space of test functions, whose semi-norms (1.3.5) are finite. For each @ E Co (Rn ) the restriction 71 $(x) to 1 s well-defined. h We shall also denote by 1 u(h,x) = v(h,x) an extension of a meshfunction h u :

<‘

<

Proposition 1.3.1. L e t $ E

S(Rn). Then

nh@ E S(Zn), V ho <

m.

Proof. _ _ First, let n = 1 . One has:

1 $,(x)

:=

(ax,,$) (x) =

I

ax@(x+ht)dt

x E 4

0 and the induction argument yields: t l +...+ tk))dtl...dtk.

As a consequence of the last formula one finds: /D:@(~I

I

6

max la:ocx+t) 04t6jh

1,

v x E

The last estimate implies for @ E s(Rn

)

%1 . and each u E Z y , la1

=

j:

Furthermore, using Peetre’s inequality (which is a consequence of the triangle inequality): (1.3.7)

< x > ~6 42<(x+t)>

2

,

one gets using (1.3.6):

where

1 \$I

Ik,

is the semi-norm:

which indices in S (Rn ) the same topology as the norms ( 1 . 3 . 3 ) . Thus,

1.3. Distribiition Theory i.e. A ~ QE s ( z " ) . For 9 E

sh

I

-h as follows: define its discrete Fourier transform 0

-h

9 (5)

97

=

Fx+5,h9 = hn

C

+(XI exp(-i<x,~>),

XE<

the inverse Fourier transform being

One has:

The following auxiliary result will be useful for further applications. Lemma 1.3.2. L e t u and v be meshfunctions and p ( x , ~ D* ) a d i f f e r e n c e x,h' x,h operator, which i s polynomial i n D D* w i t h c o e f f i c i e n t s depending on x,h' x,h x E 5 .

Then t h e f o l l o w i n g L e i b n i t z ' formula holds: (1.3.8) =

P(X,D~,,,D:,~) (uv) =

z -a !1B ! (P(a,B) (XrDx,h,D:,h)@:,h@x,hu(x,h)) -B +

(D:,h(D*

R )v(h.x),

)

x,h

where we have denoted:

P (a,R) (x,c,c*) = a~a~cp(x,crc*),

a and 5

being t h e d e r i v a t i v e s w i t h r e s p e c t t o 5 and 5 * , r e s p e c t i v e z y .

Proof. We write DU, respectively D*

when Dx,h, respectively D* U' x,h: applied to u(h,x) and use the similar notation Dv, respectively D

V'

Dx,h'

respectively DZ,h, is applied to v(h,x). It is readily seen that

is when

1. Manifolds, Functional Analysis, Distributions

98

The last formulae yield: P(X,D~,~,D:,~)(ux) = p(x,DU+(l+ihDU)Dv = = p(x,DU+(l+ihDU)Dv, D*+(l-ihD*)D*) u v (uv). NOW

using Taylor's formula

I

one gets (1.3.8).

Remark 1.3.3. Applying (1.3.8) with P(x,Dx,h,~i,h)= D:,~,

Y E

zy, one

gets the finite difference analogue of Leibniz' formula: (1.3.9)

Y!

Dy (uv) = 1 x ,h u4y

a!(y-a)!

(DY-u00.

U)

x,h x,h

(D:,hV).

: . € a

Now we can give the characterization of the space s(Zn) of the discrete Fourier transforms of elements in s(z").

If u

Proposition 1.3.4. -h transform u ( 5 ) :

€ S(zn) t h e n one has f o r i t s d i s c r e t e Fourier

Proof. It suffices to show __

B

a-h

( i(h,S)aSu

where the constant

C

u,B

One h a s :

Now, ( 1 . 3 . 9 )

yields:

(5) may depend only on its subscripts

1.3. Distribtrtion Theory

99

where we have also used the inequality:

We shall need the following Poisson formula. Lemma 1.3.5. L e t 4 E S ( n n ) and denote by G ( 5 ) and 6h(5) i t s i n t e g r a l and

d i s c r e t e F o u r i e r t r a n s f o r m , r e s p e c t i v e l y . Then one h a s :

t h e s e r i e s on t h e r i g h t hand s i d e of (1.3.11) b e i n g u n i f o i m l y c o n v e r g e n t , V B E , :Z

V h E (O,ho].

Proof. First, we show that the series on the right hand side of (1.3.11) ~

is uniformly convergent. Since 4 E

s (IRn ) , one

also has :

E

s (IRn ) ,

so

that

The last inequality yields with any k t 0: lac@(e+2iTh-la) B1 5 Ckhk (l+lal)-k, V a E Zn \ { O } ,

V

5 E Tn

5,h'

and the latter ensures the uniform convergence of (1.3.11). NOW, using the inverse Fourier transform, one finds, for x E R t :

xB9 ( x )

=

(2ir)-" J

((ia5)'mh(5))exp(i<x,5>:dE

=

Tn =

Note that for x E

J

IF? =

(2ir)-n

<

I

IR*

one has:

exp(i<x,E>)(ia

1

5,h 6((ia5) $(S))exp(i<x,<>)dc.

5

exp(i<x,S>)

Tn

)

8$(S)dE

Z

=

(iac)B6(5+2~h-1a)d5.

aE2"

5 rh The last formula and the uniqueness of the discrete Fourier transform imply (1.3.11). Corollary 1.3.6. L e t (1.3.12)

E S(lRn). Then one has:

1Cy(h,5)ac(9 B -h( 5 ) - $ ( 5 ) ) / 2 Ca,B,khk, V y,6 E Zy, V k t 0 ,

1. Manifolds, Functional Analysis, Distributions

100 where t h e constants

do not depend on 5 E T~ and h E ( 0 ,hol. h,S ~,8,k

C

Indeed, one has:

c Y ag(+ 8 -h- 4 )

=

i: gY(h,S)a~~(S+znh-’a). OfaEZ”

Note that \<(h,5)I 5 151 and

iE

S(Rn),which along with the last identity

implies (1.3.12). it is readily seen that for each $ E S(Rn) and for every

Also,

integer N 2 0 one has for the discrete Fourier transform of 4 (restricted to

<):

Indeed, (1.3.13) is an easy consequence of the Poisson formula (1.3.11) (with B = 0 ) . The same inequality (1.3.13) is also true for any 4

:

<

+

Cp,

4 E S(Zn), as it is readily seen. Definition 1.3.7. DJG t e s t meshfunctions $

k

E

s(<

1, k = 1,2,.

.. ,

said t o beZong t o t h e same equivalence c l a s s if $Jl(h,x)- $2(h,x) t h e topology defined b y t h e norms (1.3.4) , a s h Example 1.3.8. Let 4 E S ( R n ) and let

x

E

+

-+

0 , V k,j = 0,1,.

are 0 in

.. .

1

L ( R n ) be such that x(x) 2 0

a.e., and

Introduce

x

h

(x)

=

h-nx(x/h) and let

where

and, as hereabove, nh is the restriction to

<.

Then 4 (x) and $J (x) belong to the same equivalence class in S(< h h Indeed, this is readily seen by using Poisson formula (1.3.11), estimate (1.3.12) and the fact that F

X+S

x

=

-

n

E C (W 5

),

).

-

~ ( 0 =) 1, the

latter being a consequence of (1.3.14). The same conclusion is true if one considers a non-negative x E L 1( 2n) such that

1.3. Distribution Theory

101

and defines the meshfunctions J, (x) as follows: h $ h ( ~ )= (Xh*Qh)(X) := where, of course, $h

=

<,

,x((x-~)/h)$~(y), x E

C

YE% 7ih$, as hereabove and

xh (x) =

h-nx(x/h).

Definitions 1.3.9. The space S o ( < )

is t h e s e t of a l l continuous l i n e a r f u n c t i o n a h on s ( < ) , V h E (O,ho], w h i l e s ' (Zn) stands f o r t h e space of uniformly continuous l i n e a r f u n c t i o n a l s on S(zn) w i t h r e s p e c t t o h E (O,ho

<,

Remark 1.3.10.It is readily seen that the elements in meshfunctions u(h,x), x E

s' (<

s' (Z")

are tempered

h E (O,ho] (i.e. with at most polynomial

1x1

growth with all their finite difference derivatives as the elements in

)

+ m)

,

while

are precisely those tempered meshfunctions u(h,x)

for which one has: ,$(h,-)>l <

I
SUP

V $

m

E s(Zn),

O
0

i.e. V$ whose norms (1.3.5) are finite. For instance, the discrete Dirac mass 6h(x)

=

h

-n

for x = 0 and

6 (x) = 0 , for x E <\{o>, is an element in S ' (zn), as well as all its h (a;,h)B6h(x), v a , E~ Z ; . However, finite difference derivatives a" X th h-lfih(x) is no longer in s ' (Z") , being still an element of s' (<1 . One can also introduce in an obvious way the discrete versions p ( < z)(Zn) of the test function space D(Rn versions

D' (<

)

, D' (Z")

)

)

,

and the corresponding discrete

of the distribution space

D' ( R n ) .

However, we

shall not need them for further applications. Definition 1.3.11. TWO tempered meshfunctions uk E s ' (< ) , k = 1,2, are said t o belong t o t h e same equivalence cZass if t h e i r d i f f e r e n c e u = ul-u2 converges to z e r o f o r h 0 i n t h e weak*-topology on S'(zn), i . e . -+



+

0 as

h

-+

0,

V $ E s(Zn).

Examples 1.3.12. Let q E (O,l), y 2 0. Consider the meshfunctions u 1 (h,x) = h-'hh(x),

u2(h,x)

=

h-l-'x+(x)

-x/h (1-q)q

I

X

E

51

-

I

where x+(x) is the characteristic function of the set P . + = {x 2 0 ) and, as hereabove 6 (x) h

=

h-l6 ox, 6oo = 1, 6ox = 0 for x E

d\{O}, is the Dirac

.

1. Manifolds, Functional Analysis, Distributions

102

mass on the lattice I+., €st(\

1

s’(Z )

k = 1,2. However uk 1 One finds for u = u1-u2 and V $ E s ( Z ) :

Obviously, uk

),

,$(h,.)> = h-’($(h,O)-(l-q)


E

if y > 0.

-x/h q $(h,x)) =

x,o =

h-y(l-q)

C

q-X/h($(h,O)-$(h,x)) =

x>o =

-hl-Y(l-q) C q-x’h x>o

E a $(h,y). O
NOW (1.3.4), (1.3.5) with k = 0, j = 1 and the last identity yield:

Therefore, ul(h,x) and u2(h,x) belong to the same equivalence class in ST(<)

if y E [o,I).

Note that v(h,x) = hyu2 (h,x) is the vanishing at infinity solution of the equation:

Thus, up to an error O(h) as h

-f

0, one may replace v(h,x) by 6h(x), the

latter having as its support just the point x = 0. As a second example consider the meshfunction

(1.3.16)

u(h,x)

:=

-1

(l+jC(h,hS)/2)-1exp(ixS)dS, x E

(2n)

\1 ,

IhS j -1 where i(h,hS) = (ih) (exp(ihC)-l), introduced yet hereabove, is the discrete Fourier transform on the lattice operator D

=

-ia

=

%

of the finite difference

(ih)-l (ox,h-l).

x .h x,h Obviously, u(h,x) is the vanishing at infinity solution of the

equation : D* +l)u(h,x) = 6h(x), (Dx,h x,h

x E X$,

1

.

Introducing n = hc and 8 = exp(iq), one can rewrite (1.3.16) in the following fashion: u(h,x) = -h(Zai)-l

i

(02-(2+h2)8+1)-I8x/hdO

iel=i so that the residuum calculus yields:

1.3. Distribution Theory

(1.3.17)

103

u(h,x) = h

where we have denoted 9+ = B+(h)

(1.3.18)

x E

Using (1.3.17) in order to extend u(h,x) to v(x)

=

m , introducing

(1/2)(exp(-lx;)), (1.3.17), (1.3.18) then imply lim (u(h,x)-v(x)) =' 0, V x E I R , h+O

the convergence being uniform with respect to x belonging to any compact subset in R

.

Introduce

It is readily seen that u(h,x) defined by (1.3.16) and v(h,x) defined by 1 (1.3.19) belong to the same equivalence class in ) . Moreover, it

s'(%

can be shown, but will not be done here, that, in fact, h-'u(h,x) h-'v(h,x)

still belong to the same equivalence class in

and

s'(4) , provided

2.

y

We shall also need several spaces of parameter depending distributions valued either in D ' (U) or 1.3.13. With

Definition

s' ( E n )

E

t h e t e s t f u n c t i o n spaces c

E (O,E~],E~ < k

((O,E~];

t h e t e s t f u n c t i o n s @(E;x) of x E iable with respect t o V

6

E (O,E

0

1.

E

for each fixed value of the parameter.

€ (O,E

and U _ C R n a n open s e t , d e f i n e

D(u)), k

u which a r e

1 and

0

m,

2 0

i n t e g e r , u s t h e s e t of

k times continuously d i f f e r e n t -

a ' $ ( c , - ) € D(u), V j = 0,1,...,k,

S i m i l a r l y are d e f i n e d t h e t e s t f u n c t i o n spaces ck ((O,E~I;S(IR~ )),

a s w e l l a s t h e t e s t f u n c t i o n spaces L~((o,E~I ;D(u)) and L ~ ( ( o , E ~ I ; s ( I R ~ ) ) , whose elements a r e t e s t f u n c t i o n s @(E,x), which a r e Lm-functions of E

E

(O,E~]

valued e i t h e r i n D(u) o r i n S(mn

).

The corresponding spaces of parameter depending d i s t r i b u t i o n s a r e denoted by ck ((o,~~l;D'(u)), ck ((O,E~];S'(IR~)), L~((O,E~I;DI(U)) and L ~ ((0 , ~ ~ l ;(mn S'

I ) , r e s p e c t i v e l y . The elements o f t h e f i r s t two t y p e s of spaces are t h e d i s t r i b u t i o n s f(E;x) which a r e k-times continuousZy

differentiable i n

E

w i t h t h e i r v a l u e s i n t h e corresponding d i s t r i b u t i o n

spaces. The elements of t h e l a s t two spaces a r e t h e d i s t r i b u t i o n s f (E;x) such t h a t

1. Manifolds, Functional Analysis, Distributions

104

\~<

sup O<€<E

-

0

f o r any @(E,x) i n t h e corresponding t e s t f u n c t i o n s p a c e . Two d i s t r i b u t i o n s f (E,x), k

Definition 1.3.14.

k

= 1.2,

a r e s a i d t o belong

ts 7-he same equivalence c l a s s ( i n t h e corresponding d i s t r i b u t i o n space) i f one has: lim 1

=

0

E-fO

for any ~ ( E , X )i n

L~((o,E~I;D(u))

or L~((o,E~I;s(IR~ ) ) , respectively.

Examples 1.3.15. First consider a very simple example. m

Let fl

=

6(x), f2 = (2~)-'exp(-jxl/~). For each $ E L ((0,~~1;S(IRR))one

has : I1 6 (i/s)i exp(-jyl) I$(E,o)-$(E,EY)idy 6

w

Hence, E-'~(x)

ElQj

0,l'

and E-Yf2(E,x) for y < 1 belong to the same equivalence

0

class in C ( ( 0 , ~l;S'(R)),the latter being just the reflection of the fact 0

that f2(E,x) converges in S'(W) to the Dirac mass 6(x), as

E +

0. HOW-

-1

ever, fl = 6(x) and f2 = ( 2 ~ ) (exp(-lx\/E))do not belong to the same I

equivalence class in C ( ( 0 ,];S'(R)), ~ since aEf, = 0 and 0 bounded in S ' ( R ) as E + 0. As

a €f 2 is

a second example consider the distributions E(E;x,Y),

y E IR given, which are the vanishing for 1x1

-t

m

E

un-

E (O,E~],

solutions of the

equation: (1.3.20)

2

(8 D

2

q (x)D +1)E = 6(x-y),

Dx = -id/dx,

m

where q(x) is in C ( R ) , q(x) t qo > 0, V x E IE? and q(x) 1x1 b r with r <

m

5

q(m) for

sufficiently large.

Introduce (1.3.21)

F(E;

x,y)

:=

-1 (2~q(y)) exp(-lx-y1/(2sq(y))).

We are going to show that E - ~ Eand E - ~ Fbelong to the same equivalence class in C((O,a ];S'(R ) ) , provided that y < 1. 0 First, note that E(&;x,y) is well-defined and, moreover, for each E

E

1 fixed is an element in the Sobolev space H 1 ( R 1 . 0 Indeed, the integration by parts and the Cauchy-Schwarz inequality

(O,E

imply that the operator

1.3. Distribution Theory

is an isomorphism for each given E E ( o , E o l ,

105

so that E(c;.,y) E H 1 ( R ) ,

.

given that 6 (x-y) E H-l ( R

Moreover, the same argument implies that

where C

>

0 is some constant.

Furthermore, an elementary straightforward computation shows that one has for u = E-F: A u

=

f (E;x,y),

where f(E;x,y) = E$(x,Y) exp(-/x-yl/(eq(y))),

(1.3.23)

with $(x,y) smooth function of x E R such that $ ( * , y ) E D ( R ) . Again it is readily seen that one has for the Fourier transform ?(€;S,y)

=

F

f the following estimate:

X-tS

(1.3.24)

[ 5 l?(~;S,y)

CE

2

-2

<ES>

,

where C > 0 is some constant which depends only on q(x). Thus, (1.3.24) yields:

Combining (1.3.22), (1.3.25), one finds: (1.3.26)

i /U(E;.,Y)I l 1

5 CE

where C > 0 is some constant. As an immediate consequence of (1.2.26), one gets the conclusion

that E-'E

and F'-E

belong to the same equivalence class in

C ( (O,Eol;s'

(Rx ) ) ,

provided that y < 1. This example is also treated with full details in Chapter 4, since it plays an important role in the theory of singular perturbations appearing in the elasticity theory. As the last example, consider the following distributions

1. Manifolds, Functional Analysis, Distributions

106

(1.3.27)

u(E,t;x)

IT)-'

:=

-'sin(

(t/E)<S>)exp(i (x/c)E)dC,

R

which solve the following Cauchy problem: = 0, (e2(at-ax)+i)u 2 2

x E IR, t

>

o

(1.3.28) U(E,o;X) = 0 ,

E

a tu(E,o;~)= A M .

First, let x E Ut = {x E I R , 1x1 > t}. We are going to show that -k u(E,t;u) belong to the equivalence class containing the distribution v

which is identically zero in D'(Ut), V k > 0. Indeed, let us rewrite u(~,t;x)in the form:

one has: (1.3.29)

1 at@+/

2 c

~ > 0, , ~v x E Ut.

Thus, using (1.3.29) and integrating by parts, one finds:

Repeating the same argument, one gets the conclusion that u+ - can be represented in the form: u+(~,t;x)=

-

E

k f v (E,~;x), V k 2 0, k

where vi are continuous functions of x E Ut, V t > 0, such that (1.3.30)

sup O<EJEO

sup lvil < Ct,k < XEUt

m,

V t > 0, V k 2 0.

m

Thus, (1.3.30) implies that u = O ( E ) for E + 0 in D'(Ut), i.e. that -ku 1s . in the same equivalence class as zero in D'(Ut), V k 2 0. E Now, let 1x1

t. Using the stationary phase method (see Chapter 3),

107

Notes one can show that the following asymptotic formula holds for U(E,t;X): (1.3.31)

-1 2 2 1/2) sin(€ (t -x )

u(E,t;x) = E1'2(~(t,x)+Ev(E,t;x))

where Ji(t,x) is a smooth function of x E V

t is a continuous function of x E Vt such that (1.3.32)

sup O<E<E~

max / v ( E , ~ ; xI )5 Ct < Ixl
m,

{x

:=

I

1x1 < t} and v(E,t;x)

V t > 0.

Thus, (1.3.311, (1.3.32) imply that E-YU(E,t;X) and (t,x)sin(E-'(t2-x2) 'I2) belong to the same equivalence class in D'(Vt), t t 0, provided that y < 3/2.

Notes __ For the concise presentation of the smooth manifolds theory have been used [Arnold, 1

1,

[Lang, 1 1, [Loomis, Sternberg, 1

1.

As far as the classical functional analysis is concerned, the brief account on basic functional analytic results needed for the further chapters (also in vol 11) is based on [Riesz-Nagy, 1 [Rudin, 1

1, [Lyusternik - Sobolev ,

1 1,

1.

Example 1.2.11

is a specific case of the general result in [Levinson, 1

1.

Example 1.2.12 illustrates how the linear theory of strongly elliptic and coercive singular perturbations ([Vishik - Lyusternik, 1 1, [Huet, 1 [Lions, 1 1 , [Frank, 15,19,22 1, [Frank - Wendt, 1,5,10

1

1,

and others) can

be extended to the classes of non-linear singular perturbations, whose linearizations are strongly elliptic or coercive singular perturbations (see also vol. I1 of this book). Example 1.2.13 is an illustration of the applicability of the Lindstedt-PoincarG method ([PoincarG, 1.2 Mitropolsky ,

1 1, see also [Nayfeh, 1

1

1

[Bogolyubov-

where formal asymptotic

expansions using this method are given for second order equations) to hamiltonian systems, which are singular perturbations of hamiltonians, describing motions with holonomic constraints. Extension of Banach's theorem (Theorem 1.2.34) to the parameter dependent situation as well as the statements concerning the commutativity of diagrammes in this case (Corollary 1.2.36) seem to be new and are useful in applications. Example 1.2.37 is an illustration of the general result concerning the stability of the index for the elliptic and coercive singular peturbations (see [Frank - Wendt, 1, 5,lO ]).Example 1.2.38 shows how the results concerning the classical

1 . Manifolds, Functional Analysis, Distributions

108

Wiener-Hopf and Toeplitz operators (see LGohberg [Gohberg

-

Feldman, 1

-

Krein, 1

1, [Krein,

1

1,

1) can be extended to one parameter families of

elliptic difference operators (see [Frank, 4,6,9-13

I).

Example 1.2.40

illustrates the complexity of spectral problems for singular perturbations; it is examined with full details in [Frank - Norde, 1 1. We follow [Schwartz,l

1, [Gelfand - Shilov,

1

1, as far as a concise

presentation of the distribution theory is concerned. Parameter dependent test functions and distributions were considered in [Frank, 8

1. The concept

of the singular support for a parameter dependent family of distributions was introduced in [Frank, 14

1, but, of course, was implicitly present in a

previous work by others on this subject.

CHAPTER 2

SOBOLEV SPACES OF VECTORIAL ORDER

3 o f v e c t o r i a l o r d e r s E IR are (5.1 i n t r o d u c e d and t h e i r b a s i c p r o p e r t i e s a r e p o i n t e d o u t . These s p a c e s p l a y I n t h i s c h a p t e r t h e Sobolev s p a c e s H

t h e same r o l e i n t h e s t a b i l i t y t h e o r y of c o e r c i v e s i n g u l a r p e r t u r b a t i o n s , of o r d e r r E IR f o r t h e e l l i p t i c

a s t h e c l a s s i c a l Sobolev s p a c e s H

boundary v a l u e problems w i t h o u t p a r a m e t e r s . 2 . 1 . s p a c e s on

nn

= ( s l , s 2 , s ) E IR3 3 Definition 2.1.1.

Let s

and l e t

E~

be a p o s i t i v e c o n s t a n t

For a f a m i l y o f d i s t r i b u t i o n s

m

(O,Eo1 x

n-1

3

( € , < I )

*

V(E,5',Xn)

E S'(IRX

)

n

whose Fourier t r a n s f o r m s a r e ZocaZZyintegrabZe f u n c t i o n s d e f ~ n ethenorms 3 (FxnlSnv) ( E , 5 ' t S n ) I I L

We say t h a t v E

v

(€,C')

H

E (O,Eo1

The f u m i l y v E H

,E,E'

x

IRn-1.

(s), < '

Obviously, H ( s ) I

(S)

. I

(IR) i f i t s norms ( 2 . 1 . 1 ) a r e f i n i t e ,

(IR)

(IR,

)

'n

if

C,(R)i s a f a m i l y of H i l b e r t s p a c e s w i t h t h e i n n e r

products

where v . = F, 7 n For si,

n

v,. l

j = 2,3 non-negative

i n t e g e r s one c a n a l s o i n t r o d u c e t h e norms

2. Sobolev Spaces of Vectorial Order

110 t h e norms

1 1.11

being e q u i v a l e n t uniformly with

r e s p e c t t o t h e parameters. P r o p o s i t i o n 2.1.2. H

(s). 5 '

i s a family of Banach spaces. s

-s

E .The

maps H (R) i s o m e t r i c a l l y 2<~5>s3F (s). 5 ' x +En 2 n o n t o t h e Banach s p a c e B ( ( O , E ~ ] L; (R))of bounded f u n c t i o n s on ( 0 ,0~1 v a l u e d 2 i n L (R). 0

'

operator E

D e f i n i t i o n 2.1.3.

For a family of d i s t r i b u t i o n s 3

(O,E01

E

-t

U(E,X)

E

S'(R;)

d e f i n e t h e norms of order s:

Again,

( O , E ~ ]3

E

+

H

(s), E

i s a Banach s p a c e , s i n c e

( R n ) i s a f a m i l y of H i l b e r t s p a c e s and H

isl < 5 > s 2 < ~ < > s 3x+< F

(S)

(Rn)

i s a n i s o m o r p h i c map of

( I R ~ ) onto B ( ( o , E ~ I ; L ~ ( I R ~ ) ) .

H (S)

Remark 2 . 1 . 4 . For c o n v e n i e n c e w e s h a l l u s e t h e n o t a t i o n [ u ] o f f a m i l i e s of d i s t r i b u t i o n s v a l u e d i n S ' H ( S )

(wn-') ,

X'

[ u ] ( ~ )f o r t h e norms (s), E l t h a t belong t o H

respectively.

Example 2 . 1 . 5 . ( i ) L e t @ E S(Rn-')

and l e t v ( ~ , C ' , x) be t h e f a m i l y

( s ) ,E

W"

2.1. Spaces on

111

-+.

i.e. i f f s2 > ( i i )L e t

(0,11 3 E

(2.1.6)

+ U(E,X)

=

2 ( 4 n ~

W e find, using t h e polar coordinates = (F

;(E,<)

=

U) (E,c)

<EE>

-2

X+E

Therefore, u

E

E

H

Further, u

H

3

(IR

)

.

i f f s +s

2 3 < + (R3)i f f t h e f u n c t i o n

( s ),E

( S )

-2s (0,11 3 E

+

IS(€) = E

1

m

I

p

2

2 s2

(l+p )

2 2 s3-2 (1+~ p ) dp

0 i s bounded on ( 0 , 1 ] . One h a s f o r

E

+

+0:

i-2s1 -2s1

,

if s

2

In E , i f s2 -3-2(s +s2) 1 , i f s2

I ~ ( E w{ )

< -3/2 =

-3/2 -3/2. 3

Hence, t h e f a m i l y ( 2 . 1 . 6 ) b e l o n g s t o H

(IR )

i f f s belongs t o t h e

(S)

s e t V,

v

3

E R , sl
= {sis

+S

1

5-3/2,

2-

s

2

+S

3


3

Notice t h a t ( 2 . 1 . 6 ) s a t i s f i e s t h e e q u a t i o n : (-E

where A =

2

A+~)u(E,x)

2

2

2

2

=

6(x),

2

a / a x l + a /ax2+a /ax23

VE > 0 , x

E IR3

i s t h e L a p l a c e o p e r a t o r and 6 ( x ) i s t h e

Dirac's &-function. Remark 2 . 1 . 6 . For e a c h ( € , E l )

E

( 0 . ~ ~X 1IRn-'

f i x e d t h e space H

(s)

(IR) c o i n c i d e s

, € , E l

w i t h t h e u s u a l Sobolev s p a c e Hs +s (IRR) and f o r e a c h E E ( O , E ~ ] f i x e d t h e 2 3 space H ( R n ) is n o t h i n g e l s e b u t H (IRn), but, i n general, there s +s ( s ), E

2

3

i s no e q u i v a l e n c e o f norms o f o r d e r s w i t h p a r a m e t e r s w i t h t h e u s u a l

S o b o l e v norms o f o r d e r s2+s3u n i f o r m l y w i t h r e s p e c t t o t h e p a r a m e t e r s .

2. Sobolev Spaces of Vectorial Order

112 m m n Hence C ( R ) and C (I3 ) 0 0 respectively.

are dense i n H

5 s2+s3t h e n f o r e a c h

I f s;+s;

imbedded i n t o t h e s p a c e s H

E

(s'),E

,€,El

( S )

( R ) and H

(s),E

(Rn),

(Wn) E (00 ,] ~ f i x e d t h e s p a c e s H ( S ) ,E (nn), b u t t h e imbedding D p e r a t o r s are

n o t n e c e s s a r i l y u n i f o r m l y bounded w i t h r e s p e c t t o

E

are

E ( O , E ~ ] . W e are g o i n g

t o p o i n t o u t n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s on s and s ' f o r t h e uniform c o n t i n u i t y of t h e c o r r e s p o n d i n g i m b e d d i n g ' s o p e r a t o r s . D e f i n i t i o n 2.1.7.

The f a m i l y os norms I 1 . 1 i ( s ) , € , E E ( 0 , c01 , a r e s a i d t o be s t r o n g e r than i 1.i I ( S s ) , E l if t h e r e exists a c o n s t a n t C , which may depend only on s, s' and

E

such , that

~

V

1 1 . I 1 (S'),E

Ve w r i t e

,E

I I .j I

E ( 0 . ~ ~ 1V , u E H ( s ) ( R n ) .

>

ll-ll(s),E Or

a r e s t r o n g e r than

I 1.1 I (s)

E

l l - l l ( s # ) , Ei f

~ ~ . ~ ~ ( S ) , E

t h e norms

( S ' ) ,E'

P r o p o s i t i o n 2.1.8.

One has (2.1.8)

l l - l l ( s # ) , E l l . l l ( s ) , E iff s; 5

sl,

si+s; 5 s l + s 2 , s ' + s ' 5 s 2 + s 3 .

P r o_ o f . I t i s immediate t h a t t h e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r _ ( 2 . 1 . 7 ) t o h o l d is t h e i n e q u a l i t y

where t h e c o n s t a n t C may depend o n l y on s , s' and

5

By p u t t i n g i n ( 2 . 1 . 9 ) , f i r s t

5

E +

0' 0 , afterwards

E

=

-1

<5>

,

E +

0 , and,

one g e t s t h e c o n c l u s i o n :hat (2.1.8) is necessary s -si s3-s3 / 2 , f o r ( 2 . 1 . 7 ) t o h o l d , w i t h C 2 max{EO1 ' , 2

finally,

E =

To show t h a t c 0 5 1.

1,

= 0,

E

+ m,

(2.1.8)

i s a l s o s u f f i c i e n t , it i s enough t o c o n s i d e r s'

In t h a t case, -s E

E~

0,

=

(2.1.8) yields:

s 1<5> 2<E5>s3

-s =

(E)

2

If

I

I

s 1<5> s +s

<€<>

s +s

fS

2<E5>s3 t

<5>

2<ES>

s -s 3

1

t

t ~ .

> 1 , t h e n t h e s a m e argument y i e l d s :

-s CE

s l<<> 2 < c p s 3

21

where t h e c o n s t a n t C depends o n l y on s and We u s e t h e n o t a t i o n s'

<

s or s

> s'

E

0'

f o r w r i t i n g (2.1.8)

i n a s h o r t way and

2.1. Spaces on

W"

113

> s') i f t h e r e i s a t l e a s t one s t r i c t i n e q u a l i t y i n

t h e one s ' < s ( o r s (2.1.8).

Equipped w i t h t h e r e l a t i o n s

<, <,

t h e norms d e f i n e d above and t h e c o r r e s -

ponding s p a c e s become p a r t i a l l y o r d e r e d w i t h r e s p e c t t o t h e s e t

i . e . t h e imbeddings H

5 H(S'),E

( s ) ,E

(En)

a r e uniformly continuous with r e s p e c t t o In p a r t i c u l a r , with Hr(Rn),

(wn)

E

E ( 0 , ~I i f f 0

(s'-s) E V .

r E R, t h e u s u a l Sobolev s p a c e , t h e imbeddings

a r e uniformly continuous with r e s p e c t t o

H r ( w n ) E H ( s ' ) ,E iff ( O , r , O ) > s ' , i.e.

i f f si 5 0 ,

si+s;

E

E

(O,E

0

I,

5 r , s'+s' 3 S r .

C o r o l l a r y 2.1.9.

For s H ( S )

>

s'

one h a s : H

( w n )5 H

( ~ , () R n )

c

(IR) 5 H ( s ' ) , E , < ' (IR) H ( s ) , < '(B) H ( s ' ) ,C' (R) and t h e imbeddings operators a r e uniformly bounded I

( S ), C I S '

w i t h r e s p e c t to t h e parameters

( € , < I )

E ( 0 . ~ ~x 1Rn-'.

P r o p o s i t i o n 2.1.10.

Let u E (2.1.10)

m3, s

3

E IR , s < u and l e t

p = s-t(u-s),

t > 0.

P r o o f . O b v i o u s l y , it s u f f i c e s t o p r o v e ( 2 . 1 . 1 1 ) f o r s -

=

0. I n t h a t case

(2.1.11) i s e q u i v a l e n t t o t h e i n e q u a l i t y :

-20 1 5

2

2u 20 2tUl I<<> 2<E<> 3+6-2tE

6

-20

o r , with A = 6

1 s X+A

20

'<€,>

E

-t

, v

20 '<E<>



-2tu

-2tu 2<E<>

',

t o the inequality

A > 0.

The l a t t e r o b v i o u s l y h o l d s , s i n c e t > 0 and inf(A+A-t)

X>O

=

t l ' ( t + l ) + ( , / t ) t ' ( t2+ I, l)

v

t > 0.

3

I

2. Sobolev Spaces of Vectorial Order

114

C o r o l l a r y 2.1.11.

With

s

< u,

g i v e n by (2.1.10) and any 6 > 0 hoZds:

p

2

I lul I ( s ) , E

(2.1.12)

2 5 6

2

-2t

I (u),E+6

IIUI

I Iul I

v

(2.1.13) Vs

*

=

3

{s'ls' E IR

,

s'

<

.

C o r o l l a r y 2.1.12. 3 * L e t s E R and u E i n t v . Then f o r V

T

E

C

u E €3

( 0 ) ,E

(2.1.11) o v e r RnT1.

5

s}, V,* = {s'Is' E R

v

constant

v

w e denote

being c a l l e d reci p ro cal t o

the set V

(p),E

E (O,Eo1,

E

I n f a c t , one g e t s (2.1.12) by i n t e g r a t i n g

For a g i v e n s E IR3

2

S

vS,

which may depend only on S,T,U,S

3

s'

>

s}

V 6 > 0 there e x i s t s a

and

cO

such t h a t

I n f a c t t h e c o n d i t i o n u E I n t v means, t h a t u -sl > 0, u +a -s -s 1 1 2 1 2 > O , a +u - s -s > 0. T h e r e f o r e , t h e r e e x i s t s p = s - t ( u - s ) < T . 0 2 3 2 3 For s E R w e d e n o t e by a V t h e boundary of V which c o n s i s t s o f t h r e e p l a n e components:

(2.1.15)

alvs a3vs

W e define 3 . V

*

= =

I S ~ / ES ~avs, t s l l s i E avs,

S'

= s,?,

si+s'

2

3

=

2

a2vs +S

3

t o b e t h e p l a n e component of

1 s b e l o n g t o t h e same p l a n e .

=

{s'Is~

E 3vs, s ' + s ' = S + S } , 1 2 1 2

1.

av

*

*

such t h a t 2 . v and 3 . v 3 s 1 s

C o r o l l a r y 2.1.13.

3 * L e t s E R and l e t u E I n t 2 . v . Then f o r V

constant

c such t h a t

I S

T

E 2.v

7 5

there e x i s t s a

(2.1.14) h o l d s .

I n f a c t , t h e p r e v i o u s argument c a n be a p p l i e d .

2.2. R e s t r i c t i o n t o h y p e r p l a n e and f u r t h e r imbeddings r e s u l t s D e f i n i t i o n 2.2.1. FOP

(o,E01

x

Rn-'3

(E,E')

--s

= E

+ ~ i ( E , s ' )

E

c let s3

1<<'>s2<E<'>

(l/I(E,cl)

I

2.2. Restriction to Hyperplane

and l e t c

( S ), E , S '

115

be t h e s e t of complex valued f u n c t i o n s $ ( E , S )

where t h e

[ * I ( S ) ,E,S"

d i s t a n c e s are d e f i n e d by Define

[$I

(2.2.2)

S,

= sup

(s), E , S '

EE(O,E ]

0

and l e t c

(s). 5 '

( 0 . ~ ~3 1E

Let e 2 , e

be a f a m i l y of Banach spaces o f t h e f u n c t i o n s

+ $ ( € , < I ) ,

E

whose norms ( 2 . 2 . 2 ) a r e bounded V 5' E mn-'.

R3 b e a s f o l l o w s :

(2.2.3)

e

With n o

: H

=

2

(O,l,O), e

( S ) , E 8 5 '

(IR)

+

= (l,-l,l).

C t h e r e s t r i c t i o n o p e r a t o r t o {xn = 01,

we state

theorem: Theorem 2 . 2 . 2 .

3 L e t s E R , s 2 + s 3> f . Then w i t h some c o n s t a n t C , which may depend o n l y

on

s

and

E~

and f o r V v E H

( s ). E , S "

(s-te2), E , s ' (2.2.4)

5 CI j v ;

I (s)

5 c ( l + / & nE

Proof. Since f o r V E , S ' -

( s ), E , 5 '

i f s2 > f , / I

f

IIvIl

5 CI

IVI

,E.S'

I ( s ), E , S '

i f s2 = f , i f s2

f.

(R) c a n b e i d e n t i f i e d w i t h t h e

(IR) , it i s enough t o show ( 2 . 2 . 4 ) f o r v v a l u e d

'2+'3 t h e l a t t e r b e i n g dense i n H ( R ) , v r

The Cauchy-Schwartz

holds

, € , E l

(S)

fixed H

u s u a l Sobolev s p a c e H

s(R) ,

E ( 0 . ~ ~x 1R

( s - f e 2 ), E , s ' [n,vl ( s - f e 2 + ( s 2 - f ) e ) , E , 5 '

in

n- 1

v (E,S')

E

IR.

V

E

For such v one h a s :

inequality yields:

where (2.2.6)

de f I ( E , ~ '=) J



-2s

-2s 2 < ~ 5 > 3d5n <

m,

> 0,

IR s i n c e s +s < f . 2 3 F i r s t , c o n s i d e r t h e c a s e : s2 > f . Making t h e change o f v a r i a b l e

5

n

= <S'>t, we g e t :

2. Sobolev Spaces of Vectorial Order

116

with

cs

=

max{/ R

-2s -2(s +s ) *dt,/ dt). IR

Now, let s2 = j . Making the change of variable

en

= E-'<~c'>t, I(E,c')

becomes : -2s

-2s

'f

I ( E , ~= ' )< E < ' >



3(E22<~c'>-2+t2)-'dt.

R One checks easily that with q

where C

depends only on s

E

s3 > 0 holds:

(0,1],

3'

Further,

v

lLn(E<e'><Ec'>-l)I 5 Ln(l/E),

E (0,ll x IRn-l

(E,C')

and 1 /Ln(E-) I 5 ClLn

v

El,

where the constant C may depend only on cO if Hence, with s

2

E

(E,s') E

0

(o,E01

x

> 1.

= j we get:

',

-2s (2.2.8)

I(E,s') 5 C(l+lLn El)

<Eel>

Finally, consider the case: s2 < f . The same change of variable

sn

=

E-'<Ecv>t

yields: 1-2(s +s

2s -1 I(E,S')

(2.2.9)

= E

$ csE

1-2(s +s ) 2s -1 2 3 <Eel>

c

max{/

with =

w

jtl

-2s -S 3(~2<5'>2<~5t>2+t2) 2dt

)

J

<Eel>

R

-2s -2s ' 3dt,

r

-2(s +s dt).

IR

Combining (2.2.5)-(2.2.9) we get (2.2.4).

0

Theorem 2.2.3. 3 Let s E R , s2+s3 > f . Then the r e s t r i c t i o n operators r0

n

( a ),E

: H (s),E (w"-l) a r e uniformly bounded with respect t o E E ( 0 . ~ ~if1

a = (sl,s2-f,s3)f o r s2 > f and a = (s +s - j , o , s 1

2

2

(wn)

+s -j) f o r s2 < f . 3

+

2.2. Restriction to Hyperplane

P r o o f . One

applies (2.2.4)

v(E,S',X

to

s q u a r e s and i n t e g r a t e s o v e r

= F

)

n

x'+S

5 ' E Rn-'

117

,u)

( E , ~ ' , X ~ ) t, a k e s

the

0

Corollary 2.2.4.

The r e s t r i c t i o n operators

( R n ) -F (R) C ( o ) , S , ,T o : H ( 5 ), S ' ( S ) ( m n - l ) w i t h a d e f i n e d i n Theorem 2 . 2 . 3 . , a r e uniformly bounded w i t h (a) r e s p e c t t o 6 ' E mn-l. :

T~

H

+

H

Remark 2 . 2 . 5 . ( R n ) and l e t s2+s3> f . Then t h e f a m i l y

L e t u ( E , x ' , x ~ )€

( O , E ~ ]X R 3 ( € , x n ) + u ( € , x ' , x n ) E with

0

E R3

H

( 0 ), E

is continuous i n x

d e f i n e d i n Theorem 2 . 2 . 3 ,

E

n'

V

E

E

(O,E~].

E iR i s u n i f o r m w i t h

Moreover, t h e c o n t i n u i t y of t h i s map i n x respect t o

(1R;;l)

E (O,E~I

I n f a c t , one f i n d s , u s i n g t h e p a r t i a l F o u r i e r t r a n s f o r m F x , + S , : [ U( E ,x ' ,x +h) -u (E , X ' ,x n ) 1 2

2

5

( 0 ), E

-2s1 2s 2s 3 I S ( ~ , S ' , h ) ~ <5> 2 < ~ S > / ; ( E , S ' , S , ) I

J

82

dS'dS,,

mn where

I s ( ~ , F , ' , h )= 2 s -1 (2~)-~/<5'>

2s

-2s

3<5>

-2s 3 2 2<~F,> l e x p ( i h F , n ) - l / dF,

R

and

i f s2

7

f

I S ( ~ , F , ' , h )= (2n)

-2

1-2s / E

*<EF,'>

2 s + 2 s -1 2 3

-2s

<<>

-2s 3 2<~<> exp (ih5,) -1

1

R

I n b o t h cases h o l d s : when h

-+

I 'a<,

i f s2

0 f o r each

1

11 ( ~ , t ' , h ) 5 C

(E,S'),

,

$.

V ~ , c ' , hand moreover, I s ( ~ , S ' , h + ) O

so t h a t L e b e s g u e ' s theorem on bounded convergence

i m p l i e s t h e c o n t i n u i t y of t h e f u n c t i o n R

3 xn

+

u(s,x',xn)

E

H ( a ) ,E (IR;,'),

The l a t t e r i s u n i f o r m w i t h r e s p e c t t o E sup I S ( ~ , F , ' , h )= I s ( E ' , h )

+

0 when h

-f

E

V u (O,s0],

E

if

H (s), E 07:).

1 lul I

0 f o r e a c h F,' E Rn-

F) .

<

m,

E

Theorem 2 . 2 . 6 .

If s2+s3> n/2 then each u E f u n c t i o n s of x E R ~ ,

H

(s),E

( m n ) i s a f a m i l y o f continuous

since

2 . Sobolev Spaces of Vectorial Order

118 (O,Eo1 3

(2.2.10)

and moreover, w i t h

E +

U(E,X) E

O

c

n

(IR

I I . I I co (IRn) t h e

)

norm on Co (R:)

one has s

'

-s -s +n/2

1

2

I

2s

CE



= E

(S)

-2s -2s 2<E€,> 3dc <

m,

v

E

E

n/2

,E'

if s

2

= n/2

if s2 < n/2,

&,

where t h e c o n s t a n t c may depend only on s , n and

where

>

2

0'

E (o,Eol,

lRn since s + s > n/2. 2 3 One finds easily when

E

+

+0:

As a consequence of (2.2.12) one gets (2.2.11).

Corollary 2.2.7.

If u E

H(s)

(Rn) w i t h s2+s3 > n/2, then

where t h e c o n s t a n t c may depend onZy on s , n and

E

~

.

Corollary 2.2.8.

Let u E H

(S)

( 0 . ~ ~31 6

+

( R ~ ), s 2 + s 3 > k+n/2, where k > 0 i s i n t e g e r . Then ,E U(E,X) E Ck(Rn) and, moreover, t h e norms l u ( E . . ) ICk(,p)

I

1

2.2. Restriction to Hyperplane

119

s a t i s f y a l t e r n a t i v e l y t h e f i r s t , second o r t h i r d i n e q u a l i t i e s (2.2.11) if s2 > k+n/2, s2 = k+n/2 and s2 < k+n/2. 2 ICk(,p) 5 cll<<>k IL1 ( p ) , and IL1(p) 5 In fact, j / u / 2 where C is the same constant as in the proof of ( s ) ,E' CEI Theorem 2.2.5 with s2 replaced by s2-k.

I bk I;

i;

luI I

Theorem 2 . 2 . 9 . ( I R ~,) s2+s3 = n/2+a w i t h a E (0,l). Then U(E,X)

Let u E H b'

E

( s ) ,E ( 0 , ~1, 0

E

E

Ca(R:),

and, moreover,

(2.2.14) E

n/2+a-s -s 1 2

where t h e c o n s t a n t c may depend o n l y on

s.

n and

E

0'

Proof. We have only to estimate /u(E,x+h)-u(e,x) 1 with lh[

I /u(E,.)

6 1, since

IIco(mn) does satisfy (2.2.14). We find:

12=

lu(E,x+h)-u(E,x)

2s

-2n

5 (2n)

E

1

(27Tr2n / / ; ( € , E l

Rn

(exp-l)dc/ 2 5

-2s -2s 3 2 2<E<> d < /l u / j ( s )

1 Iexp-1I2<<>

IRn

For s2 2 n/2+a write:

which proves the first of the inequalities (2.2.14). NOW consider the case: s2 < n/2. Write

J

" -2s -2s lexp-l/L<<> 2 < ~ < > 3 d ~=

2

= E

5 CE

1 lexp(ihs-lS)-ll Rn 2s -n 2

{E

(E

-s -2s 2 + / < 1 2 ) 2 3d5 5

-1 -2 IEIh' n+l 2 2 -52 h 1 P (E +p ) (l+~~)-'~dp+ 0

,E.

2 . Sobolev Spaces of Vectorial Order

120

0 n-1-2s

J

+

-2s

2

1P

€14-

2 s -n-2a 3dp] 5 C E

(l+p)

2

lh!2n,

which p r o v e s t h e second i n e q u a l i t y i n ( 2 . 2 . 1 4 ) .

0

Remark 2 . 2 . 1 0 . If u

E

H

( s ), E

( R ~ ) ,s2+s3= n/2+a,

s2

=

n/2+8, w i t h 0 < 8 <

Q:

< 1 , then

t h e s a m e argument i s p r e v i o u s l y shows t h a t

v where t h e c o n s t a n t C may depend o n l y on u , B , n and If s2 = n/2

(8

= 0),

s

(O,E01,

E

0' = ( n / 2 ) t a , t h e n t h e l e f t hand s i d e i n

+s

2 3 ( 2 . 2 . 1 5 ) h a s t o b e r e p l a c e d by E

2.3.

E

E

- s l ( l + / h

lu(E,.)

El)-'[

I IcQ:(Rn)

.

Spaces of d i s t r i b u t i o n s s u p p o r t e d by a h a l f - s p a c e

with s

(s), E , C "

(s),S"

o+

H-

(s), E l (5) ( f o r t h e p r o p e r t i e s of t h e corresponding spaces with s c a l a r

E R3

index s

; i

? ;

o+

W e a r e g o i n g t o d e f i n e t h e s p a c e s H-

E R

w e r e f e r t h e r e a d e r t o [Li-MI,[V-E],[E].

D e f i n i t i o n 2.3.1.

A f a m i l y of d i s t r i b u t i o n s ( O , E ~ x] xn-l 3 ( E , S ' ) u + ( E , ~ ' ) E s'(R) i s o + s a i d t o belong t o H ( S ) ( R ) and supp u+ 5 +, if u + H(s) V (E,s') E ( O , E , ] x R"-l; i t i s s a i d t o b e l o n g to H (s), 5 ' i f u + E H ( s ) , t 1 ( R ) -f

, & , E l

and supp u +

~ x +V 5,' E

, € , E l

+

mn-'.

De finition 2.3.2.

A f a m i l y of d i s t r i b u t i o n s (O,E

to H:~,

0

A +(s). E

if u + E

I

3

-f

u+(E,.)

and supp u + 5 R : ;

H(s),E(IRn)

i f u+ E H ( s ) (En) and supp u + 0 -

E

0 -

5

E E ' ( R ~ ) is s a i d to belong

it <s s a i d to belong t o

q. 0-

0 -

H H t h e c o r r e s p o n d i n g s p a c e s of (s), E , C " (s) ,C" H ( s ,)E l (s) ( f a m i l i e s o E ) d i s t r i b u t i o n s whose s u p p o r t l i e s i n 5-and respectively.

W e d e n o t e by H

z!,

Remark 2 . 3 . 3 . For e a c h

E

E ( O , E ~ ] , 5' o+

H ( s ,)E , 5 '

( R ) and H-

( 5 ), E

n-l

E R

o+

f i x e d , H-

( s ), E , S '

a r e c l o s e d s u b s p a c e s of

a r e c l o s e d s u b s p a c e s of H

( 5 ), E

(Rn)

. Besides

2.3. Spaces of Distributions Supported by a Half-Space o+

and Cm(IR:)

C i ( R -+ )

0

o+

and H-

a r e d e n s e i n H-

-

121

( 5 ), E , S '

respectively

( S ) ,E

I n d e e d , t h i s i s a consequence of t h e f a c t , t h a t f o r V E , S ' f i x e d

H

(s)

Hs

,€,el

(IR) c o i n c i d e s w i t h H

2

+

+ s ( 1 ~ " ) . S i n c e H-

s +s

( s ), S '

n

=

+

H;

E>O

(R)and H

coincides with

(IRn) ( 5 ) ,E

3

n

S),E,C"

2 3 conclusions hold f o r these spaces, a s w e l l .

Hf = H+ (s) E>O ( s ),El

t h e same

Remark 2 . 3 . 4 . where s + s 3 = m+y w i t h m > 0 i n t e g e r and IyI < 2 ( 5 ), E ' a consequence of Theorem 2 . 2 . 3 , t h e r e e x i s t s t h e r e s t r i c t i o n

Let u E H+

n u E H

0

( u ) ,E

o f u on t h e h y p e r p l a n e {x

(Rn-')

=

01,

!.

w i t h u E lR3

Then a s

defined

i n t h e f o r m u l a t i o n o f t h e theorem. k- 1 ( R n - l ) , 1 5 k 5 m , is + D u(€,x',xn) E H xn ( s - ( k + ? ) e 2 ),E c o n t i n u o u s (however, n o t u n i f o r m l y w . r . t . E ) , a n d D k - l ~ ( ~ , ~ ) ' , ~0 f o r xn x < 0 , one g e t s t h e c o n c l u s i o n t h a t 71 Dk-lu = 0 , 1 5 k 5 m , V E > 0 . Xn Example 2 . 3 . 5 . S i n c e IR 3 x

The f a m i l y u ( E , x ) s +s

2

3

<

+. In

)exp(-xn/E) -i = (icEn+l)

= E-~H(X

fact, ;(E,<,)

On t h e o t h e r hand,

u E H+

E

H+

,

so t h a t

( S ) ,E

if s 1 5 0 , s2 < - f ,

(W)

(S)

compatible with t h e f a c t t h a t l i m u

=

6 ( x ) E H+

E+O

With a > 0 , one h a s :

U(E,X

)

(R), v

52

n

E

(o,E01,

iff

s 2 + s 3< f , which i s

(IR) , V s2 < - f .

= xaH(x ) e x p ( - x n / ~ )E H

n

E

+ ( 5 ) ,E

( R ), V E E ( 0 . ~ ~ 1 ,

i f f s 2 + s 3< a + + . F u r t h e r , u E H I s ) (IR) i f s 1 < l + a , s2 < -?,

s + s < a++. 2 3 Moreover, n Dk-lu = 0 , 1 5 k 5 m, where t h e i n t e g e r m i s s u c h t h a t 0 xn a = m+y, w i t h IyI < + . I n f a c t , one h a s : ;(E,c

)

l+CY =r(a+l)E ( i E C + ~ ) - l and - ~ t h e following asymptotic

f o r m u l a h o l d s i n S ' (R): =

1 +a 2+a 6 ( x n ) + ~ ( E 1,

r(a+i)E

N o t i c e , t h a t i f I$ E H

+ ( o ),E

(En-')

u = $ x D k 6(xn) E H<s),E(IRn)

x,

supp u c {xn=O} and

,

+

0.

with 02+u3 < 0 , t h e n

with s

=

u-(k+t)e2. In fact,

2. Sobolev Spaces of Vectorial Order

122

R 2.4.

S p a c e s on I R ~ -

For s .

I'

j = 2,3,

I lul I

(2.4.1)

where

CY

non-negative

i n t e g e r s d e f i n e t h e norms

I +

=

, E , c I

= (a',an),

6

(6',Bn),

=

= -ia/ax

D~

.

n We d e n o t e by H

(R+) w i t h s .

J' f a m i l y of s p a c e s of a l l d i s t r i b u t i o n s ( S ),

j = 2 , 3 non-negative

-

& , E l

( € , < I )

+

integers the

u E S'(R+) w i t h norms

(2.4.1) f i n i t e . I n t h e s a m e way o n e d e f i n e s t h e norms

t h e s p a c e s of a l l d i s t r i b u t i o n s i n S'(lR:) b e i n g d e n o t e d by H

w i t h norms ( 2 . 4 . 2 ) f i n i t e

($1.

fs) . E

For u E C m ( q ) a n d N > 0 i n t e g e r i n t r o d u c e t h e f o l l o w i n g e x t e n s i o n O m -n o p e r a t o r tN : c O ( R + )+ cN-l ( n n ) , 0

(2.4.3) where H(x

k u = H(Xn)u(E,X',X N

c U(E,X',-PX

)+ff(-Xn)

lSp5N

is t h e Heaviside's f u n c t i o n and C P' s y s t e m of l i n e a r e q u a t i o n s :

(2.4.4)

)

Z (-p) 16p6N

j-1

C

P

= 1,

1 2 p 5 N,

satisfy the

1 5 j 5 N.

Lemma 2 . 4 . 1 .

For s 2 , s3 n o n - n e g a t i v e i n t e g e r s , t h e operator t N w i t h

N 2 s2+s3can

be

extended a s a continuous ( u n i f o r m l y k i t h r e s p e c t t o t h e parameters) mapping from H fs) . € , E l (lR+) ( r a p . H H

( s ) ,E

(S),E(~n+))

into

H ( S ), E L '

(IR) ( r e s p .

(Rn)J.

P r o_ o f . The s t a t e m e n t of Lemma 2.4.1 _ inequalities

i s a n immediate c o n s e q u e n c e of t h e

2.4. Spaces on F?:

where t h e c o n s f a n t C depends o n l y on N. For a f a m i l y of d i s t r i b u t i o n s ( E , S ' ) with s

I'

123

0 +

E

u

S ' ( I R + ) and s

3 E R

2,3 n o n - n e g a t i v e i n t e g e r s , i n t r o d u c e t h e norms:

j =

where t h e infinimum i s t a k e n o v e r a l l e x t e n s i o n s Q of u t o R o r t o IRn, respectively

.

Lemma 2.4.2. 3

Let

5

E R and l e t s .

=

j

3'

2.3 be non-negative i n t e g e r s

. Then

the n o m s

(2.4.1) ( r e s p . (2.4.2)) and (2.4.5) (resp. (2.4.6)) a r e e q u i v a l e n t

uniformly w i t h r e s p e c t t o t h e parameters

E (O,E

0

I

x i ~ ~ - ' .

0

P r o o f . The norm ~

I 1 . I I (s)

d e f i n e d by ( 2 . 1 . 3 ) b e i n g e q u i v a l e n t t o

, € , < I

I I * / I (s), € , E l

uniformly with r e s p e c t t o

where t h e c o n s t a n t C does n o t depend on

( € , E l )

E,<'

one h a s :

and u .

Using t h e o p e r a t o r Q

d e f i n e d by (2.4.3), (2.4.4) w i t h N 2 s +s N 2 3'

one f i n d s : m -

where C d o e s n o t depend on

E,

5 ' and u.

Therefore, C m ( s + ) being dense i n H

0

Now w e a r e g o i n g t o i n t r o d u c e t h e s p a c e s H (S),E,c'(R+)' 3 V S E R .

" ( S ) , €

for

L e t K + b e t h e F o u r i e r t r a n s f o r m of t h e m u l t i p l i c a t i o n o p e r a t o r w i t h 'n t h e c h a r a c t e r i s t i c function of t h e halfspace

2. Sobolev Spaces of Vectorial Order

124

= I x = (x',xn)

For v E

s(nn),

I

x'

E

En-1,xn

' 01

one has

Indeed, with 6 > 0 one has:

If @,$

E

1

then L (En),

Hence, one finds: F ~ ( H+(x,~ ) exp (-6x ) F-l v) = (27ii)-'I (Sn-nn-i6)-lv(S',nn) dnn. n 5-t~ R

Further,denote: v+(5',cn+i6) and let V+

=

V(

)

-1

=

(27ii)-'J (Cn-nn-i6) V(C',nn)dnn R

+V (2) where

It is obvious that

On the other hand, one finds for V

(2).

.

2.4. Spaces on

One can rewrite

v( ' )

W$

125

as follows:

so that

=

V.P. 1 iSn-~n)-1v(5'.~n)d~n. IR

Denote by Il- the Fourier transform of the multiplication Operator With 'n the characteristic function u f the half-space IR: = { (x',xn) x' E IRn-i,

I

< 0).

x

Then for v E S(Rn) , one has:

and the same argument, as previously, shows that

Hence,

TI+ v + n '

n-

v

=

v,

v

v E S(IRn).

' n

Since the multiplication by H ( f x

z

2

n

)

is an orthogonal projection of -TI

L (XIn) onto the functions in L (Rn)with their support in R+ , + 2 n respectively, TI- are projectors of L ( R ) onto the subspaces of En functions in L 2 ( R n ) which can be extended analytically onto the half

plane: Im 5

En

+

< 0 and Im

Cn

> 0, respectively. For neromorphic functions

v(C',Cn) which have no singularities on IR, one has:

) a closed Jordan curve in the half-plane {Im t where ~ ( 5 ' is

> 0 )

which

2 . Sobolev Spaces of Vectorial Order

126

encloses all the singularitles of the function

nn

w(E',on).

+

Definition 2 .4.3.

A f a m i l y of d i s t r i b u t i o n s (o,Eo]

X

Rn-' 3 (E,'')

*

U(E,<')

E s'(R+)

,'

( R ) i f f o r each ( E ) t h e r e e z s t s an e x t e n s i o n i s s a i d t o belong t o H ( s ) ,E," + E S ' ( R ) of u t o JR such t h a t Lu E H(s),E,c,(R). The nomi i n H

&u

(s), € , < I (IR+)

i s given b y (1)

(2.4-7)

I IUI I (s), E , "

=

inf 9.

1 I L U ~1 (s), E , ' "

where t h e infimum i s taken over a l l e x t e n s i o n s Lu t o R. A f a m i l y (E,S') + u E S ' ( I R + ) i s s a i d t o belong t o

One can introduce on H (s)r E , "

(m

)

H(s),E,

(a+)i f

another norms:

+

These norms do not depend upon the choice of the extension Lu. In fact, if

-

L.u, j 3

=

1,2, are two different extensions of u to R ,then supp(LlU-L2u)~R-

Therefore, the function

being analytic for Im 5 Hence, H

(S)

E,E'

> 0, one has:

Il

;

v(s)

5 0,

v

s2' s3'

( R + ) is isomorphic €0 the factor-space

n ( s ), E , S ' (JR)/qs),E,5,. The projection II+ n '

2 being orthogonal in L ( R ' n

),

one has

The closed graph theorem implies that there exists a constant C > 0 such

2.4. Spaces on R: that uniformly with respect to ( E , C ' )

I l u l I ((1) S),E,c'

5 CI

We shall use

I 1. I I+

E ( 0 . ~ ~X 1IRn-l holds:

I I(1) I lull (.-) , < '

I U I /;s),E,F'r

127

5

lul

l;s),C,-

as working norms for establishing a priori

( S )

, € , E l

estimates for the s o l u t i o n s of coercive singular perturbations. Definition 2 . 4 . 4 .

A f a m i l y of d i s t r i b u t i o n s t h e space H

( s ) ,E

(BY)

En such t h a t LU E H

( 0 . ~ ~3 1E

+

u(E,.)

E S'(mn) i s s a i d t o belong t o

i f t h e r e e x i s t s an e x t e n s i o n ku E S '

( s ), E

( R ~ ) . The norm on H

( s ) ,E

(HI:)

(nn)o f

u to

i s g i v e n by

where t h e infimum i s taken over a l l t h e e x t e n s i o n s L, or e q u i v a l e n t l y ,

A family of distributions

belong t o

H ( s ) (R+)i

( o , E ~ I3

E + U(E,.)

E s ' ( R ~ )is s a i d t o

f

We shall denote by k o the extension by zero on Rn of functions : defined on R

.

The following result will be needed later on for establishing twosided a priori estimates for elliptic (coercive) pseudodifferential singular perturbations. Theorem 2.4.5. Let s E

n3 be such t h a t

Is2] <

f , lsz+s3j < f . Then the

extension operator .toi s a continuous l i n e a r mapping uniformZy w i t h respect to

E

from H

( S ), E

ko : H

(m:)into H

,

( s ) ,€

(Rn),

i . e . there e x i s t s a constant c > 0 such t h a t

Proof. Without restriction of generality one can assume that s = (0,s2,s3). Obviously, it suffices to show that

2 . Sobolev Spaces of Vectorial Order

128

(2.4.13)

I b0ul I

v

(S),E,<'

( € , < I )

E (0,ll

n-1 x R

I I Ts)

,

is defined by (2.4.8), L! 0 is the extension by zero: where I * I ,€,' I 1 u = = for x < O and where the constant C may depend only on s 2 and s 3 ' 0 n ( R ) be an extension of u E H ( s ) ,E," (W+) which Let !Lu E H (S)

re,"

satisfies the inequality (2.4.14)

I /lulI ( s ) ,E,5,

5 CI lul

I;s),E,su

with a constant C > 0 which does not depend on u,E,S'. Then one has:

one finds also:

Using the same argument as in [Ste], we are going to show that

5n where the constant C does not depend on u , s , < ' .

2.4. Spaces on

W;

129

Introducing

one can write: T(!Zu) = Kw, where

After the substitution q n = tSn, one finds: (Kw)( € , S ' , S , )

J kl(E,S',Sn,t)W(E,S',tSn)dt,

=

R

where kl(E,S',Sn,t) = Silk(E,<',Sn,tSn). Further, one finds using the triangle inequality:

Furthermore, one has:

Denote

Therefore, there exists a constant c

0

> 0

such that

2. Sobolev Spaces of Vectorial Order

130

1 s2 1

Since

< f,

I s 2+s3 1

< f , the last inequality yields:

B

and that proves (2.4.13).

It can be shown, but will not be done here, that c1 in the last inequality can be chosen as 1 6 ( ( f - ~ ~ ~ ) ) - ~ + ( f - l s ~ +(see s ~ ~ )[Fr-Hei]). -' Remark 2.4.6. With s = ( 0 , s 2 ,O),

Is2) < f , Theorem 2.4.5 has as a

consequence the continuity of the extension by zero .Lo in classical SobolevSlobodetsky spaces, i.e. Lo mapping for 1s21 <

: Hs

(R:)

B 2

t.

+

HS (Rn) is a continuous linear 2

The following result will be needed, as wel1,in the elliptic theory of pseudodifferential singular perturbations. U

E H(s)

,E,c,

( R + )tvith s2 2 0 , s 3 t 0 and let

$1. Then holds:

where t h e c o n s t a n t

Proof. Im

En

c may depend o n l y on def

Since Lou(S',S

< 0 , one has:

)

=

Fxn+Sn

s2,s3

and 6 .

9. u(S',x ) is analytic for

0

2.5. Dual Spaces

131

where

Using t h e s u b s t i t u t i o n ri

= tlIm

E n / , one g e t s t h e f o l l o w i n g estimate

f o r I(E,S): I(E,s)

I nI


26-2

-2 2 <,-t)) ( t + j I m ~ n l - 2 < < l > 2 ) 6 - f d 5t

I (1+(11mEnl-'Re R

Furthermore ,I one f i n d s :

H e r e , w e have u s e d t h e c o n t i n u i t y o f t h e e x t e n s i o n by z e r o to from

(R+) i n t o H

H(o,t-s,o) ,E,S'

(R) g i v e n t h a t

1 f-61

< f

(see

I

Remark 2 . 4 . 6 ) .

2.5.

(O,t-6,0) , € , S t

Dual s p a c e s Let H

fixed H

*

( R n ) b e t h e d u a l s p a c e of H ( I R n ) . For each E > 0 ( s ), E (s), E ( R n ) b e i n g isomorphic t o t h e H i l b e r t s p a c e w i t h t h e i n n e r

( s ) .E

product

__ -2s

(u,v)

(2.5.1)

( s ), E

=

( 2 ? ~ ) - * ~1 Rn

;(E,~)V(E,~)E

2s

<5> 2 s 2 < ~3d5 p

1

The R i e s z theorem s a y s t h a t any c o n t i n u o u s l i n e a r f u n c t i o n a l @ ( u )on H

(s), E

(2.5.1)

(Rn) and

i s g i v e n by a n e l e m e n t v

I

1

=

ilvli

(s), E

E

H

( s ),E

(IRn)

as an i n n e r product

= (v,v)'

(s), E '

Denote -2s

(2.5.2)

w

= 6

so t h a t w

E

H

2s '2s2<~i;> G,

(-s),E

and

2. Sobolev Spaces of Vectorial Order

132 for V u E H Hence,

(Rn), V w E H (2). (-s) , E (s , E ( 2 5 . 2 ) i s a n isomorphism between H*

u E

H

(mn)

( B n ) and H

( s ) ,E

t h e v a l u e of a f u n c t i o n a l w E H

a n d , moreover

(mn) on

,

(-s) E

(-s).E a function

( R n ) i s g i v e n by t h e f o r m u l a ( 2 . 5 . 3 ) . W e s h a l l u s e H

( s ), E

a s an isomorphic r e a l i z a t i o n o f H

*

(mn)

B e s i d e s , t h e form ( 2 . 5 . 3 )

(El").

( s ),E

(-S) ,E

c a n b e a l s o v i e d a s a c o n t i n u o u s e x t e n s i o n of t h e form -2n

J

(u,$) = ( 2 n )

;(~,E)$(e,c)d;, V u E H

mn

( R n ) , V @ E S(IRn).

( s ),€

Theorem 2.5.1.

Let

(i:s) *

isomorphic t o n O +

on u + E

H

(s).E

!L : H

)

* is

(-s), E

(my)

( ~ T ) - ~ " (IE , S ) ! L V ( E , S ) ~ S ,

=

Rn

where

(s). E t h e vaLue of a f u n c t i o n a l v E H

(my) and

(-s),E

w i t h s E m 3 . Then

,€

i s g i v e n by t h e formula:

(u+,v)

(2.5.4)

i:s)

be t h e dual of

,€)

+

i s any continuous Linear e x t e n s i o n

+ H(-s) ,E

(-s) , E

operator; b e s i d e s ( u + , v ) i s we22 d e f i n e d by (2.5.4), i . e .

t h e r i g h t hand

s i d e i n ( 2 . 5 . 4 ) does not depend on t h e e x t e n s i o n Lv of v t o Bn. ~

"-

Proof. F i r s t , n o t i c e t h a t i f u + H;s) , E and E -2n = 0 and, conversely, ( u + , v - ) = (2.rr) (u+,v-)

-

( 0 ) $E

and ( u + , v - ) = 0 f o r V u

+

E H

t h e n v- E H ( - s )

(S),E'

then

,€

if v

-

E

€3

t o t h e d e f i n i t i o n of t h e s u p p o r t of a d i s t r i b u t i o n , one h a s :

.

V $ E C:(Rn)

I

m

i s d e n s e i n H-

S i n c e C0(.(") -2n

(u,@)I S ( 2 n )

I lul I (s), € I 141 I ( - s )

one g e t s t h e c o n c l u s i o n t h a t ( u + , v ) C o n v e r s e l y , l e t v- E H Hence, i n p a r t i c u l a r , supp v-

5

-

.I!,

Let f E H (2.5.5)

(-S)

,E

HT-~),~.

(By), if

(u+,if)( 0 ) , E

=

E

H

(-S) ,E

(2lTY2"1 Rn

v

(IRn),

(u+,$)

=

0,

, € I

v

,€,

and l e t ( u + , v - )

(lRn)

(W")

and

0 , V u + E H;s)

=

(-s),E (v-,$) = (@, v -) = 0 ,

i . e . v- E

(-s) ,E

(-S),E

I n f a c t , according

,€'

=

v- E H-

(-s), E '

0 , V u+ E H+

m

(S)

$ E Co(my)

u+ E H;s)

,E'

, i.e.

,€.

Then t h e form

(E,<).e^f(E,c)d< +

d o e s n o t depend on t h e e x t e n s i o n k .

E H

I n f a c t , i f k,f

( - s ) ,E

(Rn) i s another extension, then ( i f - L l f ) E

H-

(-s), E

and ( u + , ! L f - k l f ) = 0 . A s a consequence of

1 (u+Jf) ( 0 ) , E l

2

( 2 . 5 . 5 ) one g e t s t h e estimate

1 / u + /1 ( S ) , E l

I

IQfI ( - s )

,E

and, s i n c e ( u + , l l f ) d o e s n o t

2.6. Sobolev Spaces of Vectorial Order on Manifolds depend on t h e e x t e n s i o n ,.P

133

one h a s :

T h e r e f o r e , any f E H (Ry) d e f i n e s a c o n t i n u o u s l i n e a r f u n c (-s), E a c c o r d i n g t o t h e formula ( 2 . 5 . 4 ) a n d , moreover, t h e r e i s t i o n a l on H+ (s), E ( R : ) and t h e one t o one c o r r e s p o n d e n c e between t h e e l e m e n t s i n H c o n t i n u o u s l i n e a r f u n c t i o n a l s on H

( S )

I n f a c t , d i f f e r e n t elements i n H ( s ), E l

i.e.

,E;1

(R ) +

(S),E

+

l i n e a r f u n c t i o n a l s on H

(-s), E

+

t h e e l e m e n t s of

(H

*

L , E )

.

d e f i n e d i f f e r e n t continuous

s i n c e t h e form ( 2 . 5 . 5 ) d o e s n o t depend on

F u r t h e r , l e t @ ( u ) b e a c o n t i n u o u s l i n e a r f u n c t i o n a l on

t h e e x t e n s i o n s 9,.

( R n ) i s a H i l b e r t s p a c e ( f o r any E f i x e d ) , w i t h (s), E ' (S),E (s), E t h e i n n e r p r o d u c t ( 2 . 5 . 1 ) a n d , a c c o r d i n g t o t h e R i e s z theorem t h e r e

H+

H+

C H

e x i s t s an element @ Let f0(E,c) fo

E

H

= E

E

H+

( 5 ) ,E

s u c h t h a t @ ( u + )= ( u + , $ + )(s)

2s2<E5,2s3

<<>

(Wn),

( - 5 ) ,E

( u + , f o ) ( 0 ) ,E

=

+ -2s 1

and

= f

T+fo

Il4I

=

-

$+(E,S), f0

= F

-

E H(-s),E my),@ ( u + )= ( u + , @ +(s) )

ll$+ll(s),E

=

v

u+EH+ ( s ), E '

-1 5+xfD. Then

l!foll(-s),E

2

=

l l f l l ( - s () 1, E ).

T h e r e f o r e , any c o n t i n u o u s l i n e a r f u n c t i o n a l on H+ can be d e f i n e d (s). E by ( 2 . 5 . 5 ) a n d , moreover, 2 ( '- )S ) , E . On t h e o t h e r hand, a s a (1) SUP consequence of ( 2 . 5 . 6 ) , one h a s : 1101 = 5 ( - s ) , E l (s) (1) sa t h a t = (-S),E'

11@1 I

1If11

I

1 I@ I

I If1 I

I n t h e same way one shows t h a t

(H

h+)I IIflI

I b+I/ *

+

(By)) = H ( S i , E ' same d u a l i t y p r o p e r t i e s have t h e f a m i l i e s of s p a c e s H (S)

H

( - 5 ) ,E,S'

Of c o u r s e , t h e

(-5) ,E

and , € , E l

(R-).

2.6. Sobolev s p a c e s o f v e c t o r i a l o r d e r on m a n i f o l d s We a r e g o i n g t o e x t e n d t h e d e f i n i t i o n of S o b o l e v t y p e s p a c e s o f v e c t o r i a l o r d e r t o t h e f u n c t i o n s d e f i n e d i n a bounded domain U C Wn m

with

m

C -boundary

a U and t o C - m a n i f o l d s w i t h o u t and w i t h boundary.

W e need two a u x i l i a r y r e s u l t s f o r d o i n g t h a t .

Let @

: En+

(2.6.1)

R n b e a Cm diffeomorphism,

and d e f i n e O * b y :

*

0 u ( x ) = (u''@)(x) = u ( @ ( x ) ) .

Assume t h a t 0 i s l i n e a r o u t s i d e some compact s e t i n R n .

Then t h e

2 . Sobolev Spaces of Vectorial Order

134

*

( R n ) with s . j = 2 , 3 non1' n e g a t i v e i n t e g e r , i s a c o n t i n u o u s l i n e a r mapping u n i f o r m l y w i t h r e s p e c t t o chain r u l e implies: @

E

: H

( s ) ,E

E ( O , E ~ ] . The d u a l i t y y i e l d s t h e same r e s u l t f o r s .

j = 2 , 3 non-positive

3'

integer. Lemma 2 . 6 . 1 .

*

( m n ) + H ( ~ , )€ ( I R ~ )d e f i n e d by ( 2 . 6 . 1 ) w i t h O(x) l i n e a r o u t s i d e a compact s e t i n mn, i s a continuous l i n e a r mapping u n i f o r m l y 3 w i t h r e s p e c t t o E E ( 0 . ~ ~f o1r V s E R . The map Q

: H

(s),E

Lemma 2 . 6 . 2 .

Let

C$ E s ( I R ~ ) .Then

i s a continuous l i n e a r map uniformly with r e s p e c t t o

E.

These two l e m m a s w i l l b e p r o v e d ( i n a more g e n e r a l s i t u a t i o n ) i n t h e Chapter 3 ) . Now u s i n g Lemma 2 . 6 . 1 ,

2 . 6 . 2 and t h e p a r t i t i o n of u n i t y t e c h n i q u e , w e c a n

d e f i n e Sobolev s p a c e s of v e c t o r i a l o r d e r f o r f u n c t l o n s on bounded domain on smooth m a n i f o l d s w i t h smooth b o u n d a r i e s . If U

C lRn

i s a bounded domain w i t h Cm-boundary a U one d e f i n e s

(U) a s t h e f a c t o r spaces:

H ( s ),E

0 -

( R n ) such t h a t i s t h e c l o s e d s u b s p a c e of a l l u E H (s),E ( s ) .E SUPP u c n n \u.

where H

The norms

1 . 1 I ( s ), E , U

a r e d e f i n e d by t h e induced t o p o l o g y of a f a c t o r

space :

where t h e infinimum i s t a k e n o v e r a l l e x t e n s i o n s Ru of u t o Rn. One can g i v e an e q u i v a l e n t d e f i n i t i o n o f norms and s p a c e s

H

( s ),E

(U) u s i n g t h e p a r t i t i o n of u n i t y .

Let ) , Q ' ( l < e s r b e a c o v e r i n g of t h e r e e x i s t diffeomorphisms

au

by open ( i n R n )

s e t s , such t h a t

~ ~ : u ~ n u +i sv p .~ z ,r where

Vp. 1s a s u b s e t of R n .

F u r t h e r , it i s assumed t h a t

x a. ( a u n

n-1

U ) c IR

p.

x

{O}.

L e t UO b e an

2.6. Sobolev Spaces of Vectorial Order.on Manifolds

CI

open s u b s e t of U such t h a t Uo C U ,

U

O
'

2

U and l e t

135

( $ 9 . ) o =-

-

<Er

be a

p a r t i t i o n of u n i t y s u b o r d i n a t e t o t h e c o v e r i n g (U.9 ) O59.6r of u . -1 F u r t h e r l e t 9. ( ( u . $ ~ )' X9. ) : I R y + C and L 0 ( u . $ 1 : R n + C be t h e e x t e n s i o n

0

0

by z e r o of t h e f u n c t i o n s (u.$ )

O

X-l and u.Q0 d e f i n e d on t h e s e t s V9. and

9 . 2 ,

Uo,

respectively.

D e f i n i t i o n 2.6.1.

For a f a m i l y of d i s t r i b u t i o n s ( O , E ~ ] 3

E +

u(E,.)

3

E 0'U.J) and f o r s E IR ,

let 2

I IuI I (s)

(2-6.3)

= €1'

and l e t

H ( ~ (u) ) be

u f o r which ( 2 . 6 . 3

1

1srsi

( I / & o ( ( u . $ 9 . ) X i-1 ) I I +( S ) , E ) 2 + I

19.0(~.$o)

I I 2( S ) , E

t h e f a m i l y of spaces of t h e ( f a m i l i e s of) d i s t r i b u t i o n s

is finite, V

E

E

( 0 . ~ ~ 1 .

Introducing

J u i 1 (s), E , U I I. I j

with

( s ) ,E

d e f i n e d by

f a m i l i e s of d i s t r i b u t i o n s u

2 . 6 . 3 ) , we denote by H ( s )

( u ) t h e spaces of t h e

f o r which ( 2 . 6 . 4 ) i s f i n i t e .

Remark 2 . 6 . 2 . Two d i f f e r e n t c o v e r i n g s and c o r r e s p o n d i n g p a r t i t i o n s of u n i t y y i e l d an e q u i v a l e n t f a m i l y of norms H

(U),

(s)

Remark . _

I I . I I (s), E , U

and t h e same spaces H

a s a consequence of Lemmas 2 . 6 . 1 ,

( s ) ,E

(U),

2.6.2.

2.6.3.

D e f i n i t i o n 2 . 6 . 1 e x t e n d s immediately t o f a m i l i e s o f d i s t r i b u t i o n s on a Cm-rnanifold M w i t h a cm-boundary aM. I n f a c t , one h a s t o t a k e s e v e r a l open

sets i n o r d e r t o c o v e r t h e i n t e r i o r of M . L e t rO d e n o t e t h e r e s t r i c t i o n o p e r a t o r t o t h e h y p e r p l a n e R n - l x

{O}.

D e f i n i t i o n 2.6.4.

For a family of distributions ( O , E ~ ] 3

E

+

$(E,.)

E U'(aU) and for s E R 3 ,

let

and l e t H

( s ),€

( a u ) be t h e f a m i l y of spaces of t h e ( f a m i l i e s of) d i s t r i b u -

t i o n s Q which have t h e norms ( 2 . 6 . 5 ) f i n i t e , V Introducing

E

E (0.~~1.

2 . Sobolev Spaces of Vectorial Order

136 we denote by H

(s)

(aU) the Banach space o f the ( f a m i l i e s o f ) d i s t r i b u t i o n s

w i t h the norm (2.6.6) f i n i t e . Again, the norms [. 1 dependonapartitionoftheunity, ( s ) ,E,au' [ - l ( s ),au but are uniformlyequivalent with respect to E E ( 0 , ~1 and define the 0 same topology and the same spaces.

au

In Definition 2.6.4 one can take as

m

the boundary of any C -manifold and

not necessarily the boundary of a domain in IRn Definition 2.6.5.

A family of d i s t r i b u t i o n s ( 0 . ~ ~3 1E belong t o the spaces ( 0 . ~ ~3 1E

supp u

+

c u, v

( s ) ,E

E H

u(E,.)

(S)

+ u(E,.)

( u ) , i f supp u

E

,€

(mn) i s said t o

5 u.A family

(w") i s said t o belong t o the space

(u), i f

( S )

E (O,EO1.

E

All the results in the Sections 2.1 - 2.5 extend in a natural and obvious way to the spaces of vectorial order on a smooth manifold or in a bounded domain U

C

XIn. We shall illustrate this by giving a detailed proof of the

version of Theorem 2.4.5 for spaces H m

( s ) ,E

(U)

with U C R n

a bounded domain

with a C -boundary a U . Theorem 2.6.6.

Let

u

s =

(sl,s2,s3),s a t i s f y the condition:

c

mn be a bounded domain wCth a cm-boundary

(2.6.7)

3,

/s21 <

3u. Let s E R

3

,

Is2+s3\ < f .

Then t h e continuation by zero Lo i s a continuous linear mapping uniformly w i t h respect t o Lo

(2.6.8)

:

H

e

E' ( 0 , l l from

H

( s ) ,E

,E ( s ) ,E (u) + H ( s )

(U) i n t o H

( s ) ,E

(XIn):

(Rn).

x 3. ' lSjSr, orientation x. E Cm(u. n u), 1 5 j 5 r, x.73 ( U3.n aU) C R n-1 { O } .

__ Proof. Let {Uj]lljSr be an open covering of aU and preserving diffeomorphisms, xj where V

1'

n

V. 7' lSjSr, are subsets of mf such that :

U.

U

+

<,
7

3

can always be extended as diffeomorphic mappings of Rn into itself which coincide with the identity outside a ball with sufficiently large diameter (cf. [Pal. 1

I),

so that supp(X.-Id) is compact. Let

set such that {Uj)05j5r

cover

Ti

3

UoCC

U be an open

and let {$j}05jsr be a C"-partition

Of

137

2.6. Sobolev Spaces of Vectorial Order on Manifolds unity subordinate to the covering {Uj}05jsr.

(U) and denote u : By + C the continuation by zero of Let u E H ( s ) ,E j -1 (u.$.) ,,xj : V . C, 1Sj6r. Furthermore, let uo = u$,. 3 7 Definition 2.6.3 yields: -+

L'u O j

Denote by L'u the extension by zero of u . O j I : IRn + c. Using the triangle-inequality in H

( s ) ,E

:

By

-f

C onto IRn

,

i.e.

(U) r times and after that

Lemma 2.6.1, one finds:

where the constant C may depend only on s,r and

x 3.'

15j5r* Thus, by Theorem 2.4.5 with (2.6.7) being valid, one finds using (2.4.12)

(with L ' instead of II ) and (2.6.10): 0 0

where the constant C does not depend on u and Theorem 2.6.6.

E.

This ends the proof of

I

Example 2.6.7. tx E B2 ,1x1 < 1 1 and denote T1 = a B , 8 E T 1 being the angle 1 coordinate. It is readily seen that g E H (T ) iff its Fourier coef-

Let B

=

E

(s) ,E

ficients gE (n),

satisfy the condition:

With I (x) = J (ix) the Bessel function of the imaginary argument, consider the function u (2.6.13)

:

B

+

C

given by the series:

u E ( x ) = (2~)-'E gE(n)I

nEZ

In1

(rc-1 )(I

(c-'))-'exp(in@), In1

r = ixi, e=arg(x +ix ) . 1 2 With gE E H ( s ) ,€(T1),s2+s3 > 0, uE(x) given by (2.6.13) is the (continuous on B)solution of the boundary value problem:

2. Sobolev Spaces of Vectorial Order

138

2 (1-E A)u (x) = 0 , X E B , u (cos 8 ,sin 8 ) = g

I),

8 E aB.

) as X = E-' In1 one can write uE(x) = vE(x)+ wc(x), where

using the asymptotic expansion for I [Grad.-Ryz., 1

(e),

-1 (rE

+ m

(see

vE(x) = X(r) (~n)-' c gE(n) exp(-E-'<En>(l-r)+in8) nEZ

x

with

E Cm([O,l]),

x

:0 for r E

[0,%1,

x

E 1 for r

E [f,11, and where

w (x) has already a better regularity (uniformly with respect to than vE(x). Furthermore, the regularity of v 0

one of vE (p,t)

where

:

B

+

E

E (0,11)

C is the same, as the

R+ x W - t C, given by:

I

0 = vE(p,t)

(2.6.14)

:

9E(S)exp(-E-1p<Eg>+itS)dS, p E R+

,

tER,

R is defined by (2.6.12) with n replaced by 5 E W .

;,(C)

Using (2.4.10) one gets: vo E H

s,

E H(s),E(R1),

gE

H(S),E(T

i.e. u

E

E H

,P+

with s +s > 0 iff 2 3 (s1+t,s2,s3+f) ) ( B ) with s2+s3 > 0 iff

(s1+f,s2,s3+f),E

) -

2.7. Spaces of one parameter families of meshfunctions Discrete analogues of classical Sobolev-Slobodetski spaces H

are introduced

here, whose elements are meshfunctions on one parameter families of grids in Rn or Ry, the parameter h E (O,ho] being the meshsize. These spaces are suitable for the investigation of corresponding classes of difference operators (see ch. 111, 3.11). For meshfunctions on the grid

$=

hZn in Bn , h E (0,h 1 introduce 0

the inner products

and the norms

the family of spaces of the meshfunctions u for which (g) the norms (2.7.2) are finite, h E (O,ho].

Denote by

H

(0),h

V

As a consequence of Parceval's identity, (2.7.2) is equivalent with

the norms (2.7.3) -h where u ( 5

u is the discrete Fourier transform. Fx+S, h Let (sl,s2 E lR2 and let 5 = ( C l , ...,Cn) with 5 k = (ih)-'(exp(ihEk)-l). =

2.7. Spaces of One Parameter Families of Meshfunctions

139

Definition 2.7.1.

For a f a m i l y o f meshfunctions (O,ho] 3 h

-f

u(h,x) C

S'

h

(Rt)

d e f i n e t h e norms of order s

I

(2.7-4)

IUI

s2 I (s),h 1 Ih-sl <5> (Fx+g,hu)(h,hS)11

.

=

L (T

5 rh

)

(37;) if its norms(2.7.4) a r e f i n i t e V h E (O,ho]. We say t h a t u E /f (s),h The meshfunction u € H (2") i f one has ( S )

Obviously,

H ( s ),h inner products:

(IRE)

is a family of Hilbert spaces equipped with the

For s2 non-negative integer one can also introduce the norms

the norms (2.7.4) and (2.7.7) being in this case equivalent uniformly with respect to the parameter

h E (O,ho], as a consequence of Parceval's

identity and the inequalities:

where the constant C depends only on its subscript s2 Proposition 2.7.2. (s)

(2") is a Banach space.

Proof. Indeed, rewriting (2.7.4) in the form ~

n with Tn = Tn 11

n,l

,one gets the conclusion that the operator

u + h

-(s +s +n/2) 1 2

I

(h2+ w ( r i )

1 2)

Fx+ri, 1u

2 n (2") isometrically onto the Banach space B((O,h 1;L (T ) (s) 0 n bounded on f 0.h 1 functions valued in L2(Tn). (I maps H

0

)

of

2 . Sobolev Spaces of Vectorial Order

140 Example 2 . 7 . 3 .

For t h e f a m i l y of m e s h f u n c t i o n s (2.7.10)

u(h,x) = h

-a

exp(-h

-1

1 lxj-/)t lSj5n

one f i n d s e a s i l y i t s d i s c r e t e F o u r i e r t r a n s f o r m :

TI

Ch(h,hC) = h n - a ( l - e - 2 ) n

cos h C . + e - 2 ) - 1 .

(1-2e-1

3

l<jln T h e r e f o r e , t h e meshfunction and b e l o n g s t o h

H

(s ) n - 2 ( o + s +s 1 2

( 2 . 7 . 1 0 ) b e l o n g s t o any

,v h

H (s), h

>

0,

iff

(Z")

JTn ( h 2 + 1 u ( r l )I

)

s2

n

16j5n

(1-2e-1c0srl,+e 3

-2 -1 )

dn
V h E (O,hol, where C

s ,ho

depends o n l y on i t s s u b s c r i p t s .

The l a t t e r h o l d s i f f h

n - 2 ( a + s +s ) 1 2 J

2 ( h + l w ( n ) 1 2 ) s 2 d n 5 C s , h o ~ V h E (O,ho],

Tn

w i t h some c o n s t a n t C

S,ho

.

I t is e a s i l y seen t h a t f o r h

0

-f

one h a s :

O(1)

( h 2 + / w ( q )/ 2 ) s 2 d q

=

Tn 17

{

O ( l n h - l ) , when s 2 n+2s2 O(h ),

so t h a t t h e f u n c t i o n ( 2 . 7 . 1 0 ) b e l o n g s t o

s +s2 5 ( n / 2 ) - o , o r s 1

= -n/2,

F o r a f u n c t i o n u E C o ( Xn )

s

when s 2 > -n/2

I

=

-n/2

when s2 < -n/2

H ( S ) (Z")

i f f s 2 > -(n/2),

< -(n/2),

s1 5 n-a. < n-a, o r s 1 2 d e n o t e by ITh u i t s r e s t r i c t i o n t o

$.

AS

usual,Hs ( R n ) s t a n d s f o r t h e c l a s s i c a l Sobolev-Slobodetski space of o r d e r s E R.

p r o p o s i t i o n 2.7.4.

If

u E

H s ( ~ n ) w i t h s > 4 2 , then n hu E H ( 0 , t ) (z"), v t

over, t h e following estimates hold:

I

'hUI

I (0, t ), h

5 IIu/lt

(2.7.11)

I lu/ I t

5 [/.a hu

'(0,t)

'

s-t +

Cs,th

Ilul

Is)

< s-n/2

and, more-

2.7. Spaces of One Parameter Families of Meshfundions

fcii:

V h . E ( 0 , h 1, where t h e c o n s t a n t c depends only 0 s,t

on i t s s u b s c r i p t s .

P r o_ o f . I t s u f f i c e s t o c o n s i d e r t h e c a s e , when u E S(IRn) _ i s dense i n

H

(IRn)

,

W e have used t h e f a c t t h a t 1

I<

(h,h<)

,

since S(Rn)

( 1 . 3 . 1 1 ) , t h e Cauchy-Schwarz

V s . The P o i s s o n formula

i n e q u a l i t y and t h e o b v i o u s estimate

141

I

< m if

6

151,

V h E R

,yj.eld:

s-t > n/2.

a Thus, t h e f i r s t of t h e i n e q u a l i t i e s NOW

(2.7.11) i s proved.

we prove t h e second cane. The s a x e argument a s p r e v l c u s l y shows t h a t

where C

depends o n l y on i t s s u b s c r i p t s . S,t

S i n c e t h e l e f t hand s i d e o f t h e l a s t i n e q u a l i t y d o e s n o t depend on h , one f i n d s :

I f one s u b j e c t s a f u n c t i o n u E

ff ( R n ) t o a smoothing o p e r a t i o n by

means of c o n v o l u t i o n w i t h a f a m i l y o f smooth d e l t a - s h a p e d k e r n e l s ( t h e f a m i l y c o n v e r g e s t o 6 ( x ) when h RE

of t h e f u n c t i o n

belong t o

H

op_ osition -P r_

(Z").

(0,s)

U,

+

O), t h e n t h e p r o j e c t i o n s o n t o t h e g r i d s

r e g u l a r i z e d by a s p e c i f i c c h o i c e of k e r n e l s , w i l l More p r e c i s e l y , t h e f o l l o w i n g r e s u l t i s v a l i d .

2.7.5

L e t u E Hs(xn) and l e t s' be any nonnegative numheu. Then t h e r e e x i s t s

2 . Sobolev Spaces of Vectorial Order

142

a f a m i l y of f u n c t i o n s v

,v

1". v E Hs+,(wn)

E yo,s) ( ) " '

20.

71 hv

3 O .

I l ~ - - v IIs-s'

h

=

E

and

v(h,x) s a t i s f y i n g t h e f o l l o w i n g c o n d i t i o n s : t ~ P h ~ < ~ 6 ~Ctlj~lls; ~ 0 'hV/] ( O , s ) 1 ;

/Iv]I~+~

R + , V t 2 0 and s

'

[I

5 Cs,hS'I IUI

Is

IuI Is

'

where ct and cS, depend o n l y on t h e i r s u b s c r i p t s . Proof. -

m

Let 4 ( 5 ) E C ( R n ) satisfy the conditions 0 0 5 (2.7.13)

aa$

5 0

(0) = 6

On

r V a with la1 5 [s'l+l,

where dOa is the Kronecker symbol and [s'] stands for the integral part of s'

-1 2 0. Introduce v(h,x) = FS+x $~~(hg)F~,~u.

Since supp 4

has the diameter 6 ch-',

0

v(h,x) has the property 1";

indeed, h2t

Property 3" is an immediate consequence of (2.7.13):

Il~--vIIs-s' 2

2 2 <5>2(s-s') ll-40(hS)I l u ( 5 ) I d5 6

-

151

2s'

-

2

I u ( E ) I d5

2 2s' 2 ' Cs,h I lulls.

Finally, 2" can be easily derived from Proposition 2.7.4. In fact, for each m

(since $o E C (W") 0 fore, for any t > n/2 the inequality

h > 0 one has v E S(IR:)

I I'hvl

and, thus,

E

S(R9) ) .

There-

t

( 0 , s ) ,h 5

I IVI Is+Cth I IVI I s + t

holds uniformly with respect to h E (O,hO1, V ho > 0. Using 1' and the last inequality one gets 2". Proposition 2.7.6.

L e t u E H ( o , s ) (z"). Then t h e r e e x i s t s an e x t e n s i o n 1 u of t h e meshfunction h

u o n t o IR"

such t h a t for any t b 0 holds:

(2.7*14) SuPO
htl

llhUl /s+t

' 't[

lul

(O,s).

2.7. Spaces of One Parameter Families of Meshfunctions

143

Proof. Introduce v(h,x)

(lhu)(x)

=

-1 (F5-x XTn

=

F

x+S,h

5 .h is the characteristic function of Tn

xn..

where

def

u) (x), x E IRn,

5 rh

'5 ,h

1,

Noticing that 151 5 ( n / 2 ) 15 (h,hS)

V h > 0,

I

={5ERn

v 5

] h S j / < lZjSn}. ~,

E Tn

5 th

,one finds

immediately that h2tl IvI

2 =

In

h

2t

2(s+t) /;h(E)12d5 Z

<5>

T

E, ,h

5 c2

t

2

<5> 2s;1 h ( 5 ) ]2d5 5 Ct[lu/l(o,s). 2

In

I

T

5 th y f R , k E Z+, the difference analogue of the

k n Denote by C ( 2 ) , k n spaces c ( w ) of meshfunctions u(h,x) equipped by the norm (2.7.15)

IUl

sup (kty)= O
Let u E

H

IuI

(kty),h

lul

(kry),h

=h-'

1.

sup 1 /DZ,hu(h,x) xcRn / a / s k

h

k n (Zn) w i t h s > n/2. Then u E C ( Z ) , V k < s-n/2. Y (Y,S)

Proof. _ _ One has: Dau(h,x)

(2n)-" In

=

u (S)dS.

ei<x15'

T

5 .h Hence, for l a / 5 k,the Cauchy-Schwarz inequality and the estimate

151

5

(n/2)/51,V h 2 0, V 5 E Tn

5 rh

l2

IDzu(h,x)

<5>

5 C

yield:

-2 (s-k) d5

2 (s-k) I;h(5)12d5 lta/2<5>

I Tn

Tn

5 rh

5 .h

and that proves Proposition 2.7.7. Remark 2.7.8. Proposition 2.7.6 is no longer true when k

=

s-n/2. In fact, the mesh-

function uo(h,x)

=

-1 F (<5>'ln<5>)-1, 5+x,h

obviously belongs to

H ( o , f ) ( 21 ) ,

n

=

but uo(h,O)

1, +

m

when h

+

0.

2

2. Sobolev Spaces of Vectorial Order

144 D e f i n i t i o n 2.7.9.

The f a m i l y o f norms

1 1 . I I (s),h,

h E (0,h0IJiss a i d t o be s t r o n g e r than

~ ~ ~ ~ ~ ( is f t t h ) e,r eh e3x i s t s a c o n s t a n t C, which might depend hO, such

(2.7'16)

' '

that

1 l u l 1 ( s # ), h

' '1

l u / 1 (s),h' V

h E (O,hO], V u E H ( s ) , h ( < )

1 1 . I I (s'), h < 1 1 . 1 1 (s), h 11.1 1 ( ~ ) , h a r e stronger than 1 1 . I I ( s ),h (s'),h-

we w r i t e

I

on s,s' and

Or

.

/ 1 * 1 / ( ~ 8 ) ,i hf t h e norms

as

The f o l l o w i n g r e s u l t c a n b e proved p r e c i s e l y i n t h e same way P r o p o s i t i o n 2.1.8. P r o p o s i t i o n 2.7.10.

One has (2.7.17)

I 1 . 1 I (s'),h <

' ' -I

(s), h

iff

s; 2 s l , s'+s' S s +s 1 2 1 2 '

W e s a y t h a t s'

<

s if

(2.7.17)

holds.

The same argument as i n t h e p r o o f o f P r o p o s i t i o n 2.1.10,

c a n b e used i n

o r d e r t o prove t h e following P r o p o s i t i o n 2.7.11. 2 L e t o E R , s E x2 and

let

Then for V 6 > 0 h o l d s :

V h E (O,hol, V V E f f ( u ) , h ( < )

.

For a g i v e n s E R 2 d e n o t e (2.7.20)

vs

=

is'

I s'ER2,

5'

< sl, v;

=

{s'

I

s'E

m2,

s'>

sj,

t h e c o n i c s e t ( s e c t o r ) V* b e i n g c a l l e d , a s p r e v i o u s l y , r e c i p r o c a l t o V C o r o l l a r y 2.7.12. 2

L e t s E IR m d Zet a E i n t

.:v

Then for V

c o n s t a n t c which may depend only on G,T,U,S

T

.

E vs, V 6 > 0 t h e r e e x i s t s a

and ho, such t h a t

2.7. Spaces of One Parameter Families of Meshfundions

I n f a c t , t h e s a m e argument as i n C o r o l l a r y 2.1.12,

145

leads t o (2.7.21).

With 2 . V , j = 1 , 2 , d e f i n e d a s p r e v i o u s l y i n ( 2 . 1 . 1 5 ) ( w i t h s = O ) , and 3 * I S 2 . V b e i n g t h e r e c i p r o c a l h a l f l i n e s , one h a s a l s o t h e f o l l o w i n g

1 s

C o r o l l a r y 2.7.13.

L e t s E n2 and l e t o E i n t a . v * . Then f o r V 1 s

T

E

a 1. vs t h e r e e x i s t s a

c o n s t a n t c such t h a t ( 2 . 7 . 2 1 ) h o l d s . The f o l l o w i n g r e s u l t t u r n s o u t t o b e u s e f u l f o r e s t a b l i s h i n g v a r i o u s k i n d s of a p r i o r i e s t i m a t e s f o r d i f f e r e n c e o p e r a t o r s ( s e e a l s o Theorem 3 . 1 1 . 1 5 ) . Theorem 2.7.14.

Let

@

E S ( Z n ) . Then for V s E R

(2.7.22)

1 l @ u lI

(O,s) , h

where t h e c o n s t a n t

C

'

,

V h E (O,hOl one has:

I 1 IuI

max XERn

s,@,ho

I(0,s), h +C s , $ , h O

'

IUI

(0,s-1) ,h

depends o n l y on i t s s u b s c r i p t s .

Proof. By d e f i n i t i o n , one h a s

where
x ,h

>' d e n o t e s t h e o p e r a t o r a c t i n g on a meshfunction v a c c o r d i n g t o

t h e formula

= [
W e s h a l l show t h a t f o r t h e commutant Bh S,@

holds

x rh

>',@I

:

h

1 IBs,$ul

( 0 ) ,h

'

's,@,h0l

lUl

(0,s-1) ,h

'

where t h e c o n s t a n t C may depend o n l y on i t s s u b s c r i p t s . One h a s

:

v(h,E)

def =

Fx+:,h

h Bs,$U

=

with

'

2. Sobolev Spaces of Vectorial Order

146

We estimate the integrand. For doing that, we notice that a routine computation shows that €or any point R = ( Q l , R ) on the unit circle 2 2 2 Q1+R2 = 1 the following inequality holds for V p E I R :

with C

=

max{l,Ip/}.

P

Since both sides of (2.7.24) are homoqeneous in R of order zero, (2.7.24) holds for V Q E n L \ { 0 1 . Applying (2.7.24) with

Q1 = <5(h,hS)>2,

one obtains

Using the identity

which along with (2.7.25) yields the estimate:

Indeed,

n2

=

<<(h,hr~)>~and

p =

s/2,

2.7. Spaces of One Parameter Families of Meshfundions

147

Further, since I$ E S(Zn), one can apply(l.3.10)with k = 0, so that (2.7.29)

sup O
Therefore, using (2.7.28), (2.7.29), one gets the following estimate:

Taking j

=

1 s I+2+n and

denoting

one gets immediately the conclusion that the family of the convolution operators Gh with the kernel Gh, (2.7.30)

(GhGh)(5) =

I

-h -h g ( < - n ) v (ri)dri,

Tn

ri ,h

is uniformly bounded with respect to h E (O,ho] in the spaces

ff

(O),h(%) . -1 Indeed, after the inverse Fourier transform F S-fxjh, (2.7.30) becomes -h so that the multiplication operator v + gv, where g(h,x) = F<+x,h '

where the constant C

depends only of n.

The latter implies that with some constant C not depending upon h the inequality

2. Sobolev Spaces of Vectorial Order

148

is s a t i s f i e d .

Using t h e o b v i o u s i n e q u a l i t y

0

one g e t s (2.7.22). c o r o l l a r y 2.7.15.

A s a consequence of

(2.7.22) t h e map u

continuous Linear map from H

(S)

+

$u f o r each '$ E S ( Z n ) is a

(z") i n t o H

(S)

(z"), v

s = (sl,s2)E

R

2

.

W e t u r n now t o t h e s t u d y o f t r a c e s of m e s h f u n c t i o n s on h y p e r p l a n e s of

codimension one,which c a n b e g i v e n i n t h e form Il w i t h some y E Z n , y

=

(yl

, _ . ,.y n ) ,

Y

= {x

E Rn

,

< y , x > = 01

yk b e i n g i n t e g e r w i t h 1 a s t h e i r

g r e a t e s t common d i v i s o r ,

(2.7.32)

{y, ,...,y

} = 1.

F i r s t , w e prove t h e following Lemma 2.7.16.

Let y ( l )

= (yll

,...,y l n )

E zn s a t i s f y t h e c o n d i t i o n (2.7.32). Then it can

be compZeted b y v e c t o r s y ( j ) way t h a t t h e m a t r i x y

= (yjl

= (y.

,..., y I,n )

jk l < j , k 5 n

belong t o S L ( n ; Z ) , i . e . d e t y

=

having

E zn, 2 5 j 5 n , i n such a y ( J ) , 1 5 j 5 n,

as i t s lines,

fl.

Proof. If y ' l )

=

setting y

e (kf

, { e l , ...,e

} b e i n g t h e s t a n d a r d o r t h o n o r m a l b a s i s i n lRn

,

then

1 5 k 5 n , one g e t s t h e i d e n t i t y m a t r i x y = I d E S L ( n ; Z ) .

= ek Now l e t y ' l ) / ek, 1 5 I( 5 n , s a t i s f y (2.7.32). I t i s immediate t h a t

(2.7.32) i s n e c e s s a r y f o r t h e e x i s t e n c e o f y E S L ( n ; Z ) h a v i n g y l i n e . ~ i r s t n, o t e t h a t if y ' l ) f ( l ) = ( f l l ,..., f l n ) E Z n ,

y").

= .f(l)

with s o m e a

as i t s

E SL(n;Z) and Some

t h e n f ( l ) s a t i s f i e s (2.7.32) i f f so i t i s f o r

,...,f I n } = d w i t h d E Z , ( d l > I , t h e 11 = d and v i c e - v e r s a , s i n c e = a f ( l ) y i e l d s { y l l , ...,y

I n d e e d , assuming t h a t { f

equality y ' l )

SL(n;Z) i s a group. Therefore,

if one s u c c e e d s t o complete f ( ' )

by v e c t o r s f

(2)

, . . ., f ( n )

i n such a way t h a t t h e m a t r i x f = ( f . ) h a v i n g f ( 1 ) a s i t s l i n e s w i l l b e l o n g t o lk S L ( n ; Z ) , t h e n t h e m a t r i x y = f a t w i l l s a t i s f y t h e r e q u i r e m e n t o f t h e lemma.

2.7. Spaces of O n e Parameter Families of Meshfunctions Hence, it i s s u f f i c i e n t t o show t h a t f o r a g i v e n y " )

=

149

( y l l , ...,y l n ) E Zn

s a t i s f y i n g ( 2 . 7 . 3 2 ) t h e r e e x i s t s a m a t r i x a E SL(n;Zn) s u c h t h a t = a e . Denote by y # 0 t h e c o o r d i n a t e o f y ( l ) w i t h t h e minimal 1 1, k o a b s o l u t e v a l u e ( i f t h e r e i s more t h a n one c o o r d i n a t e w i t h t h i s p r o p e r t y ,

y(l)

w e t a k e anyone o f t h e m ) . L e t y y"),

Irjo

,

j,

# k o , b e any o t h e r c o o r d i n a t e of

= with integer p . , r . , t h e 1, l o 'j,Y1 r k o + r j o lo lo s a t i s f y i n g t h e i n e q u a l i t y lr , < I Y l , k o ' . Introducing

t h e n one c a n w r i t e y

remainder r .

I

JO

'0

a y ' l ) w i t h a = ( a . ) such t h a t a , , = 1 , Ik I1 1 5 j 5 n , a . = 0 f o r j # k , j # j, o r k # ko and a . = - p . , so Jk 10'kO Jo t h a t d e t a = 1 and a E S L ( n ; Z ) . it i s e a s i l y seen t h a t f ( l )

=

C o n t i n u i n g t h i s p r o c e s s (which i s p r e c i s e l y t h e E u c l i d e a n a l g o r i t h m f o r t h e c o m p u t a t i o n o f t h e g r e a t e s t common d i v i s o r o f i n t e g e r s y l l , ...,y l n ) , one g e t s f i n a l l y a l i n e c o i n c i d i n g w i t h some f e k , 1 5 k 6 n . B e s i d e s , e a c h s t e p o f t h i s p r o c e s s r e p r e s e n t s t h e p a s s a g e from some v e c t o r f E Zn w i t h t h e property (2.7.32)

t o a n o t h e r one f ' E Zn w i t h t h e same p r o p e r t y , f '

w i t h some a E S L ( n ; Z ) . F i n a l l y , it i s q u i t e o b v i o u s t h a t e l

o b t a i n e d from +e i f o n e a p p l i e s t o +e a n a p p r o p r i a t e CY E S L ( n ; Z ) . k k A s a consequence o f Lemma 2 . 7 . 1 6 ,

= af

can be

I

one c a n w i t h o u t r e s t r i c t i o n o f

g e n e r a l i t y , c o n s i d e r t h e t r a c e s o f m e s h f u n c t i o n s on t h e h y p e r p l a n e

II

=

{x

=

01.

AS

usual x

Dznote by (Sh(R;;l a n d , i n t h e s a m e way,by

th

=

) )

(x*,xn), 5 = (6',cn), 5 = ( c ' , ~ , ) . k t h e d i r e c t p r o d u c t o f k c o p i e s o f Sh (IRn-l x',h)'

II l5jSn

'(s.,h) 1

(IRn

x,h

t h e d i r e c t p r o d u c t of s p a c e s

ff(s,),h,

1 6 j 5 k. 1 Theorem 2.7.17.

L e t k > 0 be i n t e g e r . F o r s2> k+f t h e one parameter f a m i l y of mappings (2.7.33)

Hts)

, h (an ) 3 u ( x ' , x n ) x,h

{Dl h u ( x ' , 0 ) j 0 5 j 5 k E x- 1 n-1 0 5 j 5 k '(s1,s2-j-f) , h ( m x ' , h ) -f

is, uniformly w i t h r e s p e c t t o t h e parameter h E [ O , h o ] , a f a m i l y of continuuus l i n e a r mappings. For any g i v e n k s,-f t h e one parameter f a m i l y of mappings

2. Sobolev Spaces of Vectorial Order

150

i s surjective, and there e x i s t s a family o f ( u n i f o m Z y with respect t o the parameter h E (O,hO])bounded linear mappings, l i f t i n g f r o m n- 1 (sl' s2- k - f ) ,h'RxRx',h) to H ( s ().I,",h'.

'

Proof. We first show the continuity of the map (2.7.33) uniformly with respect to h. One has:

Cauchy-Schwarz

n

inequality yields:

Further, obviously, one has for j < s -4: 2 2(1-s2) dS 6 C 1 (2.7.36) I1 <<(h,hC)> n 1 T1 T Cn,h 2 (1-s2)+l 5 C2<S'(h,hS')>

2 2 -2 (<<'(h,hS')>+En)

dSn 6 2 ( j - s )+1

dEn 6 C3<5' (h,hS')>

1R

En where the constants C . may depend only on s2,and n. -2s

J

(h,h5')>2(s2-'-f),

Multiplying (2.7.35) by h

using (2.7.36) and

integrating over Tn-' , one finds that 5 ' ,h

1 ID;

1j

hu(x~'o)

1

(sl,s2-j-f),h 5 C I ( u ( (s),h

,V

h E [O,holt

n' where the constant C may depend only on s2 and n. We now prove the surjectivity of the map u(x',x ) each k < s2-4. Let a meshfunction a(h,x') E H

+

k D

xn

hu(x',O) for

(B?:T1h) be ( ~ 1 ~ ~ 2 - k,h -f) given. We construct a meshfunction u(h,x',xn) E H(s),h(Rz,h) , such that

k

I

I

'

with s o m e D hu(h,x',O) = a(h,x') and lul (sl,s2-k-f) ,h (s),h xn8 constant which does not depend on a, u and h E (O,hO].For doing that, put

la'

<<'(h,h<')>* and let @ (t) E Cm(R) be such that supp @ 6 0 6 @ (t) 0 for It/ 5 & / 2 , and such that 6

X

=

(2.7.37)

(2n)-'

I @ 6 (t)dt =

1,

R

where 6 > 0 will be chosen later. Let us define

Gh(E',<

)

as follows

C

[-6,61,

2

2.7. Spaces of One Parameter Families of Meshfunctions

151

-1 -h u s a t i s f i e s t h e r e q u i r e m e n t s above W e show t h a t u ( h , x ' , x ) = F n E+x.h I n d e e d , one h a s :

=

(27I

t h e i n t e g r i n d A-l$(h-l
15

2 0

i n e q u a l i t y 6 < rr(h + 4 ( n - 1 ) )', V h E (O,ho], V

5 ' E Tn-l

o:e

1

5 6 h . Choosing 6 t o s a t i s f y t h e f i n d s i m m e d i a t e l y t h a t h6X <

IT,

T h e r e f o r e , i n t h e i n n e r i n t e g r a l one can

E',h' so t h a t t h e s u b s t i t u t i o n n n = A - 1 instead of T i n t e g r a t e o v e r IR1 'n Cnrh' 'n k hu(x',O) = a ( h , x ' ) . and t h e c o n d i t i o n ( 2 . 7 . 3 7 ) y i e l d D xnr F u r t h e r , u s i n g P a r s e v a l ' s i d e n t i t y and t h e o b v i o u s i n e q u a l i t i e s

151 5 151 5 (71/2)1<1,V h L 0 , V 6 E Tn variables

5

A-1

-

nn,

5,hr

and making t h e change o f

one f i n d s

where t h e c o n s t a n t C may depend o n l y on s 2 , n and

I$

6( t ) .

I

C o r o l l a r y 2.7.18.

A s a consequence of Theorem 2.7.17,

integer m

> 0:

(2.7.38)

/ D i , h u ( h . O ) I 5 C1 IuI

one g e t s for n

I (0,m),h8

V j

=

=

1

and any g i v e n

O,l, . . . ,rn-1,

where t h e c o n s t a n t c does not depend on h and u.

2 . Sobolev Spaces of Vectorial Order

152

Remark 2.7.19. I n t h e same manner, one c a n show t h a t t h e f a m i l y o f mappings

(2.7.33) i s

o n t o as w e l l , and c o n s t r u c t e x p l i c i t l y a ( u n i f o r m l y bounded w i t h r e s p e c t t o t h e parameter h

E (O,ho]) family

of l i f t i n g o p e r a t o r s .

W e s h a l l now i n t r o d u c e a n e q u i v a l e n t d e f i n i t i o n of norms i n s p a c e s

'

which i n c e r t a i n c a s e s t u r n s o u t t o b e more c o n v e n i e n t , s i n c e i t s ( s ), h ' d e f i n i t i o n d o e s n o t u s e t h e F o u r i e r t r a n s f o r m of m e s h f u n c t i o n s . For i n t e g r a l s2

>

0 s u c h a d e f i n i t i o n h a s a l r e a d y been g i v e n . T h e r e f o r e , w e

now c o n s i d e r t h e c a s e 0 < s ,

c 1 . F u r t h e r , w i t h o u t r e s t r i c t i o n of

g e n e r a l i t y one c a n t a k e s1

0 , and we s h a l l w r i t e

L

=

1.11

s ,h

i n s t e a d of

I'"'(O,s),h* I n t r o d u c e t h e norms o f o r d e r s = ( s 1 , s 2 ) ,s 2 E ( 0 , l ) f o r m e s h f u n c t i o n s 1 on R x,h

..

where 0 u t

=

u ( x + t ) is the s h i f t operator.

Theorem 2.7.20.

The f a m i l i e s o f n o r m

I . 1 1 s,h ~

and [ . ]

s,h

d e f i n e d by ( 2 . 7 . 3 9 ) a r e m i -

f0mZ.V e q u i V a k 2 t w i t h r e s p e c t t o h E (O,ho], i . e . t h e r e e x i s t s a c o n s t a n t

c

>

0 such t h a t h o l d s :

It s u f f i c e s to c o n s i d e r t h e case s1

= 0,

so t h a t w e s h a l l w r i t e s

i n s t e a d of s

2' Parseval's identity yields:

1 I (Ot-l)ul 1 '

=

(277)-1

0,h Introduce g(a) =

C

J1 T

5 ,h

k - ( 2 s + 1 ) s i n 2 ka.

k> 1

One h a s w i t h some c o n s t a n t C > 0

I n d e e d , f o r any N 2 0 one h a s :

leitE-l12\;h(E)

I2dE.

2.7. Spaces of One Parameter Families of Meshfunctions

153

m

k - ( 2 s + 1 ) sin2 ka < J t - ( 2 s + 1 ) d t = ( 2 s ) -1 N - 2 s

C

RN+l(a) =

k>N+1

N

one gets:

( 2 s ) -1/ ( 2 s ) l a l - l ,

With t h e l e a s t i n t e g e r N 2 (2.7.42)

.

2s

0 5 RN+l(a) 5 ' a /

.

Now we e s t i m a t e t h e sum

x

k-(2s+1)sin2ka5 15k5N =

0

2

kl-2s

2N

< a / t

dt =

0

16kZ-N

-1 2 2 - 2 s (2(1-s)) a N

s i n c e , a s a consequence of t h e c h o i c e o f N ,

1-2s

5

Cia/ 2 s ,

one h a s : N

-

1al-l.

I n t h e r i g h t hand s i d e o f t h e l a s t i n e q u a l i t y t h e c o n s t a n t C may depend o n l y on s. F u r t h e r , i f M i s a n i n t e g e r s u c h t h a t n/4 5 M i a 1 5 71/2, t h e n

g(a)2

2

2 2 kl-2s X k-(1+2s)sin2ka2 (2/n) a 16k<M 15k5M ~

(2/71)2 a 2

f: t l - 2 s d t

2

2 Cja12s,

1 where a g a i n C may depend o n l y on s . Hence, P a r c e v a l ' s i d e n t i t y a n d ( 2 . 7 . 3 9 ) y i e l d : [uI2

0,h

=

=

(2nI-l

( 2 n I - l J1 T

5th

where t h e e q u i v a l e n c y

5 rh

(1+4h

Jl T

-

1 O
(l+h

k>0

t- ( 2 s + l ) 4sin2

3 2

t,h h - ( 2 ~ + l ) -(2s+l) . 2 h5 k s i n ( k -)) 2

i s u n i f o r m w i t h r e s p e c t t o h E [O,h

0

1.

For n > 1 t h e norms ( 2 . 7 . 3 9 ) a r e d e f i n e d i n a n a t u r a l way:

Also u s e f u l i s t h e f o l l o w i n g d e f i n i t i o n o f t h e norms o f o r d e r s s2 E ( 0 , l ) :

=

(sl,s2),

2 . Sobolev Spaces of Vectorial Order

154

Theorem 2.7.21.

I I . 1 I ( s ),h are

ivomns (2.7.44) and t h e norms

uniformLy e q u i v a l e n t w i t h

respect to h E [ O , h o l . Proof. __ It suffices to consider s1 ~

.1

,h instead of

Putting y

=

=

0, so that we shall write s instead of s 2 and

1 .1

(0,s),h' xtz and using Parseval's identity, one can rewrite (2.7.44)

in the following fashion:

where, as usual, 0 u

=

u(x+z) is the shift operator.

Denote G(n)

=

X

lkl

-(n+2s) i 2 le -11 .

kEzn\{ 0} Again one shows that with some constant C > 0 holds:

the argument being exactly the same as in the case n

=

1.

Using (2.7.45). (2.7.46) one gets immediately the equivalency of the norms

1.

Is,h

and

11.1

Is,h uniformly with respect to h E [O,ho].

Remark 2.7.22. For V nonintegral s = k c e

> 0, k

are defined in a natural way:

E Z + , 8 E ( 0 , l ) the norms (2.7.43), (2.7.44)

2.7. Spaces of One Parameter Families of Meshfunctions

155

and, correspondingly,

The p r o o f o f e q u i v a l e n c y o f norms ( 2 . 7 . 4 7 ) ,

( 2 . 7 . 4 8 ) and t h e norms

I 1 . I Is,h

( u n i f o r m l y w i t h r e s p e c t t o h E [O,h 01 ) d i f f e r s i n no way from t h e case s E ( 0 , l ) . Remark 2 . 7 . 2 3 . One h a s i n t h e s p a c e

'

H

(Z"),

s = (sl,s2),three different equivalent

(S)

norms, which a r e d e f i n e d a s supremum w i t h r e s p e c t t o h E ( 0 . h 01 o f t h e n o r m s

I I.

(s),h' h - s 1 [ . ] s 2 , h

and h - S 1 / .

Is2,h.

<,+ <,

Now w e s h a l l c o n s i d e r s p a c e s o f m e s h f u n c t i o n s o n t h e g r i d s n , + de f Zn-l x z+ = { a = ( a ' , a n ) , = {x E x 2 0 ) . Denote Z

a'

E

zn-l,

an E

Denote by

x+;.

H (o) , h ( < f +

)

t h e c o l l e c t i o n of m e s h f u n c t i o n s u ( h , x ) s u c h

that

and by

H(o)

(Zn'+)

t h e space o f a l l meshfunctions u : ( 0 , h

01

x

Znt+

+

C,

f o r which h o l d s

<,+

a and e a c h m e s h f u n c t i o n u on the f o r w a r d d i f f e r e n c e d e r i v a t i v e s Da a r e w e l l d e f i n e d . Hence, one c a n d e f i n e x,h f o r e a c h i n t e g e r m t 0 t h e norms o f o r d e r ( s l , m ) :

N o t i c e t h a t f o r each multiindex

and r e s p e c t i v e l y ,

W e d e n o t e by

H

( s l , m ), h

(<,+)

t h e s p a c e s o f m e s h f u n c t i o n s u f o r which

2. Sobolev Spaces of Vectorial Order

156

are finite, V h E (O,ho], and by

the norms ( 2 . 7 . 5 1 )

H ( 3 1 ,m) ( Z y )

the space

of all mesh functions u for which the norm (2.7.52) is finite.

proposition 2.7.24.

Let

+

.

E c ~ ( R ~ )Then

where cm depends only on m. Proof Using the Leibnitz formula

one gets immediately (2.7.53). For any meshfunction u extension operator 1 u m

:

where the coefficients C

R : +

P

(2.7.56)

Z

(-1)J

lspsm

c

:

<,++

C and a given integer m > 0 define an

C, as follows :

satisfy the system of equations

pJ = 1,

0 2 j < m,

P

and where H(x) z 1 for x 2 0, H ( 0 )

=

0 for x < 0 is the Heaviside function.

We shall need the following auxiliary assertion concerning any meshfunction on

%,+

:

Lemma 2.7.25.

Let u(t)

:

d,+* c and

let

where lm is defined by (2.7.55), (2.7.56) with xn Then the following formula is valid:

=

t.

157

2.7. Spaces of One Parameter Families of Meshfunctions

where a Qj,p

=

(al,...,a , ) is a muZti-index, / a / =

=

{a

CL

1 1 (al,...,a , ) , 0 5 ak < p, 1 5 k 2 j}.

=

and +...+a. 1'

7

Proof. -

1 It is obvious, that v . (t) 3 u .(t), V t E %,+, V j 2 0. 1 1 We prove (2.7.58) by induction on j. We shall first check it for j = 1. For t < -h, one gets using the definition (2.7.551, (2.7.56) of lm: (2.7.59) v (t) = h-l 1

C C (u(-pt-ph)-u(-pt)) = lspsrn

-

Correspondingly for t = -h, using (2.7.56) (with j (2.7.60) v (-h) = h

-1

1

C

=

01,

(~(0)C C ~ ( p h ) =) - C C C 16pSm '~EQ l
the latter coinciding with (2.7.59) when t

=

C

C

l
ul(-pt-ph+ah) ,p

we find ul(ah),

1 .P

-h, and (2.7.59) being

precisely (2.7.58) with j = 1. Now, let us assume that (2.7.58) holds for j, and we shall prove it for j+l (j+l i m). The formula (2.7.58) being true for j, let us take u to be the polynomial t1/j!, so that using (2.7.571, we obtain the formula: (2.7.61)

1 = 1,

(-111 C C I: lsp<m '~EQ

v

t

E Rt,h, t < O

1 rP hlal?pt+jph

tl/j! one has: u.(t) = H(t) and v.(t) Z 1. 1 Substracting (2.7.61) from (2.7.56) and using the identity

Indeed, for u

=

1, we get the formula: pl = ' a E Q . 18P (2.7.62)

(-1)'

C C C l
1 = 0, v t E

m t,h ' t < O .

3 rP

h/a)
7+1

(t) for t < -h. Putting a = (a',aj+l), we have

v . (t) = h-l(v.(t+h)-v.(t)) = 1+1 1 1 h-l(-l)l+l( I: c c (~.(-pt-jph+la'Ih)-u.(-pt-(J+l)phf/a';h))) = lsp<m ~ ' E Q , , ~1 1

(2.7.63) =

=

(-1)J + l

z c lspsm

h 1a

+

(-1)j+l

x

c

lsp<m

h-l ( u ,(-pt-jph+(a'Ih)-u,(-pt-(j+l)ph+/a'Ih)

C

3

1

cr'EQi rP

1 Lpt+ ( j+1 ) ph

z

h-lu.(-pt-jph+la'/h) =

~ ' E Q ~

tP

h(a'I
1

+

2. Sobolev Spaces of Vectorial Order

158

=

(-1)j+l lspsm

+

(-1)j+l

z

+

c

lspsm

=

(-1) j+l

x

Z u. (-pt-(j+l)ph+(a'lh+a. h)+ l+l 3+1 ~ ' E Q,p , 053 l

1

c

lspsm

c c ~ E Q ~ + ~ , ~

u , (-pt-(j+l)pi>+1 a 1111 + 3+1

hln' (>pt+(j+l)ph + (-1)j+l

c

c

z

z

h 1 a '+ t :1

u.

OSkh<->t-jph+ I a ' 1 h-h 1+1

lsp<m P ~ ' E Q .

(kh).

( j+l)ph

We used here formula (2.7.621, while substracting u . ( O ) in the corresponding 3 sum. We make a change of summation index in the last sum on the right hand

side of (2.7.63), putting kh = -pt- ( j+l) ph+ 1 a

I h+aj+lh,

after which this sum is transformed as fOllOWS:

Substituting this last expression into (2.7.63), we finally obtain for t

-h

which proves formula (2.7.58) for t < -h. We now prove (2.7.58) for t = -h. Using (2.7.61) with t = -h and the fact that u .(t) 3

0 for t < 0, t E lRL,h

,

we have

2.7. Spaces of One Parameter Families of Meshfundions

=

cp c

c

( - 1 ) j+l

lSpSm

a'EQ.

I #P

1011

vj+l

u . (kh). z O
(j-l)p

-jp+la'I+a. and again taking into consideration that 3+1 0 for t < 0, we obtain for the last sum the expression

Putting k u . (t) 3+1

IZ

159

=

=

(-h) = (-1) j+l

=

(-1)j+l

=

(-1)j+l

c

c

c

c

lspzm '~'EQ. p>a 3 rP / a ' It (j-l)p

. (-jph+lalh) = 1 1 u 1+1

jp- a

+

z c u . (-jph+/a/h) = 1:p6mCpa~CQ, O
z cpaEQ. c

u . (-jph+lalh), ,p 3+1

lSp5m

1+1 which proves (2.7.58) for t

=

I

-h.

Although the proof of Lemma 2.7.25 is based on a rather tedious computational work, the idea which is behind is quite transparent: if t 5 -jh then the inward sum on the right hand side of (2.7.58) is over full cube Q .

I tP

and it is easily seen, that in that case (2.7.58) can be

rewritten in the following fashion (2.7.64) v.(t) = c c 3 l
ai,hu(-pt),

v

t 6 -jh.

The same formula (2.7.64) is valid even for all t < 0, if u(kh) k

=

O,l,

...,m-1.

=

0,

Hence, via interpolation procedure (one can use Newton

interpolation polynomials) the general case is reduced to theone of the k polynomials t /k!, k = 0.1, m-1, which have beenusedinorder toderive the

...,

relation (2.7.61). NOW we are in the position to prove the following theorem: Theorem 2.7.26.

For t h e e x t e n s i o n operator im d e f i n e d by (2.7.55), (2.7.56) hoZds (2.7.65)

1 I1,ui 1

(sl,m),h "rnl

lul

I;sl,m) ,h'

V h

E (O,hol,V

s1

E IR,

where t h e c o n s t a n t cm depends o n l g on m. Proof. u(x',xn), one gets for v . = ( 3 1 3 x,.h using Cauchy-Schwarz' inequality, the following estimate

Applying (2.7.58) with u

=

1 u) (x',xn), m

2 . Sobolev Spaces of Vectorial Order

160

Therefore, with some constant C' the following estimate holds: m

'

j ,h'il~O),h' /lVjll(0),h CAjlUjll(0),h = CAjlax Furthermore, for any tangential difference derivative Da' with x',h ja')+j 5 m one gets:

Now.

I

(2.7.65) is an immediate consequence of (2.7.67).

Corollary 2.7.27. For lm d e f i n e d by (2.7.551, (2.7.56) one has uniformly w i t h r e s p e c t t o h E (O,hol, V ho (2.7.68)

' 0:

'

H(sl,m),h(RE,+ ) + H(sl,m),h(I( ,) v s1 E R ,

:

H(sl,m)( 2 : )

m '

and (2.7.69) lm

-f

H

(sl

,m)

(Z"),

v

s1

E R.

Theorem 2.7.28.

The one parameter f a m i l y of mappings Rh' (2.7.70)

H

(sl

,m),h

(R;")

3 u(h,x',xn)

%

i s continuous uniformly w i t h r e s p e c t t o h E (O,ho], i . e . t h e norm of Rh

i s bounded by a c o n s t a n t which may depend only on m and ho,

V h

E !O,hol.

Moreover, t h e map R h i s s u r j e c t i v e , and t h e r e e x i s t s a continuous ( u n i f o r m l y w i t h r e s p e c t t o h E (O,hO1) f a m i l y of l i f t i n g o p e r a t o r s Lh,

Proof. ____ The f i r s t part of the statement above is an immediate consequence of Theorems 2.7.28 and 2.7.17. The same Theorems guarantee the surjectivity of

2.7. Spaces of O n e Parameter Families of Meshfundions

L

i n d i v i d u a l map

h,j

with D j hu(h,x',O) = a ( h , x ' ) . x , I t n r e m a i n s t o show t h e s u r j e c t i v i t y o f

<= j

( 2 . 7 . 7 0 ) and t o c o n s t r u c t a

Lh a s i n ( 2 . 7 . 7 1 ) . Thus, l e t a . ( h , x ' ) E ff

l i f t i n g operator be g i v e n , 0

161

(sl,m-j-?) ,h(m:-l) I m. L e t u s c o n s t r u c t a f u n c t i o n u ( h , x ' , x n ) t H ( s l , m ) , h (iRn h , +1

<

s u c h t h a t D 1 h u ( h , x ' , O ) = a . ( h , x ' ) , 0 5 j < m. W e s h a l l f i n d f i r s t 1 Xnf

,

f u n c t i o n s U ( h , x ' , x ) C ff (<,+) k ( s l ,m) ,h conditions: (2.7.72)

D1

U

(h,x',O)

=

6.a.(h,x'),

xn,h k

ik 1

0 5 k < m, which s a t i s f y t h e

0 i k , j < m.

Then it i s e a s y t o p r o d u c e a f u n c t i o n u ( h , x ' , x ) a s r e q u i r e d , which i s j u s t

u

C U (h,x',xn). k 06kSm 1 Again, a s i n t h e p r o o f o f Theorem 2 . 7 . 1 7 , p u t X = q '( h , h E ' ) > 2 and m 1 d e n o t e by p ( t ) a C - f u n c t i o n on R t , which s a t i s f i e s t h e c o n d i t i o n s : =

Denote x

=

t and i n t r o d u c e

vk ( h , S ' , t )

=

X -k-h ak

-h where ak = F x , + S , , h a k and C r k ,

C C r k ( h , 5 ' ) pr t / h ( h X ) , l5rSm

1 S r 5 m,

0 5 k < m ,

0 5 k < m, a r e t o be chosen t o

satisfy (2.7.72). One g e t s f o r Crk t h e f o l l o w i n g s y s t e m o f e q u a t i o n s : (2.7.74)

C

((pr(hh)-l)/(ihX))'Crk(h,S')

=

Sjk,

0 5 j , k < m,

l
Denote jlr

=

jlr(hh)

=

( i h A ) - ' ( p r ( h A ) - l ) . F o r e a c h k we have i n ( 2 . 7 . 7 4 ) a

subsystem of l i n e a r e q u a t i o n s f o r Crk, determinant

1 5 r 5 m , w i t h Vandermonde

2. Sobolev Spaces of Vectorial Order

162

which d o e s n o t depend on k .

S i n c e A(hh) =

TL lSp
) , the question p g I of a bound f o r A ( h h ) r e d u c e s t o a n e s t i m a t e o f t h e d i f f e r e n c e s i(, -$ q P q > p . S e t t i n g t = hA a n d n o t i n g t h a t 0 5 t 5 C < m, V h E [ O , h O ] ,

V 5'

E

TnT1 , u s i n g t h e f o r m u l a 5 rh

hq-i(,pl= I t - l ( l - p ( t ) ) l J p ( t ) l n ;

c

P 1( t ) !

O
(2.7.75)

0 < C1 5 / $ q ( h h ) - ~ p ( h A ) 5j C2 <

m,

V h,S'.

The l a t t e r i m p l i e s t h a t A ( h h ) a l s o s a t i s f i e s t h e i n e q u a l i t i e s > 0 and C 2 > C1. 1 Therefore, Crk(h,S') E C ( h h ) , where C ( t ) a r e smooth bounded rk rk 1 f u n c t i o n s f o r a l l t E R+ T h i s c o n c l u s i o n a l l o w s u s t o show t h a t

(2.7.75), perhaps with d i f f e r e n t constants C

.

I n d e e d it s u f f i c e s t o v e r i f y t h e u n i f o r m boundedness i n h E (O,hOl o f the expression

W e have:

D ~ - ' v ( h , E ' , t ) = A m-j-k-h ak(h,5') t,h k

C

Crk(hA)pr t / h ( h X ) z

l5rSm

so t h a t

s i n c e ~ ' ( 0<)0 , p ( 0 ) = 1 and 0 5 hh 5 C <

m,

V h,S'.

Summation i n j from 0 t o m i n ( 2 . 7 . 7 6 ) and i n t e g r a t i o n o v e r T lead t o the inequalities:

n- 1 5',h

2.7. Spaces of One Parametcr Families of Meshfunctions

1 1 . 1 1 ;m-k-i)

where w e have d e n o t e d by

M u l t i p l y i n g ( 2 . 7 . 7 7 ) by h-'1, (2.7.78)

I lUk/ I (s,,m)

,h

'

, h t h e norm i n

H (0,m-k-t)

163 n-1 ,h(%

) '

one g e t s t h a t

;sl,m-k-f) ,h'

lakl

O S k < r n ,

X Uk, s i n c e t h e c o n s t a n t s C i n ( 2 . 7 . 7 7 1 , 05kcm I ( 2 . 7 . 7 8 ) do n o t depend o n h and a k , 0 5 k < m.

which p r o v e s ( 2 . 7 . 7 1 ) w i t h u

=

Remark 2.7.29. As

a c o r o l l a r y of Theorems 2 . 7 . 2 7 and 2 . 7 . 2 8 a n d o f t h e f a c t t h a t t h e

r e s t r i c t i o n o f a meshfunction i n in

H (sl,m)

(I$+)

,

H (sl,m)

t o Rt,+ i s a m e s h f u n c t i o n

(<)

w e o b t a i n the s u r j e c t i v i t y o f map ( 2 . 7 . 3 3 ) a n d t h e e x i s t e n c e

1 ) bounded l i f t i n g o p e r a t o r from O n (sl,m),h(%) f o r each i n t e g r a l m 2 0.

of a (uniformly w i t h r e s p e c t t o h C ( 0 , h n O - ; j < m H ( s l , m - j - t ) , h ( %n-1)

into

H

Now s e v e r a l e q u i v a l e n t d e f i n i t i o n s of norms i n s p a c e s H (s), h ( < , + ) w i l l b e g i v e n , a l l o f them b e i n g e q u i v a l e n t w i t h t h e norms i n

H

,m),h(<,+)

g i v e n p r e v i o u s l y , when s 2

m > 0 is an i n t e g e r . Further-

=

(sl more, a n e x t e n s i o n o p e r a t o r 1 : H ( ~ , )h ( < , + ) + H ( ~ , )h ( I?;) w i l l b e s2 c o n s t r u c t e d and t h e t r a c e t h e o r e m on h y p e r p l a n e s o f c o d i m e n s i o n o n e w i l l b e p r o v e d f o r e a c h s2 1 0 . We s t a r t w i t h t h e d e f i n i t i o n which u s e s ( 2 . 7 . 4 7 ) . Namely, l e t s 2 = m+€I w i t h m 2 0 a n i n t e g e r and 8

C (C,l).

Define

and

The s p a c e s o f m e s h f u n c t i o n s and t h e norms [ u ] + (S)

f o r which t h e norms [ u ] + V h E (O,hol (s), h ' H (s),h(<,+) and H ( s ) (zl),

a r e f i n i t e , a r e d e n o t e d by

r e s p e c t i v e l y . W e s h a l l c o n s t r u c t f o r each s2 2 0 an e x t e n s i o n o p e r a t o r 1

s2

which w i l l b e u n i f o r m l y bounded i n h E ( 0 , h

0

1.

,

2. Sobolev Spaces of Vectorial Order

164 For s2

m t 0 i n t e g e r it h a s y e t b e e n done.

=

W e s t a r t with t h e case 0 <

S2

?.

<

Theorem 2 . 7 . 3 0 .

Let u E H ( s ) , h ( ~ : , + )

and l e t 0

+ . Then

< s2 <

for t h e e x t e n s i o n operator

uniformly w i t h r e s p e c t t o h E ( O , h o l . Proof. -

Denote v ( x ) = l 0 u ( x ) , ~ € 4 L. e t { e j } l s i s n b e t h e s t a n d a r d b a s i s i n IRn. One h a s t o show (2.7.82)

-2s

- (2s2+1)

1

h

1

O < t .ER I xirh -2s 5 Ch

h

tj -

1 c o
1 XE<

t. 1

(2s2+1) h

2 n Iv(x+t.e.)-v(x)Ih 6 1 1

Z

2 n lu(x+t.e.)-u(x)I h , 15 j s n , 1 1

,h

w i t h a c o n s t a n t C which d o e s n o t depend on h E (O,ho] and u . For 1 5 j < n one h a s e q u a l i t i e s i n ( 2 . 7 . 8 2 ) w i t h C

=

1. Therefore,

f o r showing ( 2 . 7 . 8 2 ) w i t h j = n it s u f f i c e s t o c o n s i d e r t h e c a s e o f a s i n g l e v a r i a b l e x n , which w i l l b e d e n o t e d by x . Also, i t s u f f i c e s t o c o n s i d e r t h e case s1 = 0 , so t h a t w e s h a l l w r i t e s i n s t e a d o f s 2 . Thus, one h a s t o show t h a t w i t h some c o n s t a n t C > 0 h o l d s (2.7.83)

t-(2s+l)

Z t ER

h,+

h

I v ( x + t ) - v ( x ) I 2 h 5 C N~ , h ( ~ ) '

Z xER

h

where w e h a v e d e n o t e d

Given t h e d e f i n i t i o n of l o , t h e l a t t e r r e d u c e s t o t h e p r o o f o f t h e inequality

o r , what i s t h e s a m e , t o p r o v e t h e estimate:

2.7. Spaces of One Parameter Families of Meshfunctions

(2.7.84)

165

Z t-(2s+1)h Z lu(x)I2h 5 C N (u). s,h t>G O<x
Changing o r d e r of summation i n ( 2 . 7 . 8 4 ) . w e c a n r e w r i t e i t i n a n e q u i v a l e n t form

:

where w e h a v e d e n o t e d

O b v i o u s l y , one h a s w i t h some c o n s t a n t C: c-l

s

-1

-2s -1 -2s (x+h) 5 P ~ , ~ ( X6 ) CS (x+hl

,V x

2 0 , V s > 0.

Thus t o o b t a i n t h e n e c e s s a r y e s t i m a t e f o r a g i v e n s E (0,+),it s u f f i c e s t o prove t h e i n e q u a l i t y :

where t h e c o n s t a n t C

may depend o n l y on s E ( 0 , i ) .

The l a t t e r w i l l f o l l o w from t h e f o l l o w i n g Lemma 2 . 7 . 3 1 .

Let u E H(O,s),h(<,+) u

, with

0 < s < f . Then

( ~ ~ + h I - ~ u

j.

i s a f a m i l y of ( u n i f o r m l y w i t h r e s p e c t to h E (G,ho]) continuous l i n e a r mappings f r o m

H (O,s),h( Rhn, + )

into H(O,O),h(<,+).

Proof o f Lemma 2 . 7 . 3 1 . S i n c e t h e v a r i a b l e s ( x l , ...,x ) p l a y t h e r o l e o f p a r a m e t e r s ( o n e can u s e n- 1 d i s c r e t e F o u r i e r t r a n s f o r m i n x l , ...,x n - l ) , one i s l e f t p r e c i s e l y w i t h proving (2.7.87) with x

=

x.

Introduce (2.7.88)

$(x)

=

-1 (x+h)

Z 05y5x

One h a s

Indeed, put

( u ( x ) - u ( y ) ) h ,$ ( x )

=

1 y-'@(y)h, ydx+h

x L 0.

2 . Sobolev Spaces of Vectorial Order

166

Z

Q(x) =

( u ( x ) - u ( y ) ) h= ( x + h ) $ ( x ) .

ocysx

*

a Y # h and a y t h

The summation by p a r t s f o r m u l a ( w i t h

t h e f o r w a r d and back-

ward d i f f e r e n c e d e r i v a t i v e s , r e s p e c t i v e l y )

gives for $(x) -1 -1 ( y + h ) Q ( y ) h= - C Q(Y)ay,hY h = y>x+h -1 * = (x+h)-lO(x)+ C y-’a* O ( y ) h = $ J ( x ) +1 Y ay,h@(Y)h. yhx+h Y .h ytx+h +(x)

=

Z

y y>x+h

-1

B u t , it i s e a s i l y s e e n t h a t a;,ho(y)

I:

=

05zSy

a

*

Q(y) Y #h

*

=

~ a ~ , ~ u (I n y d)e e. d ,

(u(y)-u(z))- c (u(y-h)-u(z)) 0 5 zSY- h

=

*

z

(u(y)-u(y-h)) 1 = yay,h~(~). Osz5y-h

=

Thus, t h e i d e n t i t y ( 2 . 7 . 8 9 ) i s p r o v e d . Usinq t h e d e f i n i t i o n of $ ( X I , o n e f i n d s a p p l y i n g t h e Cauchy-Schwarz inequality: /$(x)

i2

-1 5 (x+h)

l u ( x ) - u ( y ) I2h,

C

05y5x and c o n s e q u e n t l y

-2s

C (x+h)

(2.7.90)

2 I$(x)I h 5

xto C

=

y>o

h

C

-(2s+l)

(x+h)

-(2s+l)

IU(X)-U(Y)~

Z

h

I: ( x + h ) x>o

IU(X)-U(Y)/

2

h =

O
h =

xty 2

I: h Z ( t + y + h ) - ( 2 s + 1 )l u ( y + t ) - u ( y ) 1 h 5

=

y>o

t>O

I t should be s t r e s s e d t h a t (2.7.90) holds f o r V s.

For $ ( x ) w e s h a l l now p r o v e t h e estimate: (2.7.91)

2

I: ( x + h ) - 2 s 1 $ ( x ) 1 2 h 5 C 1 ( x + h ) - 2 s l $ ( x ) I h . xzo

x>o

Denoting G ( x + h ) = ( x + h ) - ’ $ ( x ) , x 2 0 and g ( x ) = x-’@(x), x > 0 , g ( 0 ) = 0 ,

2.7. Spaces of One Parameter Families of Meshfunctions we c a n r e w r i t e

167

( 2 . 7 . 9 1 ) i n t h e e q u i v a l e n t form:

One c h e c k s e a s i l y t h a t

I n d e e d , one h a s j u s t t o u s e t h e d e f i n i t i o n o f G(x), g(x) and ( 2 . 7 . 8 8 ) f o r $J

(x-h)

.

Therefore, putting x

=

nh, y = k h , G(x) = G

and g ( y )

=

g k , one f i n d s

W e s h a l l d e n o t e by n ( y ) t h e s t e p f u n c t i o n d e f i n e d by t h e f o r m u l a

a(y) =

l g k l , f o r k 5 y < k t l } and by p m

(2.7.92)

p

1

(x) =

1 (li)'

a(y)

X

*, Y

1

(x) the function

x > 0.

Noticing t h a t kl-s

k+ 1 J yS-'dy k

t ks-l(l+sk-'-s(l-s)

(2k2)-l-l) >

+,

V k 2 1,

we f i n d

NOW,

we show t h a t m

(2.7.93)

J 0

m

p

2 1

(O,t),

may depend o n l y o n s .

where t h e c o n s t a n t C L e t us put x =

2 a (x)dx, V s E

J

(x)dx 5 C

t

,

y

=

e',

B(t)

=

et12a(et)

and R ( t ) = et12p1 ( e t ) .

Then ( 2 . 7 . 9 3 ) r e d u c e s t o t h e f o l l o w i n g e q u i v a l e n t i n e q u a l i t y :

-m

-m

where R ( t ) i s t h e f o l l o w i n g c o n v o l u t i o n

Ks being the following kernel: K s ( t ) function.

=

H(-t)e

t(f-s)

where

H ( t ) i s theHeaviside

2. Sobolev Spaces of Vectorial Order

168 Since m

[ks(c)I

1

=

-it6

J’ K s ( t ) e

-1

(C2+(t-s)21-’ 5 ( 5 s )

dtl =

,

m

w e a t once o b t a i n ( 2 . 7 . 9 3 ) w i t h C

= (4-s)

-1

.

P ( x ) 1s monOtoniCally 1 (2.7.93) y i e l d s :

A s a consequence o f i t s d e f i n i t i o n ( 2 . 7 . 9 1 ) ,

d e c r e a s i n g f u n c t i o n of x E lR+

.

Hence, m

2

2 C

5 4 1 I p l ( n ) 1 2 5 4 1 p ( x ) dx 5 n>Om o 1 2 2 lgnl , 5 4 C J’ a(x) dx = 4Cs C ntO

lGnl

n>O

and t h a t p r o v e s ( 2 . 7 . 9 1 ) . Now, t h e e s t i m a t e s ( 2 . 7 . 9 0 ) ,

( 2 . 7 . 9 1 ) and t h e i d e n t i t y ( 2 . 7 . 8 9 )

p r o v e t h e s t a t e m e n t o f Lemma 2 . 7 . 3 1 and Theorem 2 . 7 . 3 0 . W e s h a l l now c o n s i d e r t h e case

f <

as well.

1.

s

Lemma 2 . 7 . 3 2 .

Let (2.7.943

with

( < I + )

H(O,s),h

-s + u /l ( 0 ) , h

1 1 (Xn+h)

where t h e c o n s t a n t c

s,ho

s

E ( + , I ) and l e t u(0)

’

Cc.,hol

l‘l

j;O,s)

,h‘

=

0.

Then hoZds:

V h E (O.hol,

depends only on its s u b s c r i p t s .

P r o_ of. _ Again, it s u f f i c e s t o c o n s i d e r t h e c a s e n = 1 , so t h a t one h a s t o show

With o ( x ) , $(x) d e f i n e d by ( 2 . 7 . 8 8 ) fact that u(0) (2.7.96)

$(O)

one f i n d s , u s i n g ( 2 . 7 . 8 9 )

and t h e

0, Q(0)= 0:

=

def =

-1

Z

y

Q(y)h,

V = 0.

y?x+h Hence, i n t r o d u c i n g

(2.7.96) y i e l d s : $ l ( x ) (2.7.97)

=

- $ ( x ) , so t h a t (2.7.89)

u ( x ) = Q ( X ) + $ ~ ( X )V, x C

A s it h a s b e e n s t r e s s e d i n t h e p r o o f

c a n be r e w r i t t e n a s f o l l o b s :

mi. of Lemma 2 . 7 . 3 1 ,

the inequality

2.7. Spaces of One Parameter Families of Meshfundions

169

( 2 . 7 . 9 0 ) h o l d s f o r V s . T h e r e f o r e , it s u f f i c e s t o p r o v e t h a t

(2.7.99)

g(x)

x-'@(x), G(x) = x

=

-1

C

g(y)h,

x > 0, g(0) = G ( 0 ) = 0,

o
i n a n e q u i v a l e n t way:

S i n c e 0 < 1-s < f , t h e p r o o f o f

( 2 . 7 . 1 0 0 ) i s t h e same a s t h a t o f

(2.7.91)

I Remark 2.7.33. 0

Denoting by

I{

(<'+) t h e s u b s p a c e of a l l m e s h f u n c t i o n s ( s l r s 2 ), h , f < s 2 < 1 , f o r which h o l d s : u ( 0 ) = 0 , one h a s f o r

(qt+) H(s1,s2)

-S

2:

t h e m u l t i p l i c a t i o n o p e r a t o r (x +h)

uniformly with r e s p e c t t o h E ( 0 , h

0

3,

a n d , a s a consequence o f

(2.7.101),

one h a s a l s o --s

(2.7.102)

where

H

:.i

(x +h)

(S1'S2)

n

v

s2 E ( f , l ) ,

i s t h e subspace o f a l l meshfunctions i n H

(Zn'+) ( s l ,s2)

which v a n i s h f o r x

(Z"'+),

(Z"'+), (Sl'S2)

=

0

Theorem 2.7.34.

The e x t e n s i o n operator lo d e f i n e d by ( 2 . 7 . 8 1 ) is a f a m i l y o f uniformly w i t h r e s p e c t to h E ( 0 , h0 1 continuous Linear mappings from H

into H

(s1rS2)

rh

(R:)

s ) ,h(<'+)

(sl' 2 f o r each s2 E ( + , I ] . Moreover, f o r each s 2 E ( f , 1 3

ho l d s :

Proof. For s2

=

1 t h e a s s e r t i o n s above a r e o b v i o u s . T h e i r p r o o f f o r s2 E ( f , l ) i s

2. Sobolev Spaces of Vectorial Order

170

e x a c t l y t h e same a s t h a t o f Theorem 2.7.30 w i t h t h e o n l y d i f f e r e n c e t h a t

I

i n s t e a d o f Lemma 2.7.31 one h a s t o u s e Lemma 2 . 7 . 3 2 . For s2 E R

d e n o t e by m t h e c l o s e s t i n t e g e r and l e t 8 = s 2 - m .

d e n o t e by ff' ( *

,s 1

,

) ,h(<'+)

2

u E H ( s 1 , s 2 ) , h ( q i n , +)

u E H

( s l,s2)

(Zn'+)

s2 2 0 , a l l t h e m e s h f u n c t i o n s

such t h a t u ( x ' , x ) = 0 f o r x

H

R e s p e c t i v e l y , w e d e n o t e by

We shall

(Z"") ( s l ,s2)

such t h a t u ( x ' , x

t h e s u b s p a c e s o f all m e s h f u n c t i o n s

= 0, x

)

= k h , 0 5 k 5 m.

=

k h , 0 5 k 5 m.

Theorem 2 . 7 . 3 5 .

The e x t e n s i o n operator lo d e f i n e d by ( 2 . 7 . 8 1 ) ( e x t e n s i o n by z e r o for x n

< 0)

i s ( u n i f o r m l y w i t h r e s p e c t t o h E ( O , h O ] ) a f a m i l y of continuous l i n e a r (%) f o r aZl s 2 z 0 such n g + ) into H mappings from H ( s l , s 2 ), h ( %

( s l , s 2 ), h

t h a t s2-f i s n o t an i n t e g e r . Moreover, one has under t h e same c o n d i t i o n on s2 t 0:

Proof. ~

Again, it s u f f i c e s t o c o n s i d e r t h e c a s e n For 0 5 s2 < 1 , s 2 - f (see Theorems 2 . 7 . 3 0 , 2.7.35

1.

# 0 , t h e a s s e r t i o n s above have a l r e a d y b e e n proved 2.7.34).

'

Further, for inteyer s

1

i s q u i t e o b v i o u s , s i n c e 1110 u ] ( s l , m ) , h

Now l e t m < s 2 < m + l , 2.7.34

=

=

=

m

0 Theorem

2 +

'uII

( s l , m ) ,h'

s2 # m+f w i t h i n t e g e r m Z 1. Applying Theorems 2.7.30,

t o va - D x , h ~ , la1 = m, one f i n d s :

(2.7'105)

['ova3 ( s l , s 2 - m )

,h 5 c [ v CY I +( s l , s 2 - m )

,h'

where t h e c o n s t a n t C d o e s n o t depend on h and v a r e d e f i n e d by ( 2 . 7 . 3 9 1 ,

T h e r e f o r e , Theorem 2 . 7 . 2 0 , (2.7.105) y i e l d :

V h E

,

(O,hol,

where [ . 1

( s ), h ' [ - ] ; s ) , h

(2.7.79).

d e f i n i t i o n s (2.7.39), ( 2 . 7 . 4 7 ) and i n e q u a l i t y

2.7. Spaces of One Parameter Families of Meshfunctions

where t h e c o n s t a n t C d o e s n o t depend on h and u , so t h a t ( 2 . 7 . 1 0 4 ) immediate consequence o f

171

i s an

I

(2.7.106).

Now we p a s s t o t h e c o n s t r u c t i o n of a n e x t e n s i o n o p e r a t o r from

H(s),h(
into

H(s),h(<)

f o r any s = ( s1 , s 2 ) w i t h s2 2 0 .

For i n t e g e r s2 2 0 it h a s a l r e a d y b e e n done (see Theorem 2 . 7 . 2 6 ) .

we b e g i n w i t h t h e f o l l o w i n g Lemma 2 . 7 . 3 6 .

,

L e t u and v belong t o H ( s ) , h ( % n r + ) s2

+ +. I n

addition, l e t u(x',O)

(2.7.107) w ( x ' , x n )

=

=

s

=

f f ( x n ) u ( x ' , x n ) + [I-H(x

beZongs t o H ( s ) , h ( ~ : )

,

o

( s l , s 2 )with

< s2 < 1 and Let

v ( x ' , O ) . Then t h e meshfunction

)Iv(x',-x

)

and, moreover,

where t h e c o n s t a n t c does n o t depend on u , v and h . Proof. O b v i o u s l y , it s u f f i c e s t o c o n s i d e r o n l y t h e c a s e n

w r i t e x instead of x

and

By v i r t u e o f Theorem 2.7.20

~ t h e norms [ w ] ~ ,and

t h e a s s e r t i o n o f Lemma 2 . 7 . 3 6 , d ef

(wI

(2.7.40) b e i n g v a l i d , i n o r d e r t o prove

C

2 Iw(x+t)-w(x)l h 5

xE Wh

t>O

5 c[N(u)+N(v)],

where w e have d e n o t e d (2.7.110) N(u)

=

C t

-(2s+l)

t>O

Rewrite M ( w ) u s i n g (2.7.107)

h

s,h being equivalent

it s u f f i c e s t o show t h a t

t-(2s+l) h

=

1 , s1 = 0 so t h a t we

i n s t e a d of [ - 1 ~ o , s 2 , , h r[ . 1 ( 0 , s 2 ) , h .

uniformly with respect t o h , i . e .

(2.7.109) M ( w )

=

2

C lu(x+h)-u(x)1 h. x>o

i n t h e following fashion:

2. Sobolev Spaces of Vectorial Order

172

Therefore, for showing (2.7.109) with N(u) defined by (2.7.110), it suffices to show that t-(2s+l)h C ]u(t-x)-v(x)1 2h 5 C[N(u)+N(v)l t>G O<x
Z t-(2s+1)h

(2.7.111)

t>O +

C

I

u(t-x)-u(x)

I 2h i

O<x
2 t-(2s+l)h I: /u(x)-v(x)(h 2 C[N(u)+N(v)l. t>O O<x
Changing the order of summation in the second sum in the left hand side of (2.7.111) (just as in the proof of Theorem 2.7.30), we come to the conclusion that for proving (2.7.111) it suffices to show that (2.7.112) C (x+h)-2slu(x)-v(x)1 2h S C[N(u)+N(v)l. x>o Noticing that u(x)-v(x) = f (x) E H ( o , s ) ,h(IF(r')

,

f (0) = 0 (by virtue of

the condition u ( 0 ) = v(O)), and applying Lemmas 2.7.31, 2.7.32, we immediately obtain that

1

C (x+h)-2S(u(x)-v(x) 2h 5 CN(u-v) 5 C[N(U)+N(V)l.

x>o It remains to show that def Q(u)

=

t-(2s+l)h t>O

1 u (t-x)-u (x)I 2h

6 CN (u).

O<x
But obviously, (lu(2t-x)-u(t-x)/2+/u(2t-x)-~(x) \ 2)h = (2.7.113) Q(u) S 2 1 t-(2s+1)h Z t>O O<x
2.7. Spaces of One Parameter Families of Meshfunctions

where P(u) s t a n d s f o r t h e l a s t sum on t h e r i g h t hand s i d e o f

173

(2.7.113).

Making t h e change t + x + t , we o b t a i n t h e f o l l o w i n g e x p r e s s i o n f o r P (u)

P(u) =

-(2s+l)

I: h C ( t + x ) x>o t > O

I2

lu(x+2t)-u(x) h,

so t h a t (2.7.114) P ( u ) 5 2

2s+l

X (2t)

-(2s+l)

t>O Now

(2.7.113),

2 2s+l h C l u ( x + Z t ) - u ( x ) ]h 5 2 N(u). x>o

I

(2.7.114) g i v e t h e r e q u i r e d i n e q u a l i t y .

Remark 2 . 7 . 3 7 . The e s t i m a t e ( 2 . 7 . 1 1 2 )

i s t h e o n l y one where t h e c o n d i t i o n s2 #

u s e d , s i n c e we had t o a p p l y Lemmas 2 . 7 . 3 1 , special choice of v

=

2.7.32.

?j

h a s been

Therefore, with t h e

u i n ( 2 . 7 . 1 0 7 1 , t h e a s s e r t i o n of Lemma 2.7.36 i s

t r u e f o r a l l s 2 , 0 < s2 < 1. Theorem 2 . 7 . 3 8 .

Let

w i t h i n t e g e r m 2 1 and E E (O,l).Thenonehasfortheextension

s2 = m-l+e

o p e r a t o r (2.7.55), (2.7.115) 1 m

. .

(2.7.56) u n i f o m Z y w i t h r e s p e c t t o h E (0,h

0

n,+) H ( ~ 1 , ~ 2, h) ( %

1

+ H ( s l,s2), h ( % ) '

and

f o r any s1 E R . Proof. O b v i o u s l y , it s u f f i c e s t o c o n s i d e r t h e c a s e n = 1 , s1 previously, we use t h e notation H ( 0 , s 2 )'

' I -I

( 0 , ~, h~' ) [

-I

Hs,h

and

=

1 1 . I I s,h, [ . I s , ,

so, l e t u E tl

( 0 , ~ , h~ * )

0, so t h a t a s i n s t e a d of

(q*+A S sth ).

p r e v i o u s l y , l e t u . and v , b e d e f i n e d by ( 2 . 7 . 5 7 ) and ( 2 . 7 . 5 8 ) , r e s p e c t i v e l y . 3 3 and Theorem 2 . 7 . 2 6 , one Using t h e d e f i n i t i o n ( 2 . 7 . 7 9 ) o f [ . I ( s ) ,h

g e t s immediately

2. Sobolev Spaces of Vectorial Order

174

where t h e c o n s t a n t C d o e s n o t depend on h and u . I t remains t o o b t a i n t h e estimate

6 C X t t>O

L

- (2e+1)

h ( [ u m - l ( x + t ) - um- 1 ( x ) [

w i t h a c o n s t a n t C which d o e s n o t depend on h and u . W e s h a l l f i r s t prove t h e f o l l o w i n g a u x i l i a r y a s s e r t i o n .

Lemma 2.7.39. =

(2.7.119)

T

=

2

c -

hZ+ and l e t D

2 \,+ . L e t t h e mapping

q llt+q12x+q13hr Y = q21t+q22x+q23h,

be i n v e r t i b l e and t a k e

5

i n t o D;

D~

<

.

r+

Assume, i n a d d i t i o n , t h a t w i t h scme c o n s t a n t

v

(2.7.120) t ( T , y ) 2 C O ( T + Y ) ,

Then fcr any meshfunction f

:

(T,Y)

E

Co > 0

one has:

Di.

xhl ‘ + + c holds:

Proof o f Lemma 2.7.39. D e n o t i n g t h e sum on t h e l e f t hand s i d e o f

( 2 . 7 . 1 2 1 ) by I ( f ) and u s i n g

( 2 . 7 . 1 2 0 ) , one g e t s i m m e d i a t e l y t h a t

I(f)

c

=

t(TrY)

-(2e+1)

If

h ) - f( Y )

.2 2

I

h

6

(T,Y)€D; -

0

(T

1

,y)ED;

c

5 c- ( 2 e + 1 ) 0

2 2 (T+y)-(2e+1) f ( T ) - f ( y ) ] h 6

c

_< C - ( 2 e + 1 )

( T + y ) - ( 2 e + 1 )l f ( T ) - f ( y ) I2h2 =

( T t Y ) EXh,

c

c

y>o

T>Y

+

(T+Y)

(2e+1)

If ( T ) - f ( Y )

I 2h 2 .

2.7. Spaces of One Parameter Families of Meshfundions Making t h e change o f v a r i a b l e s

T

noticing t h a t t+2x b t , V ( t , x )

E

= t + x , y = x i n t h e l a s t d o u b l e sum, and 2

R h , + , we obtain the required inequality

( 2 . 7 . 1 2 1 ) , and t h h t e n d s t h e p r o o f of Lemma 2 . 7 . 3 9 . Now w e c o n t i n u e t h e p r o o f o f Theorem 2 . 7 . 3 8 . Formula ( 2 . 7 . 5 8 ) y i e l d s : (2.7.122) v

(X+t)-Vm-l(~)

m- 1

=

~ ~ + ~ ( x + t ) -(x) u + m- 1

z

um-l(x+t)-um-l(x),

I f x 2 0 then, of course, v

m- 1

(x+t)-vm-,(x)

T h e r e f o r e , f o r proving (2.7.121)

5 c

v

t b 0.

i t s u f f i c e s t o show t h a t

I: t - ( 2 e + i t>O

F i r s t , l e t x 5 -t-(m-l)h

L e t u s p u t z = -x-t-(m-1

h , s o t h a t z b 0. For

such x formula (2.7.122) y i e l d s :

Setting

T

=

pt

y = p z + l c t ] h and n o t i n g t h a t t h i s map t a k e s t h e s e t

{ t > 0 , z 2 01 i n t o i t s e l f , we f i n d

The l a t t e r i m p l i e s

175

2 . Sobolev Spaces of Vectorial Order

176

w i t h some c o n s t a n t C which depends o n l y on m. m Now l e t -t 6 x < 0. L e t u s p u t z = -x, 0 < z 5 t . For s u c h x formula (2.7.122) y i e l d s : v

m- 1

(X+t)-vm-l(x) = u

m- 1

(t-2)-

c c

- ( - 1)

16p';m P

u

I: a'Qm-

m- 1

(pz-(m-l)ph+\alh).

1 ,p

Denoting, as u s u a l , by H(x) t h e H e a v i s i d e f u n c t i o n , H ( x ) : 1 , x 2 0 and

0 , x < 0 , one c a n rewrite ( 2 . 7 . 6 1 )

H(x)

(2.7.125)

(-l)J C C 16p<m

i n t h e following f asi on:

C ~ ( p z - j p h + l a l h )5 1 , V z > ~ E Q .

0, 0 5

j 5 m-1.

I rP

Using i d e n t i t y ( 2 . 7 . 1 2 5 ) w i t h j = m-1, w e o b t a i n t h a t € o r -t 5 x < 0 holds : v

m- 1

(x+t)-v

m- 1

(x)

The t r a n s f o r m a t i o n D

h,arP

=

=

= t-z,

7

y

=

p z - ( m - l ) + l a ( h maps t h e s e t

{ t > 0 , -pt+p(m-l)h-lalh 6 pz < p (m- l ) h- l al hj ,

c D h,a,P

x2

hr+

Moreover, s i n c e 0 5 la\ 2 p ( m - l ) , one h a s \,+. t ( 7 , y ) = ? + y / p + ( m - l ) h - \ a l h / p 2 (T+h)/m, f o r 1 5 p 5 m. Applying Lemma 2.7.39,

we find t h a t

The l a s t i n e q u a l i t y i m p l i e s t h a t t-(2e+i)

(2.7.126)

Z -t<x
t>O

where t h e c o n s t a n t C

m

1

Ivm-l ( x + t ) - v ( x ) 2h2 6 m- 1

depends o n l y on m .

into

2.7. Spaces of One Parameter Families of Meshfundions

171

< x < -t. P u t z = - ( x + t ) , so t h a t 0 < x < (rn-l)h.

F i n a l l y , l e t -t-(m-l)h

I n t h a t c a s e formula (2.7.122) y i e l d s : ( 2 . 7 . 1 2 7 ) ~ ~ - ~ ( x + t ) -( v x) m- 1

=

(-1)

m

z

c

c

16pSm P

+ (-1) m

c

=

z

c

16p6m

(In-')

1 (1-H ( X I

( x + p t )11

aEQm- 1 , p

The e s t i m a t e o f t h e t e r m s

+

( x ) 11

[ H( x ) u m - l ( x + p t ) - u

aEQm- 1 ,p

x=pz-(m-l)ph+lalh

I

x = p z - (m-1 ) p h + a1 h

H ( X ) U ~ - (~x + p t ) - u m - , ( x ) , w i t h x = p Z - ( m - l ) p h + l a l h

i s o b t a i n e d j u s t a s above i n t h e c a s e x + t + ( m - l ) h 5 0 . I t r e m a i n s t o e s t i m a t e t h e l a s t sum o n t h e r i g h t hand s i d e of

( 2 . 7 . 1 2 7 ) . Denote

H- ( x )

= l-H(x),

and n o t i c e t h a t f o r m u l a ( 2 . 7 . 6 2 ) (2.7.128)

(-1)J

1

H - ( p z - j p h + l a j h ) E 0 , V z > 0 , 0 2 j 6 m-1.

1

C

c a n b e r e w r i t t e n i n t h e form

~ E .Q 1 .P

l
Using i d e n t i t y ( 2 . 7 . 1 2 8 ) w i t h j = m-1, on t h e r i g h t hand s i d e o f

Z

(2.7.129) ( - l ) m C C lsp<m The t r a n s f o r m a t i o n

T

w e can r e w r i t e t h e l a s t sum

( 2 . 7 . 1 2 7 ) i n t h e form:

ff - (x)r um-

( x + p t )-u

m- 1 ( 2) 1

aEQm- 1 ,p =

1

I I

x=pz- (m- 1 ) ph+ a h

pt+pz-(m-l)ph+/alh, y = z takes the Set

2 c Moreover, it i s e a s i l y s e e n t h a t t ( T , y ) I- C o ( ~ + y ) , o n t o some D' h,a,p w i t h some c o n s t a n t Co > 0 , which may depend o n l y on m. b' ( T , Y ) E Di,a tP Applying a g a i n L e m m a 2 . 7 . 3 9 , w e o b t a i n t h e r e q u i r e d e s t i m a t e f o r t h a t sum

%,+.

( 2 . 7 . 1 2 9 ) , which t o g e t h e r w i t h ( 2 . 7 . 1 2 4 ) ,

(2.7.126) g i v e s t h e estimate

( 2 . 7 . 1 1 8 ) , and t h a t e n d s t h e p r o o f o f Theorem 2 . 7 . 3 8 .

1

2. Sobolev Spaces of Vectorial Order

178 Remark 2.7.40. Let s

2

=

m+B w i t h i n t e g e r rn t 0 and 0 < 0 < 1. A s a consequence of

Theorem 2.7.20, one h a s u n i f o r m l y w i t h r e s p e c t t o h

E ( 0 , h01 t h e

e q u i v a l e n c y of norms

where, a s p r e v i o u s l y d e f i n e d ,

W e s h a l l d e n o t e by

H(s)

,h(IF()

to

TI+

IF(".

t h e r e s t r i c t i o n o p e r a t o r of meshfunctions i n Since, obviously,

1 1 '+" 1

+ (s

l , O ) ,h

' I Iv'

(s1,0) ,h'

t h e e q u i v a l e n c y (2.7.131) y i e l d s

where t h e c o n s t a n t C d o e s n o t depend on u and h . I n o t h e r words one h a s u n i f o r m l y w i t h r e s p e c t t o h :

and c o n s e q u e n t l y ,

'+

:

(Z")

H(s1,s2)

+

fi

(sl, s 2 )

$ I + ) ,

v (Sl,S2)E

R

x

z,

C o r o l l a r y 2.7.41.

The spaces

H

(sl,s2) ,h

(R:'+)

and H

(z""),

(S1'S2)

s2 L 0 , c o i n c i d e

( a l g e b r a i c a l l y ) with t h e spaces of r e s t r i c t i o n s to

from H(sl,s

)

2

(R:)

and H

(Sl'S2)

< I +

of meshfunctions

( z n ), r e s p e c t i v e l y .

I n d e e d , i t immediately f o l l o w s from Theorem 2.7.38 and Remark 2.7.40 t h a t each meshfunction u i n

H (s),h(IF('+) o r H

(2"")

can be r e p r e s e n t e d

(S)

( R t ) and H (Z"), 1 u with 1 u i n H + m m ( s ), h (S) W e s h a l l i n t r o d u c e a n o t h e r d e f i n i t i o n of norm i n H

i n t h e form u

as follows:

= ~i

respectively. n,+ (sl,s2) ,h(% ) '

2.7. Spaces of One Parameter families of Meshfunctions

179

w h e r e 1 i s a n y e x t e n s i o n operator,

F u r t h e r , w e d e f i n e a new norm i n ti

(zn'+)

a s follows:

(Sl'S2)

Theorem 2 . 7 . 4 2 .

Norms

( 2 . 7 . 1 3 2 ) and ( 2 . 7 . 7 9 )

E

1, s o t h a t norms 0

h

(0,h

are equivaZent uniformZy w i t h r e s p e c t t o

(2.7.133)

and ( 2 . 7 . 8 0 ) are e q u i v a l e n t , V

s1

E R,

Proof. U s i n g t h e e x t e n s i o n operator 1 f r o m Theorem 2 . 7 . 3 8 , m that

one a t once o b t a i n s

+ ( s l r s 2 ,)h ' w h e r e t h e c o n s t a n t C does n o t d e p e n d o n h a n d u , so t h a t

Since

ti

(Z"")

p r o v i d e d w i t h norm ( 2 . 7 . 1 3 3 )

i s a Banach space ( i t i s l e f t

(S)

t o t h e r e a d e r t o show i t ) , ( 2 . 7 . 1 3 5 ) e q u i v a l e n c y o f norms (2.7.133) (2.7.134),

and Banach's theorem y i e l d t h e

and (2.7.80).

shows t h e e q u i v a l e n c y o f

N o w t h e s t a t e m e n t s o f Theorem 2 . 7 . 2 8

H

(<'+)

w i t h a n y s2 >

The same a r g u m e n t , u s i n g

(2.7.132)

and (2.7.79).

I

g e n e r a l i z e t o t h e space

t.

(Sl'S2)

Theorem 2 . 7 . 4 3 .

Let s2 (*

>

f . Then t h e farniZy o f mappings h

. 7 . 136)

ti ( s1,

+

\,

s2), h

i s u n i f o m Z y continuous w i t h r e s p e c t t o h E ( O , h O l . Moreover, t h e map Rh is s u r j e c t i v e and t h e r e e x i s t s a continuous

2 . Sobolev Spaces of Vectorial Order

180

(unifomnly w i t h r e s p e c t t o h) f a m i l y o f l i f t i n g operators Lh '

Proof. __ The f i r s t p a r t o f Theorem 2.7.43 c o n c e r n i n g

Rh, immediately f o l l o w s from

( e x t e n s i o n theorem) and Theorem 2.7.17 ( t r a c e t h e o r e m ) . I t

Theorem 2 . 7 . 3 8

r e m a i n s t o prove t h e e x i s t e n c e o f a l i f t i n g o p e r a t o r a s s t a t e d i n t h e second p a r t of Theorem 2.7.43. Without r e s t r i c t i o n of g e n e r a l i t y , one c a n assume t h a t s1

1 1.1

0 , so t h a t w e s h a l l w r i t e s i n s t e a d of s 2 and

=

Hs, Hsej-+,

i n s t e a d of v e c t o r i a l s u b i n d e x n o t a t i o n . F u r t h e r , w e u s e t h e

notation

1 1 . I 1' 0 ,h

f o r t h e norms o f m e s h f u n c t i o n s on

a . ( x ' ) EHs-j-t,h 1

(IR~-')

,o

<-'.

Thus, l e t

5 j < s-+,

b e g i v e n . W e have t o e x h i b i t a meshfunction u ( x ' , x n )

E Hs(<'+)

such t h a t

and s u c h t h a t

where t h e c o n s t a n t C d o e s n o t depend on u, a . and h . 7 We a r e g o i n g t o u s e t h e m e s h f u n c t i o n s U ( x ' , x ) , 0 5 k < s - ? , i n t r o d u c e d k i n t h e p r o o f of Theorem 2.7.28, and w e w i l l o n l y have t o show t h a t

where t h e c o n s t a n t C d o e s n o t depend on ak and h , s i n c e , a c c o r d i n g t o t h e d e f i n i t i o n of U .

I'

0 5 j < s-t,

i n t h e p r o o f o f Theorem 2.7.28, one c a n

d e f i n e a m e s h f u n c t i o n u as r e q u i r e d i n Theorem 2 . 7 . 4 3 , i n t h e f o l l o w i n g fashion

L e t s = m+B

w i t h i n t e g e r m t 0 and B

so t h a t it s u f f i c e s t o g e t t h e e s t i m a t e s

E

( 0 , l ) . Then

2.7. Spaces of One Parameter Families of Meshfunctions

181

and

w i t h some c o n s t a n t C , which d o e s n o t depend on h a n d a . k The e s t i m a t e ( 2 . 7 . 1 3 8 ) h a s a l r e a d y b e e n p r o v e d ( s e e ( 2 . 7 . 7 8 ) o f Theorem

i n t h e proof

2.7.28).

We s h a l l v e r i f y ( 2 . 7 . 1 3 9 ) . F i r s t c o n s i d e r t h e e x p r e s s i o n

u 1 ( S ' , x n ) , one and i n t r o d u c i n g V ( S ' , x ) = (F k n x'+fS',h k g e t s u s i n g P a r c e v a l ' s i d e n t i t y , t h e f o l l o w i n g e x p r e s s i o n f o r N (U ) : 1 k Setting a

=

(a',a

)

A s h a s b e e n shown i n t h e p r o o f of Theorem 2.7.20

(see (2.7.41)), the

following i n e q u a l i t i e s hold:

where C e depends o n l y on

e

and, a s u s u a l , t . = r h w i t h r = 1 , 2 , 3

Hence, u s i n g a g a i n t h e i n e q u a l i t y 2 / 5 . 1 / 7 1 6 l C . ( C , ) l ,

1

and r e c a l l i n g t h a t s

m+B,

=

I

1

... .

V hC.E[-n,nI,

I

one g e t s u s i n g t h e l a t t e r e x p r e s s i o n f o r N (U 1 , 1 k

t h e f o l l o w i n g estimate:

where, a s p r e v i o u s l y , w e have d e n o t e d 2

X (h,h5')

=

1 +

C

ICj(h,h5.)I

15j < n Using a g a i n t h e i n e q u a l i t i e s ( 2 . 7 . 7 6 )

I

2

.

i n t h e p r o o f o f Theorem 2 . 7 . 2 8 ,

2. Sobolev Spaces of Vectorial Order

182 a

a p p l i e d t o D nVk (E,',xn), and a l s o t h e P a r c e v a l i d e n t i t y , one g e t s f o r n N (U ) t h e estimate: 1 k (2.7.140)

N

1

(U k

1'

5 CI lakl s - k - f , h '

w i t h a c o n s t a n t C which d o e s n o t depend on a k and h . Now, l e t u s c o n s i d e r

U = F x'-tc',h k' Using t h e f o r m u l a f o r Vk (see t h e p r o o f o f Theorem 2 . 7 . 2 8 ) ,

where, a s p r e v i o u s l y , Vk

we easily

find that

a (2.7.141)

D

x , h

=

( V ( 5 ' , x +t ) - V ( 5 ' , x n ) ) = k n n k

a -k - h X ak(S')

r t /h

CL

C

Crk(hX)$rn(p(hh)

rxn/h -1)p

(hi).

lSr<s+f

W e prove t h e i n e q u a l i t y :

where C e depends o n l y on 8. The l a s t i n e q u a l i t y i s e q u i v a l e n t t o t h e f o l l o w i n g o n e :

where D ( t )

has t h e p r o p e r t i e s (2.7.73).

b s t i m a t i n g q ( t ) i n p r e c i s e l y t h e same way p r o o f o f Theorem 2.7.20

as the function g ( a ) i n the

( s e e ( 2 . 7 . 4 1 ) ) , one g e t s ( 2 . 7 . 1 4 3 ) .

I n d e e d , i n t h e p r o o f o f ( 2 . 7 . 4 1 ) f o r g ( a ) o n e u s e s o n l y two p r o p e r t i e s of 2 2 2 2 2 t h e f u n c t i o n s i n ka : s i n ka $C and s i n ka 5 Ck a 'rhe f u n c t i o n

.

I p ( t )rk-l I

h a s t h e v e r y same p r o p e r t i e s

Ip(t)rk-l12

6 Ck2t2, since p ( 0 )

=

:I p ( t )rk-l l 2

1. Using ( 2 . 7 . 1 4 1 ) ,

i n t h e same way as i n ( 2 . 7 . 7 6 ) , t h a t

5 C and ( 2 . 7 . 1 4 2 ) . we f i n d ,

2.7. Spaces of One Parameter Families of Meshfunctions

183

The l a s t i n e q u a l i t y a l o n g w i t h P a r c e v a l ' s i d e n t i t y y i e l d s t h e r e q u i r e d e s t i m a t e f o r N (U 1 , 2 k (2.7.145)

N (U ) 5 C / l a 2 k k

Combining (2.7.140)

1 I s-k-f,h-

and ( 2 . 7 . 1 4 5 ) , o n e f i n d s t h a t

w i t h some c o n s t a n t C which d o e s n o t depend o n a k and h. Finally, (2.7.146),

(2.7.138)

( e s t a b l i s h e d i n t h e p r o o f o f Theorem 2.7.28)

and

along w i t h t h e formula

I

prove (2.7.137). Remark 2.7.44.

Theorem 2 . 7 . 3 8 and 2 . 7 . 4 3 a l l o w o n e t o a s s e r t t h e s u r j e c t i v i t y o f t h e

t r a c e mapping (2.7.147)

u(x',xn)

+

ID1

X

I

hu(x',O)}osj<s

which i s a c o n t i n u o u s l i n e a r map from

n

(Zn-').

05j<s - + " ( s l f s 2 - j - * ) 2 of l i f t i n g o p e r a t o r s

R

-+,

2

They a l s o

H

) (2") o n t o

ee t h e e x i s t e n c e o f a f a m i l y

h'

which a r e u n i f o r m l y bounded w i t h r e s p e c t t o h , s o t h a t o n e a l s o h a s

Obviously,

Rh

i s n o t u n i q u e l y d e f i n e d , s i n c e t h e map (2.7.147)

n o n - t r i v i a l k e r n e l , which c o n t a i n s m e s h f u n c t i o n s u

for

X

=

kh, 0 5 k 5 m .

E H (s

) (2")

has a

vanishing

1' 2

Now w e s h a l l e x t e n d t h e r e s u l t s c o n c e r n i n g t h e t r a c e mapping (2.7.147)

2. Sobolev Spaces of Vectorial Order

184

to the case, when j > s2-f. Denote

T

0

u = u(x',O).

'Theorem 2.7.45.

Let

s = ( s l , f ) , w i t h s1 E R .

Then t h e follow
uI 1 ' 6 C < l r . h>l lul j (s),h ' 0 (s,,O) .h

(2.7.148)

where t h e c o n s t a n t

C

does n o t depend on u and h.

3ne has

one easily finds that

where the constant C

n,ho

depends only on its subscripts.

Integrating (2.7.149) over T (2.7.148)

.

1

n- 1 and using (2.7.150), one gets S',h

2.7. Spaces of One Parameter Families of Meshfunctions where t h e constant

C

185

does n o t depend on h and u.

Proof. With t h e n o t a t i o n i n t h e p r o o f o f Theorem 2 . 7 . 4 5 , Schwarz

u s i n g a g a i n t h e Cauchy-

i n e q u a l i t y , one f i n d s t h a t

The same argument, a s p r e v i o u s l y , y i e l d s 1-2s2

I1

-2s

<5>

h

1-2s

2d5n 6 C

h

I W n 1 (7

T Snrh

depends o n l y on s

where C

I

s2

2'

2 2(<<'>

2 -s

+\<,I

)

2d5n

=

I

The d e f i n i t i o n o f t h e norms [ . I ( s ),h F o u r i e r t r a n s f o r m and d e f i n i t i o n (2.7.132)

(see ( 2 . 7 . 7 9 ) ) , d o e s n o t u s e t h e

I I1.11 I

is o f t h e norms ( s ) .h n o t c o n s t r u c t i v e , which makes b o t h of ther:i n o t always handy i n a p p l i c a t i o n s .

W e a r e g o i n g t o i n t r o d u c e a n e q u i v a l e n t d e f i n i t i o n o f norms i n t h e s p a c e s

of m e s h f u n c t i o n s , which seems t o b e p a r t i c u l a r l y s u i t a b l e f o r t h e e l l i p t i c t h e o r y o f one p a r a m e t e r f a m i l i e s o f d i f f e r e n c e o p e r a t o r s .

.

W e s h a l l f i r s t c o n s i d e r meshfunctions u E H (R') (0), h h For e a c h meshfunction u ( t ) i n H ( 0 ) , h ( R k ) t h e d i s c r e t e < - F o u r i e r t r a n s f o r m -h

(5)

I

1 is a w e l l defined function i n L (T 1 , where T S , h = { < E C ll+ihCl = 1 ) 2
I

being, respectively, (2.7.154) u ( t )

=

(27)-'

and

t/hh (l+ihC) u (<)dg,

I T

< gh

2.Sobolev Spaces of Vectorial Order

186

where idrl

=

dc is the Lebesgue measure on Ti,h, <

=

(ih)-'(eih'-l)

being

the corresponding parametrization (angle coordinate) on T 5 ,h. With H(t) = 1 for t L 0, H(t) = 0 for t < 0, the Heaviside function 1 introduced above, each u E H (0),h(Rh) admits the orthogonal decomposition u(t) = u+(t)+u-(t), u+(t) = H(t)u(t) and u-(t)

=

(l-H(t))u(t)

1

where u (t) has its support in Rh,+ and u-(t) is supported by 1

=+{t E lRh , t < 01. R1 h,We denote by :r the corresponding projection operator h (2.7.156) (n+u)(t) = H(t)u(t). Thus, for each h the spaces H (2.7.157)

1 H (0),h(Rh)

=

'TO)

( 0 ) ,h

,h

--

1 decomposes into a direct sum 2 ,h '

'TO)

1 of the meshfunctions supported by R h,+ and lRh , - , respectively, the subspaces H+(0),h being orthogonal to each other with respect to the inner 1

product (2.7.1) in H

( 0 ) ,h* The proof of the following Paley-Wiener type result for the mesh-

functions (in fact, for the Fourier coefficients of periodic functions) is left to the reader, as an easy exercise. Proposition 2.7.47.

A meshfunction u belongs t o H+

iff i t s d i s c r e t e <-Fourier t ra n sf o rm

(0),h -h u (i) e x t e n d s anaZyticaZZy o n t o t h e e x t e r i o r o f T i . e . o n t o t h e domain 5 ,h' T+ < ,h = {iE C1 [ Il+ih 11, ? ( < I + 0 for 5 + m, V h and

(2.7.158)

J lzh(r)/21dgl5 C , V p 2 1, (l+ih
where t h e c o n s t a n t c does n o t depend on

p.

iff its discrete <-Fourier Analogously, a meshfunction v E H(0),h -h transform v ( 5 ) extends analytically onto the disk 1 T= {< E C I Il+iht/ < 1 1 and (2.7.158) holds for all p S 1 with a irh constant C which does not depend on p . With the orthogonal decomposition (2.7.157) one associates the corresponding orthogonal decomposition of the space L ( T ) of periodic 2 5th functions on T which are <-Fourier transforms of the meshfunctions <,h' in H(0),h(mh) ,

2.7. Spaces of One Parameter Families of Meshfunctions

187

( 2 . 7 . 1 5 9 ) L ~ ( T < , ~= )L ~ ( T ~ , L ~ ; )( T ~ , ~ ) . W e s h a l l d e n o t e by

Il;

+

t h e corresponding pr oj ect i on oper at or s onto

L+(T ) ,respectively. 2 C,h One v e r i f i e s e a s i l y t h a t f o r m e s h f u n c t i o n s u

E H

1

(0), h

(IRh)

with

compact s u p p o r t , t h e f o l l o w i n g f o r m u l a i s v a l i d

= ( f ) y h ( 5 ) + i / ( 2 * ) v.p.

where v . p .

-h u (u)(L-u)-'dli,

J T

V 5

E

Tr,h,

li , h

s t a n d s for t h e Cauchy p r i n c i p a l v a l u e o f t h e c o r r e s p o n d i n g

singular integral. A s it h a s been a l r e a d y shown (see Theorem 2 . 7 . 3 0 ) ,

t h e f a m i l y of

o p e r a t o r s n h a c t s c o n t i n u o u s l y ( u n i f o r m l y w i t h r e s p e c t t o h ) from 1 1 I s2 E ( o , t ) . H ( ~ ,s2) l , h ( I R h ) into ( s 1 , s 2 ), h ( R h , + ) P f o r s1

I t is

q u i t e o b v i o u s t h a t i t i s a l s o t r u e f o r s2 = 0 , so t h a t one h a s u n i f o r m l y w i t h r e s p e c t t o h: (2.7.161)

where

Jli

7(s), h ,

:

( s l ,s2),h

+

7'( s l,s2). h ' v

s1

E R,

v

s2

E ro,t,,

iss,

,h are t h e spaces of t h e d i s c r e t e 5-Fourier transforms 1 1 H ( R h ) and H (s), h ( I R h , + ) , r e s p e c t i v e l y . (s),hNh One f i n d s e a s i l y t h a t i f u (5) i s smooth, t h e n I li i s e x p r e s s e d by a

o f t h e meshfunctions i n

f o r m u l a , a n a l o g o u s t o ( 2 . 7 . 1 6 0 ) . Namely

Of c o u r s e , one h a s a l s o t h e f o l l o w i n g f o r m u l a e :

I t i s c l e a r t h a t e v e r y t h i n g a s s e r t e d above i s c a r r i e d o v e r verbatum

:,+,

t o m e s h f u n c t i o n s d e f i n e d on IR X'

=

( x 1 , ...,xn- 1) E IR;-'

since the tangential variables

p l a y t h e r o l e o f p a r a m e t e r s and one c a n p e r f o r m

t h e d i s c r e t e Fourier transform i n x ' . L e t u s now t u r n t o t h e d e f i n i t i o n o f a new norm i n

H(s)

(R:,+)

, with

2. Sobolev Spaces of Vectorial Order

188

s = (sl,s2),s 2 2 0. A s previously, let

Denote by A+(h,S') (respectively by A-(h,C')) the zero of the quadratic equation

A

2

+ (l+ihA)X2 = 0,

which satisfies the condition Il+ihA-l > 1).

Further, put

(2.7.164) 5 -

Cn-A-(h,C'),

=

1 l+ihA+1

< 1

(respectively the condition

5 , = (Cn-A+(h,C'))(l+ihcn)-'.

It is obvious, that for each real s2 the functions

are analytic and non-vanishing. (R:) , s2 2 0 , be H(sl,s2),h an extension of u onto R E . Theorems 2.7.26 and (2.7.38) guarantee the

let lU

For each u E H (sl ,s2)th

existence of such an extension. Let

Introduce (2.7.165) and

Notice, that norms (2.7.165), (2.7.166) do not depend on the extension operator 1. Indeed, if l . u , j = 1,2, are two different extensions of a meshfunction u E ff

since the function

( s ),h

cn

+

3 (RE")

,

then

s2 5 - (l1u-l2U)- l h

is analytic in T-

2.7. Spaces of One Parameter Families of Meshfundions

1 u - 1 u i s s u p p o r t e d by R 1 and 5'' 1 2 h,-

S i n c e t h e norm o f nh

+

. H(0),h(~

189

vs i s a n a l y t i c i n TLnrh' 2' 1 h ) + H ( 0 ) , h ( ~ h , + ) i s equal t o one,

1

t h e same c l a i m i s correct f o r I I ~: L ~ ( T < , , ~ + ) L+(T h ) . Using t h i s 2 5,. f a c t and ( 2 . 7 . 1 6 5 1 , o n e f i n d s a t o n c e t h a t

where t h e c o n s t a n t C d o e s n o t depend o n u and h . I t is l e f t t o t h e r e a d e r t o check t h a t t h e space

c o m p l e t e w i t h r e s p e c t t o norm ( 2 . 7 . 1 6 5 1 ,

H

( s l'S2)

( 2 . 7 . 1 6 6 ) . Moreover,

(z"'+)

is

(2.7.167)

yields:

Hence, by v i r t u e of B a n a c h ' s t h e o r e m , one h a s a l s o

where t h e c o n s t a n t C1 d o e s n o t depend on u . Thus, we have p r o v e d Theorem 2.7.48.

The norms ( 2 . 7 . 1 6 5 ) , ( 2 . 7 . 1 6 6 ) and ( 2 . 7 . 7 9 ) , ( 2 . 7 . 8 0 ) are e q u i v a l e n t . I n conclusion, using d e f i n i t i o n (2.7.165),

H(s)

(Z""),

( 2 . 7 . 1 6 6 ) of t h e norms i n

w e s h a l l p r o v e one u s e f u l f a c t ( l o c a l i t y p r o p e r t y ) c o n c e r n i n g

these spaces. Theorem 2.7.49.

Let 4 E

~ ( 2 ' )Then .

where t h e c o n s t a n t c S ( $ ) depends o n l y on 4 and s2. Proof. Without r e s t r i c t i o n o f g e n e r a l i t y , o n e can assume t h a t s1 = 0 . F u r t h e r , it s u f f i c e s t o v e r i f y t h e i n e q u a l i t y

2. Sobolev Spaces of Vectorial Order

190

we find

=

where A-(h,D

X'

)

L

1 1 (Dxn,h-n-(h,Dx,))sl(@u)1 ,:I

,h,

is the difference operator defined by the formula

-1 A-(h,D ,)u = FE; 1

,h

'IFx'+5 ,hU'

A-

I

Just as in the proof of 2.7.14, one shows that

where the difference operator B admits the estimate

where the constant C1 does not depend on h and u, as a consequence of Theorem 2.7.38. Since l u E u for x E R n r + , (2.7.171), (2.7.172) yield h the required inequality (2.7.170). I Example 2.7.50. 1". Consider the meshfunction Eh(. ,y) (2.7.173) Eh(x,y)

=

(?)

:

1 Rh

+

C,

(l+h2/4)-if3+(h) Ix-yl'h,

9+(h)

=

l+h2/2-h(l+h2/ 4 ) f , h

>

0.

It is easily seen that E (x,y) is the discrete fundamental solution for h -l), which vanishes the difference operator l+Dx,hD:,h, Dx,h = (ih)

-'

(@X,h

at infinity. Indeed, an easy computation shows that (2.7.174) Fxt5 ,hEh

=

<<(h,hS)>-2eiYS

Hence,

'

2

IEh' (sl,s2) ,h

-2s =

h

1

-4+2s

l1 T

5.h

'dS

-

2.7. Spaces of One Parameter Families of Meshfunctions

-

C

{

191

3-2(sl+s2) h ;If:I
Therefore. Eh E f f ( s

h>

,

if s2

,

if s2

=

3/2

, if s2

<

3/2.

3/2

iff (sl,s2) < (0,3/2).

) (Z)

1’ 2

2”. Consider the meshfunction v (2.7.175) v+(x)

=

e+(h)x/h,

+

:

R1’+

h

x E R;”,

where 0 (h) is given by (2.7.173). An easy computation shows that for any integer m 2 0 holds:

where C depends only on m. m Further, one finds that 2

(2.7.177)

1 lv+(x+t)-v+(x)1 1’ (sl.O) ,h =

-2s h

-2s1 + 1 2 -1 (1-8+(h) ) = h (1-8+(h)t/h)

1 le+(h) (xct)’h-8+(h)x/h/2h

xeo

-

-2s

C h

=

(1-8+(h)t/h) 2

where the constant C does not depend on h. In fact

Therefore, using (2.7.177) and (2.7.79) with n

=

1, m = 0, 2‘

E (O,l),

one gets that

Since lim e+(h)t/h = e-t, h+O one can rewrite (2.7.178) in the form 2 (2.7.179) [v

3’

+ (Slr8),h

-2s1

m

0

t-(2e+i) -t 2 (1-e ) dt

- Clh

-2s

1

given that the integral in (2.7.179) is convergent when 0 < 8 < 1. Combining (2.7.1761, (2.7.179) and using (2.7.79) with n = 1, one gets the conclusion that

2. Sobolev Spaces of Vectorial Order

192 where C

depends only on

S

s2

2'

given by (2.7.175) belongs to all ff

Thus v

(sl , s 2 )

(Z')

with

s 1 6

0.

Now we are going to use norms (2.7.165) for showing that (2.7.180) holds 2 for all s E R . One finds easily that for n = 1 (2.7.164) takes the form

<+

(2.7.181) 5 - = <-A_,

=

<-A+, with A+-

2 f 8 (h) = l+h2/2 3 h(l+h /4) f

=

-1 (ih) (8+(h)-l),

.

Hence, using (2.7.165) one finds for v+ given by (2.7.175)

1 Taking as 1 the extension operator lo which extends v+ onto Rh by putting

it zero for x E R L f -

,

one finds immediately that -1

(2.7.183) (lov+)-'h where

<+

=

-i<+ '

is given by (2.7.181).

Thus, (2.7.182), (2.7.183) yield

If 1s21 <

<52<,1

+, then

operator L2(T1

)

Now assume that

s

5 th

,

2

E L (TS,h) so that

n;

being a projection

one finds immediately that

= m+8

where m

>

0 is integer and 181 <

f . Using the

formula

and observing that s -k-1

(2.7.187) IIh< +

(l+ih$) = 0 , k = 0,l. ...

we reduce the case considered to the previous one, when Now, let s

2

=

-m+8 where m > 0 is integer and 181 <

I s2 I f.

< f

.

2.7. Spaces of One Parametzr Families of Meshfunctions

193

Using t h e formula

I s2I

a g a i n t h e c a s e c o n s i d e r e d i s r e d u c e d t o t h e o n e , when Thus, f o r a l l s

=

so t h a t v

(Z+),

The m e s h f u n c t i o n

<

t.

(sl,s2)E R2 holds

v

Gh(. ,y)

(Sl,S2)

E

:

n2

+

c1

i s t h e d i s c r e t e G r e e n - f u n c t i o n f o r t h e d i f f e r e n c e o p e r a t o r 1+D D* on x,h x,h R1 r + h . S i n c e Eh g i v e n by (2.7.173) b e l o n g s t o a l l H (Z) with s < (0,3/2), (S)

( 2 . 7 . 1 9 0 ) y i e l d s t h e same r e s u l t f o r G, 3". Let q = (3-/5)/2.

:

Gh E H

( S )

(2')

i f f s < (0,3/2).

The m e s h f u n c t i o n

i s t h e d i s c r e t e f u n d a m e n t a l s o l u t i o n f o r t h e d i f f e r e n c e o p e r a t o r 3-0 -0-1 h h ( w i t h O h t h e s h i f t o p e r a t o r ) , which v a n i s h e s as 1x1 + One f i n d s e a s i l y

-.

that Fx+S,hEh = l + / ~ ( h t ) =1 1~ + 4 s i n 2 % 2 ' F u r t h e r , a n e a s y c o m p u t a t i o n shows t h a t

Hence, Eh

E

H(s)(Z) iff s

< (O,-f).

F u r t h e r , c o n s i d e r t h e m e s h f u n c t i o n v+ : RL'+ v+(x)

=

qx'h,

x E

+ C,

nl,+ h .

Again, u s i n g t h e argument s i m i l a r t o t h e one i n 2',

o n e shows t h a t

2. Sobolev Spaces of Vecto&l Order

194

Therefore, for the Green function

-1

of the difference operator 3-0h-Oh

on RL'+

the following inclusion

holds :

2.8. Some spaces of two parameter families of meshfunctions Two parameter families of meshfunctions appear after the approximation

by finite differences (or finite elements) of singular perturbation problems affected by the presence of a small parameter

1 0

parameter h E (0.h

E

E

( 0 . ~ ~ 1the , second small

being the meshsize.

Consider mesfunctions with complex values: (2.8.1)

u ( E , ~ , x ): %(

+

C,

(~,h) E li

EO'ho where we have denoted

and shall write Il instead of I[ when it does not lead to confusion. &OTh0 Sometimes u(~,h,hk),k E Zn, will be interpreted as a function of

(E,h) E TI whose values are in some Banach or topological sequence space. With S'(Zn) the Frechet space of sequences introduced in Chapter 1 for each meshfunction u : Il + s' (Z") , its di-Crete Fourier transform -h u = FX+[ h~ is well defined, and moreover, > I

2.8. Spaces of Two Parameier Families of Meshfunctions where, a s p r e v i o u s l y , 5

IId/

2

Define H u

:

iI

-t

n

(T<,h)

ck

=

195

(exp(ihck)-l)(ih)-' and

=

i s t h e norm in L ~ ( T )~. trh

(mn) t o be t h e c l o s u r e o f t h e s e t of t h e meshfunctions (s).E,h h 1 (zn) w i t h r e s p e c t t o t h e norms (2.8.3) f o r each ( ~ , h ) E 1T. 1

Further, l e t H ( S )

(Zn) be t h e s e t of a l l meshfunctions u E

H ( s ) .E,h(m;)

such t h a t

It is quite obvious that for nonnegative integer s2,s3 the norms (2.8.3) are equivalent (uniformly with respect to (&,h) E Il

)

to the

€0rho

following ones:

where, as previously, a and 6 are multiindices and D

x,h

=

(DX1,hr.. - ,Dx ,h) n

with iD the forward difference derivative in the direction of x k' xk,h = (Ox -l)/h, (0 u) (x) iD xk'h k Xk The following

=

u(x+hek).

Proposition 2.8.2. H(s) (z") i s a Banach space.

is proved in exactly the same way as Proposition 2.7.2. For u E

H

(s)

(Z") introduce the seminorm

and denote by H o

(S)

(Zn) the set of all u E ff

( S )

(Z") such that ( u ( (s) = 0.

Proposition 2.8.3. HYs) (z") i s a c l o s e d subspace of H ( s ) (z"). Proof. -

The statement is an immediate consequence of the triangle inequality for the seminorm (2.8.6). Definition 2.8.4.

Define t h e space H (2.8.7)

(s)

H ( ~ (z") ) =

(z") t o be

(zn)/HYs)(z"),

H (S)

2. Sobolev Spaces of Vectorial Order

196

the latter being provided with the quotient norm induced.by the structure of the quotient-space, i.e.

where u E ff class

u

E H

(Z") is any meshfunction which belongs to the same equivalence (S)

(z").

(S)

Example 2.8.5. With H(x) the Haeviside function, the meshfunctions (2.8.9)

u(h,x) = (l+h)-(X+h)/hH(x), v(x,h) = (l-h)X'hH(x),

x E

1 (Z ) belong to the same equivalence class.

as elements of H (S)

Indeed, an elementary calculation yields

0 (Z'). (0) Notice that u and v are solutions of the difference equations

so that (u-v) E H

(az,h+l)u = 6h (x) and

( a x,h+l)v

= S

h (x+h), respectively, where 6h ( x )

=

the Kronecker symbol, is the difference &-function. = h-16x,0 with 6 x,o Moreover, u and v are the only solutions of the corresponding difference equations above, which vanish at infinity. Using the discrete Fourier transform, one checks easily that the meshfunction w,

is the only solution of the difference equation ($(ax,h+a*x,h)+l)w = 6h (x), 1 + m , x E IR h' Indeed, via discrete Fourier transform one gets the formula

which vanishes as 1x1

w(h,x)

=

-'

i ~~'~(e~+2h+i)-lde,

(Ti)

jBI=l

so that the residuum calculus yields (2.8.10). One has

+

t

when h

-f

0.

Hence, u and w do not belong to the same equivalence class in H

(Zl),

(0)

2.8. Spaces of Two Parameter Families of Meshfunctions

197

although the accuracy in the approximation of the differential operator a +1 by the finite difference operator f ( a x,h+a*x,h)+1 is O(h21 , while the +1 and a* +1 approximate the same differential difference operators a x,h x,h operator with the accuracy O(h). Also the essential difference in the behaviour of u and v, on the one hand, and w on the other, is reflected by their asymptotics when h + 0 . One has u(h,x)

while w(h,x)

=

e-X H(x) (1+0(h)), v(h,x) = e-xH(x) (1+0(h)),

-

e-IX'[ff (~)+(-l)~/~H(-x-h) 1

Example 2.8.6. With p = h/E consider the meshfunctions (2.8.11)

u(E,h,x) = (l+p)-(X+h)/hH(x), v(E,h,x) = (l-p)X/hH(x).

Assuming that p < 2 , one finds that

+

when

p

0 when

E +

0, h

+

0 and

p < 2,

1 H ( 0 ) ,E,h(lRh)

but v $?

t 2.

With (2.8.12)

w(E,h,x)

=

e-X'EH(x)

one finds that

where C

=

sup( (2p+p2)-1+(l-e-2P)-1-2(l+p-e-P)-1)

<

m.

pd0

Hence, u and w belong to the same equivalency class in H

1 (0) (z )

.

Notice, that w = exp(-x/E)H(x) is the solution of the differential equation

(Ea

+l)w

=

E~(x),which vanishes at infinity, while u in (2.8.11)

is the solution of the difference equation

(€a*

+ l ) u = sAh(x). x rh Notice that the meshfunctions E - ~ U and E - ~ Wwith u and w given by

(2.8.11) and (2.8.121, respectively, do not belong to the same equivalence 1 1 nor do they in any H (Z 1 . However, the meshfunctions

class in H ( o ) ( Z

E - ~ Uand (E+h)-'w

(S)

do belong to the same equivalence class in

H

(z')

(S)

2 . Sobolev Spaces of Vectorial Order

198

with any s = (sl,O,O) such that s1 <

-+. Indeed

a straightforward

computation yields:

where ~(p= ) ((l+p)2 (2p+p2)-1+(~-e-2')

-'-2(1+p) (l+p-e-')-')

5 cl, V p t O .

Example 2.8.7. m

Let f E C (IR) and let the meshfunctions u ( E , ~ , x )and v(E,~,x)be given, 0

as follows: -1 -ihS -1 u = FS+x,h~(l+p-e ) Fx+E,hf

(2.8.13)

It is quite obvious that u and v are the solutions of the difference equations with small parameter

which vanish when

x

I

+

E

m.

Since the difference equations (2.8.14) approximate the same differential equation (Ea +l)w = f(x), x E I R , one should expect that both 1 u and v belong to the same equivalence class in some H (s)( Z 1 . Since m

f E C O ( R ) ,it turns out that this holds for V s

=

isl,s2,s3)with s1 < 1.

Indeed, since

I

( l+p-e-ihc)-1- (

(

I-e-P-ihS) -1 1 5 C / ~ - e - ~ ~ ' l *

one finds, using (2.8.3) (and assuming for simplicity

I Iu-vI

2 (s),E,h

Applying (2.7.11) to

'''-v"

-2s 5 C(E+h)

=

-1

1 l?rhf/I (0,s + s +l),h

(s),E,h

E

0 2s 2s 1 2<(E+h)p 3

2-s3 1 5 C(E+h) h(l If

one gets finally the estimate

Is

+s +1 +ChllflJs1+s 2+2 1

1

2

2.8. Spaces of Two Parameter Families of Meshfunctions

SO

199

that u and v belong to the same equivalence class in any H ( s ) ( Z

1

)

with

s1 < 1. Example 2.8.8. Consider the meshfunctions u(E,h,x) = [(e

-AX

-x/E)+~~-x/E -e 1 H(x),

-p+Ah -1 -Ax - X / E -X/E v ( E , ~ , x )= [(l-e-’) (1-e ) (e -e )+@e where A > 0 and 4 are given, p = h/E, h E ( 0 , 1 1 ,

E

1

H(x),

E (0,(2A)-1].

It is easily seen that p-Ah 2 Ah, so that

Hence,

- hx-e-x/ E )H(x), (e -(A+iS)h-e-(p+ihS))(l_e-(~+iE)h-1 -(l+iEc)p -1 h(e ) (1-e ) .

A straightforwardcomputation yields for w

-h wA r E ( 5 )

=

(FX+S,h~A,E) (5)

=

X,E

=

One checks easily that with some constant C > 0 holds

Further, assuming that s2+s3 < 3/2 and that either s1 < 1 , s 2 5 $ , or that f < s2 < 3/2, sl+s2 < 3 / 2 , one gets using the last inequality and carrying

out a straightforward computation the following inequality:

where y = 1-s -max{O,s - 4 ) > 0 . 1 2 Hence, the meshfunctions u and v above belong to the same equivalence 1 class in any H ( s ) ( Z ) with any s as described above. However, they are not in the same equivalence class when, for instance, s the difference derivative of w

is of order X,E

=

E - ~ in

(0,2,0).Indeed 1

H (0),E,h(Rh), since

2. Sobolev Spaces of Vectorial Order

200

aXwx,€ =

(E

e-X/E-he-hX)H(x).

Notice that u(~,h,x)is an asymptotic solution of the boundary value -AX

problem: (Eax+l)y = e

,

x E lR+

,

y(0)

=

4 , extended by zero on R - ,

while

v(E,h,x) is the solution of the difference boundary value problem: (l-e-')-'(l-e-'O-')v

=

e-Ax , x € R h +\{O},

v(E,h,O)

=

4,

extended by zero on R the difference problem being an approximation h,- ' of the differential one with the accuracy O(h). The fact that u and v belong to the same equivalence class in

H

(s)

(Zl), with s, as described above

(i.e. either s < 1, s2 4 t , s2+s3 < 3/2,or s +s < 3/2, t < s 2 3/2, 1 1 2 s 2 + s 3 < 3/2) (and also the fact that I ly-ul I 0, as E + 0, h + G ) ( s ) ,E,h means that v, as a solution to the approximating difference scheme -f

converges to the solution y of the differential problem with small para1 ( 2 ) with s as described above. meter E in any ff (S)

It is easily seen that the meshfunction w(&,h,x), solution to the difference boundary value problem (€8 +l)w = e , x E %,+ , extended x,h does not belong to the same equivalence class as u and v by zero on IR hr-l' abc-ve in tf ( Z ) , V s , although this difference boundary problem too is an -AX

(s)

approximation to the differential one with the same accuracy O(h). However, this scheme differs essentially from both the differential problem for Y(E,X) and difference problem for v(E,~,x),since the corresponding has the solutions of the form $(l-p) x/h , homogeneous equation on 37 h.+ which do not vanish when p > 2 , & + 0, h + 0 , x 2 xo > 0 , i.e. they are not of boundary layer type. Definition 2 . 8 . 9 .

1 1 . 1 1 ( s ) ,E.h a r e s u i d to be s t r o n g e r than t h e n o r m 1 1 . 1 1 ( s ' ) ,E,h i f t h e r e e x i s t s a constant c = c which may depend onZy on i t s

The noms

' ,cO ,ha'

s ,s

s u b s c r i p t s and such t h a t

It is quite obvious that a necessary and sufficient condition for (2.8.15) to hold is the inequality s -s'

(2.8.1G)

(E+h)

'
s'-s

s;-s3

2<(&+h)p

< cS,S'.

2.8. Spaces of Two Parameter Families of Meshfunctions

20 I

Propcsition 2 . 8 . 1 0 .

I I . j I ( s ),E.h are

he norms

stronger than j

1 . 1 I (s'),s,h'

if? s > s!.

Proof. first 151

Taking alternatively in ( 2 . 8 . 1 6 ) c = h =

l€,l-',

=

1, E

0, h

h

immediately the conclusion that the condition s (2.8.16)

-f

c * 0 and finally c = 1 , h =

>

0, afterward

-+

one gets

+ 0,

s' is necessary for

(and, therefore, for ( 2 . 8 . 1 5 ) ) to hold.

On the other hand, it suffices to show (2.8.16) with s' = 0, s

>= 0 .

One gets in this case: -S

(E+h)5>s3 =

(E+h)

(

(E+h)<<>)- s 1 < p s 1 + s 2 <

? C2'

2 C,,<(E+h)G>

where

C2

depends only on

(E+h)5>s3 2

s2+s3

s -s

(for c O 5 1 , ho 5 1 , one can take C2 = 1 ) . #

E O,ho

The following statement (interpolation inequality:, is proved in exactly the same way, as Proposition 2 . 1 . 1 0 . Proposition 2 . 8 . 1 1 . Let u E lR3 (2.8.17)

,

s E lR3

,

s

iu

and let

p = s-t(~-s), t > 0.

Then with V 6 > 0 holds:

In Chapter IV boundary value problems for one-dimensional difference singular perturbations will be given a special attention, so that in the rest of this section it will be assumed that n = 1 . Of course, the results tc be discussed can be easily extended to the case of or a layer

$-'

x

<,+

= m;-l nh,+ 1 Ih, with Ih = lRhn I, I = (a,b) c R being a finite

interval, or to Tn-' h on the torus Tn-'.

Ih, where TE-l is a one parameter family of meshes

X

Denote by n u the restriction of a meshfunction u E 0 point x = 0 : 71 u = u ( ~ , h , O ) .

H

( s ) ,E,h

0

For a function 4

:

*

I! EO'ho

C,

@ E Co (II

)

€0th0

introduce

(R) to the

2 . Sobolev Spaces of Vectorial Order

202

(2.8.19)

-1 ,h = (E+h) l$(c,h)\,

[$I1,

[$I1

sup [bll,E,h, 1 E R . E,hEli

=

the family of complex planes equipped with 1,Erh and by C1 the Banach space of all functions the family of norms [-I 1,E,h : Jl -f c with the norm [$Il finite. We shall denote by C

Again, one can introduce the seminom 1

[$Il

-

=

lim ,h)+(O,O)

V

["1,E,h'

$

E c1

(E

and, with Co = { $ E C1 1 [$I; = 01, consider the factor space C1/Co 1 with 1 the quotient topology induced by the norm in

5-

Example 2.8.12. Let f E Ci(IR) and let $(E)

= (2111-l

where, as usual,

p =

2 2 -1 I, (ICE E )

f(E)dS,

h/E.

With U(E,X) and v(E,~,x)solutions of the problems:

it is easily seen, that

$(E)

= T ~ U , + ( ~ , h )=

n 0 v.

Further, Cauchy-Schwarz' inequality yields with V s > 3 :

where

I If I I s

is the norm of f in the usual Sobolev space H s ( R )

we have denoted: 2 2 -1- fl-e-P)211-e-P+ihEI-2. @(~,h,c) = ( 1 + ~ 5 ) A

straightforward computation shows that

where C is a constant. The last inequality yields:

and where

2.8. Spaces of Two Parameter Families of Meshfunctions

203

provided that s > 3/2. Hence, $(E) and $(E,h) belong to the same equivalence class in C1/C:

with V 1 < f.

Lemma 2.8.13.

L e t u E ti

( s ) ,E,h

1 (nh) with

s > f , s2+s3 > f . Then 2

i s a continuous l i n e a r mapping u n i f o m Z y w i t h r e s p e c t t o

.

E Il

(E,h)

€0rho Proof. Applying Cauchy-Schwarz' inequality to -1 u(E,h,O) = ( 2 ~ )

one gets

-2s

(E+h)

1

-h u (~,h,5)dS

J1 T

5 rh

lu(E,h,O)1'

Using the inequalities 151 5 151 S

where the constant s2+s3 >

C1

-2s

2

J I I' I I ( s ) ,e,h T1

5 (21r)-~

<5>

-2s

2<(c+h)p

3d5.

5 ,h 1 (1r/2) 151, V h > 0, V 5 E T5,h,

depends only on

s2

, ~ 3 , ~and 0 ho, given that s 2 > f ,

n

f.

A slight generalization of the previous lemma stated here below, is proved

by the same argument, Lemma 2.8.14.

Let u E H

1

(wh)w i t h

s > f , s2+s3 > 2 k < minIs2+f,s2+s3+f} t h e mapping

(2.8.21)

( s ) ,c,h

1 3 u ti ( s ) ,c,h (Rh)

+

t . Then f o r each

n 0 {DPx,h -'} I
-

-'s

1

1

x.--xcs 1

.

i s uniformly continyous w i t h r e s p e c t t o (E,h) E II

k

€0nho

Lemma 2.8.15.

Let u E H

( s ) ,E,h

1 with (mh)

s2 <

f , s2+s3 > f . Then

1

(2.8.22) l f ( s ) , ~ , h ( ~ h )

+ *OU

Csl+s2-f

i s a l i n e a r continuous mapping uniformly w i t h r e s p e c t t o (E,h) E II EO'ho

2. Sobolev Spaces of Vectorial Order

204 Proof. -

Using again the cauchy-Schwarz inequality, one gets

-

1 Again using the equivalency 15 I 151, v h > 0 , V 5 E T5,h, one can estimate the integral on the right hand side of the last inequality, in

the following fashion: f1 T

2Sl -2s -2s (E+h) 2<(E+h)p 3d5 S

5 rh 6 C

2Sl

I(E+h) 2(s

5 Cl(E+h)

-2s -2s <<> '<(E+h)E> 3d5 6

+S

-t) JICj R

-2s2 -2s 2 (s1+s2-f) <<> 3d5 = C (E+h) . 2

I

Remark 2.8.16. I f u E H (s),E,h(4) with s2 =

4,

s +s

2

3

> f , then holds

where the constant C depends only on s3,E0,h0. Indeed, this follows immediately by the previous argument. Remark 2.8.17. Obviously, the statements of Lemmas 2.8.13 - 2.8.15 and Remark 2.8.16 are also true for the meshfunctions in H(s) (Z") and H at x

=

(Z"), and their traces

(s)

0.

4

{x E I x 2 0 ) will be Now spaces of meshfunctions on %,+= 1 with introduced. For s2 5 0, s3 2 0 integer define the norms 1 I . I I + ( s ),E,h subscript s = (sl,s2,s3) in a natural way:

( R 1 ) the family of spaces of the meshfunctions u and denote by H (s),~,h h,+ with norms (2.8.24) finite for each 1 Analogously denote by H(s)(Z+) the space of the meshfunctions u such

that

2.8. Spaces of Two Parameter Families of Meshfunctions + 1 and set, a previously H ( s ) ( ~ + ) =

1 + O 1 H+( s ) (Z+)/H(s) (Z+),

205

+O 1 where H ( S ) ( Z + ) is the

subspace of the meshfunctions u in H+( s ) ( 2 ; ) such that

We are goint to use the extension operator 1 defined by (2.7.55), (2.7.56) m with m

=

s +s

2 3’ The following statement is an immediate consequence of Theorem 2.7.26.

Lemma 2.8.18.

For s2 t 0,

s3

t 0 i n t e g e r the. e x t e n s i o n operator lm d e f i n e d by (2.7.551,

(2.7.56) w i t h m = s 2 + s 3 i s a family of continuous l i n e a r mappings: (2.8.26)

Is2+s 3

’

1 H ( s ,~,h(=h,+) ) -f

uniformZy w i t h r e s p e c t t o (s,h) E Il

1 ff

( s ) , ~ , h ( ~ h’ )

.

EOfho Using the extension operator 1 = lm, one can define spaces H ( ~ ,) E , h ( ~ L , + )f o r v s E w3 . Definition 2.8.19.

With (2.8.27)

I I’I

1;s)

,E,h = inf 1

I IlU/I ( s ),E,h

where t h e infinimum is taken over a l l possibZe e x t e n s i o n s 1 of u t o R 1 h’ 1 1 d e f i n e t h e spaces ff of all meshfunctions u : mh,+ + c w i t h ( R ~ , + ) ( s ) ,E.h norms (2.8.27) f i n i t e f o r each (E,h) E Il EO

A n a l o g o u s l y , t h e spaces H

,h0’

1

( z + ) are d e f i n e d t o c o n s i s t of a l l mesh-

(S)

f u n c t i o n s u such t h a t

1

and t h e spaces H ( s ) ( z + ) t o be t h e f a c t o r - s p a c e s H

HYs)

1

(Z+)

1

1

( Z + ) / H ; ~ ) ( z + ) , where is t h e subspace of a l l meshfunctions u E H ( z + ) such t h a t (s)

( S )

Remark 2.8.20. 1 With H- (S),E,h(%$,)

the subspaces of all meshfunctions in 1

which vanish for x E R h , + , one has the isomorphism

H

1 ( s ) ,~,h(”h)’

2. Sobolev Spaces of Vectorial Order

206

Remark 2.8.21. One can show that with s2 t 0,

s3 2 0

introduced by (2.8.24) and the norms

integer, the norms

1 1 . ! I+

(s),E,h'

1 11.11

,E,h defined by (2.8.27),

are equivalent uniformly with respect to (E,h) E II

EO,ho' It is useful to have a more constructive definition of norms in 1 spaces H (I?$,+) , which would not require the minimization over all ( s )rE,h possible extensions. To that purpose we shall use the discrete <-Fourier transform and carry over constructions introduced by the end of Section 2.7 More precisely, we are going to use the discrete Mellin transform M x+8 ,h defined as follows:

-1 The inverse discrete Mellin transform Ms+x,h has the form

where the upper + means that the unit circle T1 is run counter-clockwise. Parceval's identity for M takes the form: x+8 ,h

(2.8.33)

1

xC4

2 -1 lu(h,x)! h = (2nh) Jl[zh(h,8) 121d8\, T 1

where Id01 = d5 with 5 the angle coordinate on T

:

8 = exp(i5).

Denote

With :n

the multiplication of a meshfunction u

1

: IF$,+

C by Heaviside's

function ff(x), we shall write down the discrete Mellin transform of

I I ' U .

h

Obviously, one has

Substituting (2.8.32) into the right hand side of (2.8.34), summing up the geometrical progression and introducing the notation:

2.8. Spaces of Two Parameter Families of Meshfunctions

207

one obtains, using the Pleijmel formula:

-h (t)u (h,8)+(2i~i)-lv.p.

=

F1 (e-A)-l Gh(h,X)dh,

1 V 8 E T .

T

i by R L and

Denoting the discrete Mellin transform of the operator a using the notation: (2.8.37)

-' lim

(0

-1 ,

(B6-A)

=

~ E T1 , X E 1T ,

6++0

where the limit in the right hand side of (2.8.37) is taken in the sense of distributions on T1, one can finally write down: (2.8.38)

(1;

xh)(h,B)

=

(hi)-' l; T -1

= (2.rri)

where <

,>

(7;)

<(8+o-.)

-1 -h u (h,A)dh

-1 -h ,U

=

>,

is the duality between testfunctions and distributions on T 1 .

Introducing the operator (2.8.39)

(B+o-X)

(h,x)

=

7-

h'

(l-H(x))u(h,x),

and using the same argument, one gets for the discrete Mellin transform Rh -

of rh the formula (2.8.40)

(n,

zh)(h.0)

=

(hi)-'

/1 (8-0-X)

-1 -h

u (h,X)dA =

T =

-(2vi)-1<(a-o--)-1 ,u -h >,

where

and the upper - means that the unit circle T1 is run clockwise. By the Pleijmel formula, one obtains: (2.8.42)

(I[h - -uh ) (h,B) = ( f ) uh (h,8)-(2ni)-l v.p. < ( 8 - A ) - '

Gh(h,X)dh,

T

so that for the meshfunctions u(h,x), which decrease rapidly as 1x1

one has of course (2.8.43)

(Ilizh) (h,B)+(IIh $1

(h,B) = Lh(h,e),

+

-,

2. Sobalev Spaces of Vectorial Order

208 and

% ((n h+-Il h- )u )(h,8)

(2.8.441

=

(Ti)-' v.p. <(O-A)-'

Yh(h,X)dX,

T

V h > O , V8ET1.

yo)

4,-

h, i.e. u E o for x E and Notice that if u E H 1 , then I[h- $ 0. Indeed, Cf"(h,A) is analytic for ] A ] > 1, -h 2 1 since uh z o for x E R and u E L (T ) for each h E W+. The function h,- ' X + (e6-h)-l being also analytic for 1x1 > 1 and V 6 < 0, the right hand

u E ff(o),E,h(\)

side of (2.8.40)

X

-t

vanishes identically in this case, since it is a limit

-0 of the corresponding integral over T1 3 X with the integrand -1 -h (e6-h) u (h,X).

when 6

+

one gets an analoguous conclusion for u E and 'I i.e. II+ Gh 5 0. u : 0 for x E W 1 h,+ h' h Hence, one has the orthogonal projections: h

'iT O ) ,E,h

(2.8.45)

-h

where H-

( s ),E,h

HTs), E , h '

h =

4

T O ) ,E,hr

H-(0),E,h'

1 i.e* '(0) ,E,h(%)

h T O ) ,E,h

= o

stands for the Mellin transform of the meshfunctions in

respectively.

One checks easily the following more general formula:

t Introduce the functions <,,h

2

2

:

1

where 8+(h) = l+h / 2 7 h(l+h / 4 ) 2 , FO;

V t

T

1 -f

C,

/8+(h)[ < 1 , /8-(h)( > 1, V h > 0.

E R , V h E R+ , the function Ct

vanishing when 181 > 1 (including 8 = analytic and non-vanishing for 18 Now introduce for each s

=

I

m)

( 8 ) is analytic and non+ rh t and the function 5- ( 8 ) is ,h

< 1.

(sl,s2,s3) E R3 the norms of order s , as

follows:

V u E H

1

(s),~,h(wh,+)'

1 where, as previously, p = h/E and 1 u stands for the extension of u on IRh 0 1 by putting u Z 0 for x E IR . h.-

'

2.8. Spaces of Two Parameter Families o j Meshfunctions

Indeed, 5 -s2hS-:p((lu)" h

" h ) -(leu)

H-h

E

209

so that (2.8.45) yields (2.8.49).

( 0 ) ,E,h'

Lemma 2.8.23.

The norms (2.8.48) and (2.8.27) are equivaZent in H

.

uniformZy with respect t o ( ~ , h E ) Il

1 ( s ), ~ , h ( ~ h , + ')

EO'ho Proof. -

-

2 1 Since Il+ is an orthogonal projection in L (T ) and, obviously 16 I uniformly with respect to ~ , Eh Il when I3 = exp(ihS),one

-

finds, using (2.8.49), that

IuliS)

,E,h 5

where the constant

C

CI

jlul I

( S )

.E,h'

V u E

1 H ( S ) ,E,h(%,+)

does not depend on ( ~ , h ) E Il

5 c inf

IUITs),E,h

1

and the extension 1. EO rho

The last inequality yields:

1 llul I ( s ) ,E,h = c I'I

'ls)

,E,h'

V (E,h) E Jl

1 V u E H (s),~,h(%,+)

EO,ho'

.

One uses the open mapping theorem and its corollary (see, for instance, [Rud, p.49, Corollary 2.12 (d)1 for showing that 1 I V u E H ( s ) , ~ , h ( ~ h , +' ) (Erh) 'Eo,h0

V h

1 jul I (s).E,h +

We emphasize, one more time, that 'lI is an orthogonal projector, 2 1 h 2 1

> 0, in L (T ) onto the functions in L (T ) which can be extended

1 analytically on the exterior of T1 in C , including the infinity: of course, II- is an orthogonal projector in L2 (T1 ) , V h > 0, onto the functions in h 1 L2(T1), which can be extended analytically on the interior of T 1 Let U = (x+,x-) C R be a finite interval and let Uh = U n % . We

.

assume that x+ E Uh.

-

Consider first the case s2 L 0, covering of

s3 L

by two open intervals U

0 integer. Let

and

U-

the corresponding partition of unity, i.e. X? E X,(X)+X_(X)

: 1,

v

x E

5,

U+ U U-

be a

and denote by {X+(x) ,X- (x)1 C

m

o (U* )

and

X+(X+) = 0. - - = 1, X+(X-) - +

2 . Sobolev Spaces of Vedorial Order

210

For a meshfunction u(E,h,-)

:

Uh

+

, denote

C, ( ~ , h )E JIE

0rh0

U+(E,h,X)

=

(X+u)(E,h,x++x), u-(E,~,x)= (X-

and define the norm of u of order s E R

[

where

1 - 1 1' ( s ) ,E,h are

3

,

U)

(E,h,x--x)

as follows

the norms (2.8.49).

Definition 2.8.24.

The space

H (s).E,h(Uh ) is d e f i n e d t o c o n s i s t o f a l l meshfunctions

n o m s (2.8.50)

.

finite f o r each ( ~ , h )E Il

The space

H

u with

€Ofh0 (u) c o n s i s t s o f a l l meshfunctions u E

(S)

H

(u

(s),E,h h

)

such

that

One defines the quotient-space H where

Ho( S ) (U)

' ''

(S)

is again the subspace of

functions u whose norms

For functions Q(e,h,-)

:

( S ),5

,h,Uh

aUh

C

+

(u)

=

ff

H ( S ) (U),

( S )

(U)/Hys) (U) as previously,

consisting of all mesh-

vanish When (E,h) -t ( 0 , O ) .

the norms of order 1 E IR are

introduced as previously

H1 , ~ ,(aU h h ) the family of spaces of functions Q with norms finite, V (&,h) E JIE

We denote by (2.8.52)

0gho

Further, introducing the norm

H1(aU) the space of Q with norm previously H (au) stands for the quotient 1 denote by

(2.8.53)

space

the subspace of functions Q whose norms (2.8.52)

finite, while, as

0 H1(au)/Hl(au)

with

0 H,(au)

vanish, when (E,h) + ( 0 , O ) .

Remark 2.8.25. The norms, defined by (2.8.50)

depend on the partition of unity {X+,X-}.

However, it can be shown that the norms, corresponding to different choice of partition of unity are all equivalent uniformly with respect to

, so that they define the same topology in the corresponding

(E.h) E Il spaces.

'0tho

Notes

21 1

Notes Weighted spaces with a small parameter of vectorial order (0,k ,k ) with 1 2 integer k l 2 0, k 2 0, were already implicitly present in the work by 2 Vishik and Lyusternik 1 1 1 , where they appear through quadratic forms related to singularly perturbed strongly elliptic operators with homogeneous Dirichlet boundary conditions (see also Huet [l]) . This kind of spaces is also useful Lions [l]).

in singularly perturbed control problems, (see

Spaces of order (0,sl,s2)with s1 E IR,

s2

E R , have been

introduced in Demidov [l] and used for the investigation of the first boundary value problem for some classes of elliptic pseudodifferential singular perturbations

.

HBlder type spaces with a small parameter have been used in Fife [11 for establishing one-sided a priori estimates for elliptic singular perturbations with constant coefficients. 3 Sobolev type weighted spaces of vectorial order s = ( s 1 ’ s 2 ’ s 3 ) E R have been introduced in Frank [221 in order to establish two-sided a priori estimates (uniform with respect to the small parameter) for the solutions of coercive (elliptic up to the boundary) singular perturbations. Results concerning equivalence of norms in spaces of vectorial order (Lemma 2.4.2) have been obtained in Frank and Wendt [lo], as well as theones concerning the extension by zero (Theorem 2.6.6.)

(see also Frank and Heijstek [l]).

Sobolev type meshfunction spaces of any vectorial order have been introduced in Frank 171 in order to establish two-sided a priori estimates (uniform with respect to the mesh-size) for elliptic and coercive finite difference operators (Frank [8,9,11,13], see a l s o Thomee and Westergren [l], where meshfunction spaces have been used in order to get interior regularity results for a subclass of elliptic finite difference operators). Sobolev type spaces of meshfunctions, which depend on two small parameters, have been introduced in Frank [16,21,23] and ussd in order to establish two-sided a priori estimates for solutions of elliptic finite difference equations, which approximate elliptic and coercive singular perturbations (Frank [15,19,22]).

This Page Intentionally Left Blank

CHAPTER 3 SINGULAR PERTURBATIONS ON SMOOTH MANIFOLDS WITHOUT BOUNDARY

Several classes of pseudodifferential and difference singular perturbations are introduced here and the elliptic theory of these parameter dependent operators developed. A special attention is given to parameter dependent hyperbolic pseudodifferential and difference singular perturbations and corresponding classes of Fourier integral operators. 3.1.

Singular perturbations with constant symbols

With any differential operator Q ( D ) , D

=

(D1,

...,D

),

Dk = -ia/axk, with

constant coefficients one can associate the polynomial Q(<), called also symbol, so that

where F

and F - l

X-tS

are the direct and inverse Fourier transforms defined

S’X

i n Chapter 1.

The splitting of Q ( 5 ) intci a sum of homogeneous polynomials Q ( 5 ) of k degree k,

with m

=

deg Q ( E ; ) the degree of Q , is a natural graduation of Q ( 5 ) where

especially the homogeneous polynomial Q ( 5 ) of highest order,called also m principal symbol, is well defined and plays an important role in the classical elliptic theory of partial differential operators. The main object of the investigation here are differential operators Q(E,D)

whose symbols may depend upon a small parameter

that their degree deg Q depends on deg Q ( E , S ) = m, V

E

E

E

E [O,E

0

1 and such

in the following fashion:

E ( 0 . ~ ~ 1deg , Q ( 0 , S ) = m 1 < m.

A typical example of such an operator is the one appearing in the

linear Elasticity theory:

3 . Singular Perturbations on Smooth Manifolds without Boundary

214

2 2 (3.1.3)

QO(E,D)

2

= E A -A,

f o r which m = 4, m

=

A

=

-

Dk

C

lZk
2.

1 The c o r r e s p o n d i n g symbol Q

5)

(E,

O

f u n c t i o n of d e g r e e 2 i n v a r i a b l e s

(E

=

-1

E

2

15 1 4+1 5 I

,F) E

(3.1.3) can be r e p r e s e n t e d as a product Q

Rn.

R + X

(E,C) 0

i s a homogeneous

Besides, Q ( E , ~ ) i n

2 2 2 151 ( 1 + ~151 ) so t h a t

=

it is n a t u r a l t o a s s o c i a t e w i t h Q i t s v e c t o r i a l o r d e r v = ( 0 , 2 , 2 ) , which i s

t h e l e a s t v e c t o r i a l o r d e r such t h a t Q ( E , D ) 0 uniformly with r e s p e c t t o E .

:

H

(s), E ( R n )

H(S-V) , E

(W")

I f one c o n s i d e r s a s l i g h t l y d i f f e r e n t d i f f e r e n t i a l o p e r a t o r

(3.1.4)

Q(E,D) = QO(€,D)+Q1(D),

then t h e r e p r e s e n t ati o n sum of homogeneous i n

v(l)

=

(E

(3.1.4)

-1

Ql(D)

= =

i s a n a t u r a l s p l i t t i n g of Q ( E , D )

,5) t e r m s

of o r d e r v(O)

> v"),

r e s p e c t i v e l y , b e s i d e s v")

(O,l,O),

1 b D l
(0,2,2) and

=

so t h a t a g a i n one c a n

view Q O ( ~ , S ) a s t h e p r i n c i p a l symbol of Q ( E , ~ ) and s h o u l d e x p e c t Q p l a y thedominant

0 to r o l e i n t h e e l l i p t i c t h e o r y of d i f f e r e n t i a l o p e r a t o r s

a f f e c t e d by t h e p r e s e n c e o f a small p a r a m e t e r .

> i s o n l y p a r t i a l , it i s n o t s u r p r i s i n g t h a t a , 5 ) principalsymboldoesnotalwaysexistforagivensymbol

Since t h e o r d e r i n g geneousin

(E

-1

4

I n f a c t , t h e polynomial Q ( E , ~ ) =

E

s p l i t i n a sum of homogeneous i n

(E

Q,

= E

4

151 -1

5

+E

2

homoQ(E,<)

4

151 + / 5 I 2 + < b . 5 > can b e s t i l l

,5) t e r m s

Q

= E

2

1514+1512

and

1 5 j 5 + < b . < > o f d e g r e e 2 and 1 r e s p e c t i v e l y , b u t t h e i r v e c t o r i a l

o r d e r s v")

= ( 0 , 2 , 2 ) and

n e i t h e r v(O)

> v"),

v(l) = (0,1,4) c a n n o t b e compared, s i n c e

n o r v(l)

> v").

A l l t h i s j u s t i f i e s t h e following

D e f i n i t i o n 3.1.1.

A function Q

:

( 0 . ~ ~x 1x n + c i s s a i d t o belong t o t h e class Pv w i t h

v = (v1,v2,v3) E R ~ i ,f ( i )Q ( E , ~ )i

s a polynomial i n 5 E

R~

w i t h c o e f f i c i e n t s belonging t o

m

c

((O,EO1).

( i i )There

e x i s t s a decovposition Q

=

such t h a t

Q+,R

Q,

can be extended a s

a homogeneous f u n c t i o n of ( E - I , ~ )E R + x mn of degree v1+v2, (3.1.5)

-1 Q o ( tE , t C )

= t

satisfying the inequality

v 1+v 2

Q,,(E,S),

t/ t

E R+, t/

(E,c)

E

R + x Rn,

3 . 1 . Singular Perturbations with Constant Symbols

and t h e remainder

R

215

s a t i s f i e s the inequality -V

(3.1.7)

151 v 2 < E c > v 3 ~ 1 5 ~ - 1 + E ) v,

/ R ( E , S L 5 CE

w i t h some p o s i t i v e c o n s t a n t

E

E

v 5

(O,Eo1,

E En,

15)

2 1,

C.

The f u n c t i o n Q ~ ( E , ~ ): IR

x

IR~.-+

c i s c a l l e d t h e p r i n c i p a l symbol o f

Q ( E , ~ ) .

- (v +u2)

Putting t =

E

in

go(€,<)=

one f i n d s :

[3.1.5),

Qo(l,~g).

E

For d e r i v i n g a p r i o r i estimates f o r e l l i p t i c s i n g u l a r p e r t u r b a t i o n s w e

i

s h a l l u s e symbols of t h e form Q ( E , ~ ; ) w i t h L(E,:',~,)

=

with

<5'>1c'/-l5',

=

<5>1
and even of t h e form

which a r e w e l l d e f i n e d f o r

6 E

En-,

i s a s u b s e t of measure z e r o .

where S c IRn

D e f i n i t i o n 3.1.2.

A function L

:

( 0 . ~ ~x 1m n \ s +

t o be a symbol i n t h e c l a s s Lv

c w i t h s a c o n i c s u b s e t of measure zero, i s s a i d

=

LV(xn) with v

= (vl,v2,v3),

i f

( i ) t h e r e e x i s t s c o n s t a n t s C,m such t h a t

v

-v (3.1.8)

'is>

/L(E,S)I 5 CE

'<EF,>'

( i i ) t h e r e e x i s t s a decomposition L a homogeneous f u n c t i o n of (E-',S) E

(3.1.10)

Lo(t

-1

3

,V

E ( 0 . ~ ~X 1l R n \ S ,

= L~+R, IR+X

such t h a t L 0 can be extended a s of degree v1+v2,

(R~,{o})

v +v ~ , t 5 )= t

2Lo(~,S), V t

E IR+,

V

(E,E)

€ IR+

x

(IR~\{O})

s a t i s f y i n g t h e ineyuazity -v (3.1.11)

iLO(E,S)l

and t h e remainder

R

5 CE

v

IR(E,S)I

x

(lRn\{O})

s a t i s f i e s t h e estimate: -v

(3.1.12)

3

'151 2 < ~ 5 > v , V (E,S) E R+

5 CE

v

'151

2 < ~ 5 > v 3 ( 1 5 1 - 1 + t ) ,V E E ( 0 . ~ ~ 1V, 5 E Rn\S,

151 21 w i t h some p o s i t i v e c o n s t a n t c . The f u n c t i o n L O ( ~ , S ) : E+ x I R ~+ c i s c a l l e d t h e principaZ symbol of L.

3 . Singular Perturbations on Smooth Manifolds without Boundary

216 Lemma 3 . 1 . 3 . (1)

If L . E l u ( j )j,

( i i )If

1 L~ E

1

=

1,2, then L1L2 E L" w i t h v

IL1(E,S)I 6 C I E

w i t h a p o s i t i v e c o n s t a n t cl, (iii)L

(1)

( i v ) 1" ( 1 )

uL

n

" (2) 5 L\,

1"

V

P r o o f . With -

u(1)+v(2).

s a t i s f i e s the inequality: -V

(3.1.13)

=

(1) 1

u(l)

<5>

<ES>

" (1) 3

,v

E (O,Eo1

(E,5)

t h e n L ~ ( E , ~ ) - 'E 1 w i t h u

with the least u

>

.(I),

=

x R

n y

-u ( 1 ) .

j = 1,2.

c 1 w i t h t h e l e a s t v =< u ( j ) , j = 1,2. (2) - I, b e t h e s u b s e t where L . i s n o t d e f i n e d , m . 1 1 and l e t L . = L + R . be t h e decomposition accor1 jrO I ( i i )of D e f i n i t i o n 3 . 1 . 2 . I t i s immediate t h a t t h e symbol

j = 1 , 2 l e t S . c Rn

3

the constants in

(3.1.9)

ding t o t h e p a r t

L = L1L2 s a t i s f i e s ( 3 . 1 . 1 0 ) w i t h v = v ( 1 ) t v ( 2 ) on Rn\S

For showing

(3.1.9)

with S

= S

1

w e e s t i m a t e L ( E , S ) - L ( E , ~ )a s f o l l o w s :

"

s2-

Further, the inequality

214<5-n>l4,

5

<S>a-a

so t h a t ( 3 . 1 . 5 ) h o l d s f o r L

m

=

= L L

1 2

v (5,n)

with u

v a E

E n n x lRn,

R

= u ( ~ ) + v ( ~ and )

rnaxtm2 'm1 + ~ v2( ~ ) l + l v ~ ~ ) l } . O b v i o u s l y , t h e f u n c t i o n Lo

=

LloLz0 s a t i s f i e s t h e c o n d i t i o n s ( 3 . 1 . 1 0 ) ,

(3.1.11). The r e m a i n d e r R (3.1.14)

IH(E,E)

One h a s

I

6

=

L-Lo

can b e e s t i m a t e d a s f o l l o w s :

lR1(E,5)R2(E,5) /+IR1(E,5)L2,0(E,c)

I+IR2(E,c)L

3.1. Singular Perturbations with Constant Symbols -v

/R1(E,5)R2(E,t)~5

cE

217

v llcj 2<Et>v3(~5~-i+i)8

v

(E,c)

E

(o,Eol

nn\S,

151 2

1

w i t h v , = v ! 1 ) + v ! 2 ) s i n c e ( / ~ l - l + E ) -5~ C ~ S I - ~ + fEo r 151 2 1 , E 5 E 0' 1 3 3 F u r t h e r , it i s q u i t e o b v i o u s t h a t t h e t w o l a s t t e r m s i n t h e r i g h t hand -V

s i d e of

15 1 u 2 < ~ c > u( 31 5

( 3 . 1 . 1 4 ) a r e bounded by C E

a s Well.

T h i s p r o v e s t h e f i r s t p a r t of Lemma 3 . 1 . 3 .

If L 1 s a t i s f i e s (3 1 . 1 3 ) t h e n o b v i o u s l y ( 3 . 1 . 8 ) h o l d s f o r -1 L ( E , ~ )= L 1 ( c , < ) with v =

-u(').

The d i f f e r e n c e L ( E , S ) - L ( E , I ? ) c a n b e

estimated a s follows:

where C 1 and C a r e t h e same p o s i t i v e c o n s t a n t s a s i n ( 3 . 1 . 1 3 ) respectively.

v w i t h some p o s i t i v e c o n s t a n t c

2

.

E

E

R+,

v

5 E nn\iO})

and (3.1.12),

3. Singular Perturbations on Smooth Manifolds without Boundary

218

Thus, t h e f u n c t i o n L (E,S) = L l 0 ( t , t ) - ' 0 (1)

with v =

s a t i s f i e s (3.1.10).

(3.1.11)

.

-w

The remainder R = L-L

0

can b e e s t i m a t e d a s f o l l o w s :

I t i s l e f t t o t h e r e a d e r t o check

( i i i ) and

I

(iv).

D e f i n i t i o n 3.1.4.

If

L 6

then t h e family of operators

L

(3.1.15)

L(E,D)u(x)

E

-f

-1

V u

= FEex L ( E , ~ )Fx+[u,

d e f i n e d by t h e formula

L(E,D)

E

S(Rn),

i s s a i d t o belong t o O P L ~ and i s c a l l e d a s i n g u l a r p e r t u r b a t i o n . Obviously,

L(c,D)

:

S(iRn)

+

S' (Rn).

D e f i n i t i o n 3.1.5.

A singular perturbation L ( E , D )

:

i f for

v

v e c t o r i a l order v E IR' L(€,D)

: H

( s ) ,E ( n n )

S ( I R ~ -f ) S'

(nn) is s a i d t o have t h e

s € 1~~ i t can be exLended as

( m n ) uniformly w i t h r e s p e c t t o + H(s-v) ,E

E

E ( 0 , ~ ~ l .

Proposition 3.1.6.

If

L ( E , ~ )

E Lv,

then

: s(#)

L(E,D)

-f

s 8 ( m n ) has t h e v e c t o r i a L order w.

P r o_ of. _

L

If L ( E , ~ )E

(3.1.8)

then

yields L(E,D)

E

V s E Rn u n i f o r m l y w i t h r e s p e c t t o E Along w i t h L ( E , D )

E OPLw

: S(R)

+

S'(R)

D e f i n i t i o n 3.1.7. 0

(3.1.17) L

0

(E

v1

<5>

-v

0

(Rn)

1.

-f

,

0

( € , e l )

-f

L(E,E',D,),

i s c a l l e d a reduced symbol of

-V

Lv

e = (l,-l,l),

( 5 ) being a reduced symbol of L ( E , ~ ) t, h e operator OpLO OPL = L (E,D).

L(E,E) E

-V

2 < ~ < > 3 L ( ~ , 5 ) - < 5 > 2 L o ( 5 ) ) E L-,,

t h e reduced operator o f

H(s-v),E ( I R n )

d e f i n e d by t h e f o r m u l a :

The f u n c t i o n L ( 5 ) E L ( o , v 2 , 0 )

if

(O,E

( s ) ,E

w e s h a l l a l s o c o n s i d e r l a t e r t h e f a m i l y of

one d i m e n s i o n a l s i n g u l a r p e r t u r b a t i o n s L(E,<',D,)

: H

= L'(D)

i s called

3.1. Singular Perturbations with Constant Symbols

219

Lemma 3 . 1 . 8 .

( i )I f f o r L E lv a reduced symbol L

0

E 1

unique. ( i i )I f L E

Lv w i t h v 3 -2 0 has t h e reduced symbol

Proof. __ ( i ) The i n c l u s i o n ( 3 . 1 . 1 7 ) y i e l d s L

def ined

0

(6) =

e x i s t s then i t i s

(0,v2,O) L

0

then

E L(o,v2,0)

l i m E"L(€,~).

Thus Lo i s w e l l

€4-0

.

( i i )The symbol cV1L-Lo c a n b e r e w r i t t e n a s f o l l o w s :

E

v1

L(E,S)-L

0

(5)

=

-V -v < p v 2 < E p 3 ( E"1 -v 2<€t;> 3L- ~ L o ) + L o ( (~<)E 5 > v 3 - 1 ) .

"

3 E Since < p V 2 < ~ < > V

1 (o,v2,v3),

( 3 . 1 . 1 7 ) and Lemma 3 . 1 . 3

imply t h a t t h e

f i r s t t e r m i n t h e r i g h t hand side of t h e l a s t i d e n t i t y belongs t o For v3 2 0 one h a s : ( < E S > ' ~ - ~ E ) L(-l,l,v - 1 ) , so t h a t +l,v -1). 3 3~ a t h e symbol L ( t ) ( < ~ S > ~ ~ - lb)e l o n g s t o l(-l,v + l , v - 1 ) , a s w e l l . 2 3 D e f i n i t i o n 3.1.9. 5-1,"

The symbol L~

L E

Lv i s c a l l e d e l l i p t i c of order v i f i t s p r i n c i p a l symbol

satisfies the inequality:

(3.1.18)

-vl

IL ( E , S ) ~ 2 C E

0

w i t h some p o s i t i v e c o n s t a n t

"

J '

151 2 < ~ E > ',

V (E,S)

E

1R+

x

(Rn\{O)).

C.

The c l a s s of a l l e l l i p t i c symbols o f order v i s denoted by OPE\,

E ~ w , hile

stands for t h e corresponding c l a s s o f e l l i p t i c s i n g u l a r p e r t u r b a t i o n s

o f order v. def {w

1

w E R n , / w I = 1 ) . Using t h e homogeneity of L 0 Of d e g r e e v +u2, one c a n r e f o r m u l a t e t h e c o n d i t i o n ( 3 . 1 . 1 8 ) e q u i v a l e n t l y a s 1 follows :

Let Q

-v (3.1.19)

ILo(p,w)I

2 Cp

v '

',

V (p,w)

E 0

R+x

I n t r o d u c i n g t h e r e d u c e d p r i n c i p a l symbol L o ( C ) ,

and t h e homogeneous p r i n c i p a l symbol L o o ( c ) ,

nn.

3 . Singular Perturbations on Smooth Manifolds without Bounda y

220

- ( v +v )

(3.1.21)

L

00

(5)

2

= l i m t

3

L0(l,tS), 5 E R n ~ i O l

tJ-m

one c a n r e f o r m u l a t e e q u i v a l e n t l y t h e e l l i p t i c i t y c o n d i t i o n a s f o l l o w s : P r o p o s i t i o n 3.1.10.

0

L

If L E

admits t h e reduced p r i n c i p a l symbol L ~ ( C ) and t h e homogeneous

p r i n c i p a l symbol ~

~

~ d e(f i n5e d ) by ( 3 . 1 . 2 0 1 , ( 3 . 1 . 2 1 ) , then

L

is e l l i p t i c

( L E E )iff t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d :

LOO(,)

(i)

( i i )~

~

~

v

# 0, #( 0 w,

) V

# 0, V

( i i i )L O ( p , w )

w E w

E R E R+x an.

(p,w)

Proof. -

L

If L E

i s e l l i p t i c of o r d e r v t h e n (3.1.19) h o l d s f o r V (p,w)

Multip1y':ng

letting

151

+

E

.

n

0 one g e t s ( i ) -(v2+v3) = 1 , m u l t i p l y i n g it by 151 and

( 3 . 1 . 1 9 ) by p V 1 a n d l e t t i n g p

Further, taking i n (3.1.18)

E R + x S2

+

one g e t s ( i i ).

m,

( E , C ) of o r d e r v1+v2 and ( 3 . 1 . 1 2 ) imply (iii). 0 -V ( i ) - ( i i ih )o l d s , t h e n ( i ) y i e l d s / L ( p , o ) /2 C p

The homogeneity of L Conversely, i f

0

E (O,po]

f o r V (p,w)

PO

Rn w i t h s o m e p o s i t i v e c o n s t a n t C which depends 0 0 PO '3-'1 o n l y on p o and y o = min IL (IJI) while (ii) i m p l i e s I L o ( p , w ) 2 C p wERn 0 p1 which depends f o r V (p,wl E [p,,-) X Qn w i t h some p o s i t i v e c o n s t a n t C x

I

1;

o n l y on p 1 a n d y o 0

=

min

IL

00

wERn

((0)

~. The

41

Qn b e i n g compact

s u b s e t [pO,pl]x

R ,

one can f i n d , u s i n g ( i i i ) a c o n s t a n t C > 0 s u c h t h a t n 2 C P - ~ ' < P > " ~which , i s p r e c i s e l y ( 3 . 1 . 1 9 ) and by homogeneity of 0 -1 L ( e , < ) of o r d e r v +v2 i n ( E ,C) is equivalent t o (3.1.18). I 0 1 in R IL

+

X

(p,w)

P r o p o s i t i o n 3.1.11.

Let

L ( E , D ) E O P L , ~ . Then L is e l l i p t i c o f order following e s t i m a t e holds f o r E~ small enough:

(3.1.22)

/ I ~ l l ( S ) , E ~ C ( / I L U / I ( S+ -" l l)u,lE[ ( s , ) , E ) ,v

v where the c o n s t a n t cdoes n o t depend on

alrs

=

is'

v ( L ( E , D ) E OPE")

1

SI

3

E R ,

S I

<

S ,

S;

E E

n3,

v

so E

E

E (O,E01,

a3,

$ 1

iff t h e

v

uEs(Rn),

< s, s' E airs,

and u and where

= s1}.

Proof. Let L(t,D)

be e l l i p t i c of o r d e r v and l e t L 0 ( € , C )

of L ( E , ~ ) .Witn

i

=

< 5 > \ 5 \ -15 t h e e l l i p t i c i t y o f

b e t h e p r i n c i p a l symbol L implies:

3.1. Singular Perturbations with Constant Symbols

22 1

1

C1. Denote L O ( ~ , D )= OpL ( E , E ) . Then, u s i n g t h e P a r c e v a l ' s i d e n t i t y and 0 t h e i n e q u a l i t y C 5 - P 5 1,one g e t s

The l a s t i n e q u a l i t y l e a d s t o (3.1.22) w i t h s'

=

s-e2 i f

E~

< C/(2C ) .

1

Now,

u s i n g t h e i n t e r p o l a t i o n i n e q u a l i t y (2.1.11) f o r Sobolev norms o f v e c t o r i a l o r d e r , one g e t s (3.1.22) w i t h V s'< s , s'

airs.

E

C o n v e r s e l y , assume t h a t (3.1.22) h o l d s w i t h s o m e s and s' = s-e2. Then t h e Parceval's i d e n t i t y implies:

for

v

(E,s)

E

(o,E01

x

(R~,s).

The l a s t i n e q u a l i t y i m p l i e s :

for (E,<)

E ( 0 . ~ ~x 1 ( R n \ S ) ,

positive constant C

16) 2

r w i t h r l a r g e enough, where t h e

depends o n l y on C , v and r .

The homogeneity o f L O ( ~ , S )i n

(E

-1

o f o r d e r v +v2 i m p l i e s t h a t t h e

,<)

1

l a s t i n e q u a l i t y , perhaps with a d i f f e r e n t p o s i t i v e c o n s t a n t , h o l d s f o r

v

(E,S)

E R+

x

I

(W"\IO}).

Example 3.1.12. The s i n g u l a r p e r t u r b a t i o n s

(3.1.3), (3.1.4) a r e e l l i p t i c o f o r d e r (0,2,2),

while t h e s i n g u l a r p e r t u r b a t i o n Q ( E , D ) =

E

2

4

I D 1 -1DI2 i s n o t e l l i p t i c .

Remark 3.1.13. I t i s l e f t t o t h e r e a d e r t o check t h a t any s i n g u l a r p e r t u r b a t i o n Q ( E , D )

w i t h polynomial symbol Q ( E , ~ )E p a d m i t s t h e r e d u c e d p r i n c i p a l symbol 0 and t h e homogeneous p r i n c i p a l symbol Q o o ( 5 )

Qo(S)

p~o,v2,01

The same i s t r u e f o r symbols which a r e r a t i o n a l f u n c t i o n s of 5 .

p(o,v2+v3,0)

'

3 . Singular Perturbations on Smooth Manifolds without Boundary

222

Remark 3 . 1 . 1 4 . some s ' < s s u c h t h a t s' < s l , t h e n f o r 1 s u f f i c i e n t l y s m a l l t h e s i n g u l a r p e r t u r b a t i o n L ( & , D ) , E € (@,€,,I, If

(3.1.22) h o l d s for

E

-

equiisomorphism, i . e . L -1 -1 L := Op L(E,S) : H

:= L ( s , O )

(s-")

: H

0 is m

(IRn ) and

( s ), E

(IRn),

E

E

(u,E@~,

,E

are b o t h

e q u i c o n t i n u o u s (see D e f i n i t i o n 1 . 2 . 3 3 ) .

D e f i n i t i o n 3.1.15.

An e l l i p t i c s i n g u l a r p e r t u r b a t i o n

L(E,D)

E

OPEv

is caZZed s t r o n g l y e L l i p t i c

of order u if w i t h some c o n s t a n t s 6 E [ 0 , 2 n ) , y E R+ hoZds: -v (3.1.23)

where

Re(eiCLo(s,F)) L

L ~ ( E , ~i )s

YE

u

' / 5 ( 2 < ~ 5 >3v, V

t h e p r i n c i p a l symboZ of

(E,<)

E

(IRn\{O)),

R+x

L.

Example 3 . 1 . 1 6 . The s i n g u l a r p e r t u r b a t i o n s order u

=

(3.1.131,

(3.1.14) a r e s t r o n g l y e l l i p t i c of

( 0 , 2 , 2 ) , t h e c o n s t a n t s i n ( 3 . 1 . 2 3 ) b e i n g 8 = 0 , y = 1; t h e

i D 2 + D 2 ) 4 ( 1 + ~ 2I: D;)" is e l l i p t i c I: k n llksn-1 l
= (

P r o p o s i t i o n 3.1.17.

Let

L(E,D)

E

OPE

be s t r o n g l y e l l i p t i c of order v and Zet 6, y be t h e

corresponding c o n s t a n t s in ( 3 . 1 . 2 3 ) . Then t h e r e e x i s t c o n s t a n t s

Co

and

Cl

such t h a t (3.1.24)

i6 Re<e

L(E.D)U,U> 2

(Y-C 0 E )

1 IuI 1

2 ( v / 2 ) ,E-Cll

V u E H

( v / 2 ) ,E

where < , > denotes t h e d u a l i t y betweenIl(u,2) , E

, 2 lU/

(Rn)

,

l((v-e2),2) v

(Bn)and H

E

,€,

E (O,EO1'

(-u/2) ,E

(En).

Proof. Without r e s t r i c t i o n of g e n e r a l i t y one c a n assume t h a t 8 L (6,F)

0

=

0, y

b e i n g t h e same a s i n t h e p r o o f of P r o p o s i t i o n 3 . 1 . 1 1 ,

u s i n g t h e same argument a s i n t h e p r o o f of t h i s p r o p o s i t i o n :

=

1.

one g e t s

3.1. Singular Perturbations with Constant Symbols

223

P r o p o s i t i o n 3.1.18.

Let

L ( E , D ) E OPE be s t r o n g l y e l l i p t i c o f order v w i t h p r i n c i p a l symbol L ~ ( E , ~ )Assume . t h a t L admits t h e reduced symbol L0 ( 5 ) E L and l e t

(0 'V2 , O )

0 L~

( 5 ) be t h e corresponding reduced p r i n c i p a l symbol. L e t

(3.1.25)

0 -1 0 P ( F , ~ =) L ( E , S ) - L ~ ( E , ~ ) L ~ ( LS ) ( 5 ) .

Then t h e r e e x i s t s a c o n s t a n t

c such t h a t

Proof. ___ O b v i o u s l y , t h e symbol P E identically: P

0 (3.1.12), P ( E . 5 )

consequence of

(E,S)

Z

Lu and i t s p r i n c i p a l and r e d u c e d symbols v a n i s h

0 , P o ( S ) E 0 . T h e r e f o r e , a s a consequence of

E L(u-el)

u L(u-e2),

(3.1.17), P ( E , S )

s i n c e P0 ( E , S )

E L(u-e),

. . T h i s l e a d s to t h e c o n c l u s i o n t h a t P ( E , ~ E) ).

e

=

L

Z

0. F u r t h e r , a s a

(1,-1,1),

(we)

"

0

s i n c e P ( 5 ) :0 .

('("-el)

"

L(u-e2)) =

I t i s immediate t h a t

w i t h some c o n s t a n t C > 0 and t h e l a t t e r h a s ( 3 . 1 . 2 6 ) a s i t s immediate

I

consequence. C o r o l l a r y 3.1.19.

Let

L(E,D)

E

OPEv be s t r o n g l y e l l i p t i c o f order u and l e t L O ( ~ , c )be i t s 0

0

principaZ symbo2, L (D) i t s reduced symbol and

L ~ ( D )i

symbol. Assume t h a t t h e r e i s a c o n s t a n t yo

such t h a t

Then t h e r e e x i s t s such a c o n s t a n t

c that

> 0

t s principal reduced

3. Singular Perturbations on Smooth Manifolds without Boundary

224

(3.1.28)

2 1 I U I ](v,2),E~

0 Re<eieL(E,D)u,u> 2 (YoY -CE)

V u E H

(v/2)

,E

(2) ,v

E

E

(O,Eo1.

I n f a c t , (3.1.27) is an immediate consequence o f (3.1.25)-(3.1.27). Remark 3. 1.20.

0

I f L ( E , D ) 6 OPE

a d m i t s t h e r e d u c e d symbol L ( D ) E OPE

(0,v2I O )

'

then

i n t r o d u c i n g t h e symbols

(3.1.29)

-1

0

R(E,<) = LO(~,S)LO(S)

R ( E , ~ )=

anddefining

,

S ( e , S ) = R(E,S)-~

R ( E , ~ ) ,; ( € , < I

= S(E,<),

i

with

=

< < > I t-1/ 5 ,

one

c a n w r i t e down: L(E,D)

(3.1.30) L

where

0

(D)

=

R ( E , D ) L 0 ( D ) + E Q(~E , D )

= ;(E,D)L(E,D)+EQ~(E,D),

Q 1 ( ~ , S )E L v l Q ~ ( E , S )E L ( O ,"*,o ) .

I n f a c t , t h e argument u s e d i n t h e p r o o f of P r o p o s i t i o n 3.1.18 i s 0 a p p l i c a b l e t o t h e symbols p l ( ~ , < ) L ( E , ~ ) - ~ ( E , < ) L( 5 ) and

p2(€,6) = L0 ( < ) - S-( E , < ) L ( E , S ) , s i n c e t h e i r p r i n c i p a l and r e d u c e d symbols 0 v a n i s h i d e n t i c a l l y : P . ( E , S ) 5 0 , ? . ( 5 ) :0 . JO 3 T h i s remark w i l l be u s e f u l l a t e r f o r t h e r e d u c t i o n o f s i n g u l a r

e l l i p t i c p e r t u r b a t i o n s w i t h v a r i a b l e symbols t o t h e r e g u l a r o n e s .

3.2. S i n g u l a r p e r t u r b a t i o n s w i t h homogeneous symbols Let

E 1". Then c l e a r l y t h e f a m i l y ( 0 . ~ ~3 1E

a(E,S)

-f

a ( € , . ) E S'(R;) i s

a c o n t i n u o u s l i n e a r map and one c a n r e w r i t e a ( ~ , D ) ua s t h e f o l l c w i n g

f a m i l y of c o n v o l u t i o n o p e r a t o r s :

(3.2.1)

a ( E , D ) u = (AE

where A (x) For V

=

E

F

-1

5-

*

v

U),

a ( E , c ) , ( 0 . ~ ~3 1E

-f

u E S(IRn)

,

Ac E S'(lR:).

E ( O , E ~ ] and V u E S ( R n ) t h e f u n c t i o n

r a p i d l y decreasing as

6

-f

-, so t h a t a ( ~ , D ) E u

<

m

C (IRn)

-f

,

a(E,S);(<)

V

E

E

is

(0,s01.

S i n g u l a r p e r t u r b a t i o n s whose symbols a r e homogeneous f u n c t i o n s o f (E

-1

,6) a r e

of s p e c i a l i n t e r e s t f o r t h e a p p l i c a t i o n s .

D e f i n i t i o n 3.2.1.

A function:

R+X

(R~\[o})3

(E,s)

+

a0(E,<)

E c is said to belong to t h e

3.2. Singular Perturbations with Homogeneous Symbols

symbol c l a s s Hm

V'

(i) . a

E c

i f t h e fo%%owingc o n d i t i o n s a r e s a t i s f i e d :

o i ~ x+ (IR~\IO}))

(ii) t h e r e e x i s t s a c o n s t a n t (3.2.2)

lao(E,~)I5 CE

-1

c

If a

o

-1

ao(t

E,t<)

=

> 0

such t h a t

151 2<E5>v3, v

(iii) ao(E,t) i s homogeneous i n (3.2.3)

225

(E,c)

E

R+X

(IRn\{O}),

( E - ~ , C )of degree u l + u 2 :

u +u 1 2 t ao(E,c), v t E R + , V

(€,
E IR+X(R*\IO}).

E H-u is a polynomial in 5 ,

then a (6,D) is the differential singular perturbation, 0

E

If a .

f , v2

> -n,

then R + 3

E

-f

a0 (E,.) E S'(R;)

i s a family of

regular distributions and the corresponding singular perturbation can be written as the Fourier integral (3.2.5)

aO(E,D)u(x) = (2n)-" 1 ei<x*s'

. a

(E,

5 ) ( 5 )d5,

Rn

v

E

E R + ,v u E S ( R n ) .

v 2 < 0, -(n+l) < v +v < 0. Let a E H: with u E IR3 such that -n 0 2 3 We shall compute the family of distributions IR+ 3 E + A. (E,. 1 E S ' (Rz),

(3.2.6)

-1

A (€,XI = F5+x ao(E,S),

0

which are the kernels of the convolutions a (E,D)u(x)= (A ( E , . ) * u ) (x), 0 0

u E

S(Rn).

The homogeneity of degree v + u 2 of a ( E , S ) implies: 1 0 - ( v +u +n) (3.2.7) AO(€,x) = 6 AO(X/E),

*

3 . Singular Perturbations on Smooth Manifolds without Boundary

226

Let

]5,/

=

E

-1

b0(5)

where S

=

and l e t

R

=

{y 1

-1

a n d , moreover,

v +v 2 3

(3.2.2)

=

N

,E),

E, E

( x O , x ) . Denote

(nfl+',s),

, 5, =_0 o r 5 = 01. E H yields: 0 v

E Wn+'

bO(ty) = t

(C0,5), x

=

a0(1501

/EO1

The homogeneity of a (3.2.9)

5

v -v

N

(3.2.8)

N

N

, 50 E

-

bo(5),

V t E R+, V

y

E (IRn+'\S),

implies:

w i t h some c o n s t a n t C > 0 . one c a n w r i t e

Obviously,

v -v

15,/

(3.2.11)

-1 1A0(1501

-

-1 -1 ,x) = Fg+xbO(So,C) = FX -tS F - - b o ( 5 ) . 0 X0' 5

A s a consequence of t h e assumption n+u2 > 0 , n + l + v 2 + v 3 > 0 , t h e d i s t r i b u -

tion b

0

E

S'

(R:+')

i s a r e g u l a r one a n d , t h u s ,

5

t h e i n t e g r a l i n t h e r i g h t hand s i d e o f

(3.2.13) being convergent f o r

N

x E (E?n+l\{O}) and V 6 > 0 , s i n c e ( 3 . 2 . 1 0 ) h o l d s and n+v2 > 0 , n+l+u +V > 0. 2 3 One f i n d s , u s i n g t h e homogeneity of b ( 5 ) of d e g r e e 0 N

-I-

where z + T(z) i s t h e r - f u n c t i o n , n+l = {w w E IR , = 11.

=

lTl-'T,?

=

V

2

+v 3

lxl-lz and

Therefore,

u -v (3.2.14)

A

0

(E,X)

= E

3

i ( n / 2 ) (u2+u3+n+l) r ( v 2 +v3 + n + l ) 1(2n)-(n+l)e

*

3.2. Singular Perturbations with Homogeneous Symbols

227

-

- ( v +v + n + l ) (<;',;>+is)

A f t e r t h e change of v a r i a b l e : xo

=

Ixlt,

dwdx

0'

(3.2.14) y i e l d s t h e following

formula: (3.2.15) -(n+l),

.

r ( v +u + n + l ) [ x / 2

-1 J

2

3

3

E

- ( v +v + n t l )

-1

.-iE

J

- ( v +v + n ) v3-vl

i ( n / 2 ) (v2+v3+n+l)

= (2n)

A (E,x) 0

dwdt,

'n+1 where y

=

/ x / - l x and t h e d i s t r i b u t i o n ( t + i O ) h

the distributions ( t + i G ) '

6

when

i s t h e l i m i t i n S ' ( R ) of

(see [Sch

J. 0

1

1,

[G-Sh 1

1).

Example 3 . 2 . 2 . Denote by A z ( c , x ) t h e f a m i l y of d i s t r i b u t i o n s E

0

A Z ( ~ , x =) F-l

0

X;(n)

X;(E~),

=

',

where z

E

-f

C,

A'(€,.) 0

E

S'(R:)

,

0 > Re z > - ( n + l ) .

X'S

Then t h e f o r m u l a ( 3 . 2 . 1 5 ) y i e l d s : (3.2.16)

- I

.

- ( n + l ) i ( n / 2 ) ( z + n + l ) r ( z + n + l~) ~ j - ( z + n ) ~ z

A'(E,x) 0

= ( 2 ~ )

-iE-l

J e

1x1 t ( t w + < Y , W > + l O )

0

- ( z + n + ld)r d t .

'n+1

-

The i n t e g r a l 1

z,n

=

( t < t > - l w 0 +-l+iO)

J

-(z+n+l) dw

'n+l -1 and y = 1x1 x E -1 Indeed, i n t r o d u c i n g y = < t > ( t , y ) ,

d o e s n o t depend on t

E

R

-

N

s u c h t h a t y becomes eo

=

(l,O,

...,0) E

an. = 1 and making a r o t a t i o n

one f i n d s :

W e s h a l l n o t compute h e r e t h e c o n s t a n t I and s h a l l g i v e o n l y t h e z,n f i n a l formula f o r A Z ( c , x ) w i t h 0 > R e z > - ( n + l ) :

0

A s a consequence of i t s d e f i n i t i o n ,

the function

3. Singular Perturbations on Smooth Manifolds without Bounda y

228

c a n b e e x t e n d e d a s a n e n t i r e f u n c t i o n of z E C v a l u e d i n S ' ( R n )

v

E

,

for

E R+. Of c o u r s e , one a l w a y s h a s : l i m A'(E,x) 0 €40

= 6(x)where 6(x) is the Dirac's

6-function. For z

=

1-n one c a n compute t h e i n t e g r a l o n t h e r i g h t hand s i d e o f

( 3 . 2 . 1 7 ) so t h a t

For z = 2m-n-lwhere hand s i d e of

m i s a n i n t e g e r , 1s m < ( n + 1 ) / 2 , t h e i n t e g r a l o n t h e r i g h t

( 3 . 2 . 1 7 ) c a n be a g a i n computed, u s i n g t h e r e s i d u u m

CalCUlUS,

so t h a t one f i n d s i n t h i s case:

(3.2.20)

Azm-n-l(~,x) = 21-n 2m-n-1

-

-1 2-E

1x1

.(n-1)/2r((n-1)/2-m) For Rez +n < 0 one c a n e x p r e s s A ' ( E , x )

-

f u n c t i o n s ( s e e [Grad.

If z

=

A;(E~)

0

Ryz.,

1

c 2' (2m- j - 2 ) OSjsm-1 c ' j ! ( m - l - j ) !

i n t e r m s of t h e m o d i f i e d H a n k e l ' s

1)

2m+a w i t h i n t e g e r rn > 0 a n d a E ( - 2 , 0 ) , t h e n , o b v i o u s l y , 2m = Xo ( E ~ ) X ; ( E ~ ) , so t h a t i n t h i s c a s e one g e t s t h e f o l l o w i n g

r e p r e s e n t a t i o n f o r t h e s i n g u l a r p e r t u r b a t i o n X'(ED):

0

(3.2.22)

XE (E,D)u (x) =

where A

= ( ~ - E ~ A In ~ )A; ( E , x - ~ ) u ( y )dy =

R 2 fnA;(~,x-Y) (I-€ A ) m u(y)dy, Y R

v

C a 2 / a x i i s t h e Laplace o p e r a t o r . 16kSn I n p a r t i c u l a r , u s i n g ( 3 . 2 . 1 7 ) one g e t s t h e f o r m u l a e =

A 0- ' ( E , x )

=

(3.2.23)

and f o r n = 3 t h e f o r m u l a ( 3 . 2 . 1 9 ) y i e l d s : (3.2.24)

.A -2

(E,x)

=

2 -1 - E - l ( 4 n ~1x1) e 1x1,

x

E

IR

3

.

u E S(IR?,

3.2. Singular Perturbations with Homogeneous Symbols Notice t h a t t h e d i s t r i b u t i o n s

E

.A

j.

-2

(E,x)

229

are fundamental s o l u t i o n s

2

f o r t h e d i f f e r e n t i a l s i n g u l a r p e r t u r b a t i o n 1-E A : (3.2.25)

2 (1-E A

-2

)Ao ( € , X I

V

6(x),

=

E R + , x E Bn.

E

We s h a l l e x i b i t a f o r m u l a f o r t h e s p h e r i c a l l y symmetric fundamental L

L

s o l u t i o n s f o r t h e d i f f e r e n t i a l s i n g u l a r p e r t u r b a t i o n E A -Ax,

appearing

i n t h e l i n e a r e l a s t i c i t y theory. Example 3 . 2 . 3 . Denote by

A”’-2

(E,x)

t h e f a m i l y of d i s t r i b u t i o n s F 2

xo- 2 ,

-2 (E,Sl

=

-1 x - 2 , - 2 0

S X ’

(E,S)

where

S i n c e one c a n a l s o w r i t e

I 1-

2

-E

2

<E5>

-2

,

one f i n d s , u s i n g ( 3 . 2 . 1 7 ) and t h e f o r m u l a f o r t h e s p h e r i c a l l y symmetric fundamental s o l u t i o n f o r t h e L a p l a c e o p e r a t o r : (3.2.26)

.A

-2,-2

(€,XI

=

T ( ( n - 2 ) / 2 ) l x 1 2 - n 2 -2 -E A. n/2

(E,X),

E

E R+,

X

E Rn.

4Tl I n p a r t i c u l a r , one g e t s (3.2.27)

-1 A 0 2 f - 2 ( ~ , x )= ( 4 ~ 1 x 1 ) ( l - e - €

-1 E

E R+, x E R

3

.

Example 3 . 2 . 4 . E -f Y ( E D ) , Y (E Dl = O P ( E 5 < E C > - ~ )w i l l be k k k c a l l e d s i n g u l a r l y p e r t u r b e d Riesz t y p e o p e r a t o r . W e s h a l l denote -1 (E E k < ~ F , > - ’ ) ,so t h a t one c a n w r i t e E j. Y (E,D) a s a f a m i l y Y (E,x) = F k S-tX k of s i n g u l a r i n t e g r a l o p e r a t o r s :

The s i n g u l a r p e r t u r b a t i o n

(3.2.28)

Y

k

(E,D)u(x)

I

= l i m

Yk(€,x-y)u(y)dy, V u E S(IRn).

640 I x - y [ > 6

W e s h a l l compute t h e k e r n e l d i s t r i b u t i o n Y

(E,x).

k C o n s i d e r f i r s t t h e c l a s s i c a l R i e s z o p e r a t o r Yk

=

Op(CklCl

-1

I.

One h a s : (Y u )

k

(x)

=

Op(lC

-1

)Dku(x).

1

The k e r n e l 12-l (x) o f t h e i n t e g r a l c o n v o l u t i o n o p e r a t o r Op( 15 - l ) c a n b e

0

e a s i l y computed:

230

3 . Singular Perturbations on Smooth Manifolds without Boundary

where the constant

Tk.e

C

can be found using the formula:

last formula yields:

Therefore, the integration by part

implies:

+

ic lim i 6-("+l) (xk-Yk)u( y "6J-0 lx-yl=6 - (n+l = -ic (n-1)v.p. J (x -y lx-yl Rn k k

+ ic lim i u(6w+x)w dw = k n6J-0 Rn /x-y~-(~+l = -icfi(n-l)v.p.i (x -y =n k k

Further, denoting, as previously,

E

=

/cO1-1 , -5 =

N

(C0,5), x

=

(xO,x), one

gets after the change of variable: xo = lxlt the following formula for the Riesz operator with a small parameter

:

-ix 5 0 On-(n+2)/2T((n+2)/2) v.p. xkiXl-(n+2)dx0 = =-i Je R

The operators Y (ED) can be also expressed in terms ofthe modified Hankel's k functions (see [Grad. Ryz., 1 1 ) :

-

3.2. Singular Perturbations with Homogeneous Symbols

The l a s t f o r m u l a c a n b e a l s o d e r i v e d from ( 3 . 2 . 1 7 ) , and u s i n g t h e r e l a t i o n : K ( r ) = K

231

(3.2.18), s e t t i n g z = 1

( r ), V v > 0 .

-V

Example 3.2.5. The s i n g u l a r p e r t u r b a t i o n A ( E D ) 0

=

O p < ~ pi s c a l l e d t h e

Riesz o p e r a t o r

w i t h s m a l l p a r a m e t e r . One f i n d s :

=

A -1 ( E D ) u ( x ) 0

-

C

i E

Yk(€D)

l5kSn

au ( x ) = axk

w y ayk The f o l l o w i n g s i n g u l a r p e r t u r b a t i o n of t h e one-dimensional

d

t

.

Riesz o p e r a t o r

(which i s t h e H i l b e r t t r a n s f o r m ) a p p e a r s i n t h e d i s l o c a t i o n t h e o r y ( s e e , f o r instance [ Hirth

-

Lothe,

1

1).

Example 3.2.6. Let

oE

= o p ( s g n c ( l + e x p ( - E / t / ) ) ,E,

E

R

. One

finds easily that

E +

OE i s

t h e f a m i l y of c o n v o l u t i o n o p e r a t o r s : OEU(X)

=

i

4v.p.

The m u l t i - d i m e n s i o n a l

(E,x)

u (y) ---dy x-Y

+

a n a l o g u e s of OE a r e t h e o p e r a t o r s :

One f i n d s t h e k e r n e l O k,n

0 k,n

1

= D 0

(E,x)

k n

o f t h e c o n v o l u t i o n o p e r a t o r s OE: k

(E,x),

where

For n

=

2 , e v a l u a t i n g t h e l a s t i n t e g r a l by means of t h e residuum c a l c u l u s ,

3 . Singular Perturbations on Smooth Manifolds without Boundary

232

so t h a t

(3.2.32)

0

k,2

(€,XI

= 2iT

-1

2 -3/2

2

x k ( c +41xI )

,

k = 1.2.

For n 2 2 one h a s : (3.2.33)

a

(E,x)

=

(n-1) ! 7

(2n)

71

J s i nn-2e ( ~ - i / ~ j ec) 1-n ~ sd e , 0

.

i s t h e area o f t h e u n i t s p h e r e i n R n - l where R n- 1 Hence, an e a s y c o m p u t a t i o n y i e l d s :

Of c o u r s e , i n any case, one h a s : (3.2.35)

lim

(E,x-y)u(y)dy

0

€40 Rn

=

2Yku(x),

V u E S(IR~).

krn

W e are g o i n g t o i n d i c a t e one m o r e c l a s s of s i n g u l a r p e r t u r b a t i o n s

with

homogeneous symbols, which i s i n t e r e s t i n g f o r a p p l i c a t i o n s . Example 3.2.7. Denote by AZ(ED',&D ) t h e s i n g u l a r p e r t u r b a t i o n w i t h t h e symbol

where y > 0 and z E C a r e g i v e n . F i r s t assume t h a t R e

2

< 0 . Using t h e formula

ni2/2 (3.2.37)

(-a+iB)'

=

r (-2)

t-z-l

-8t-iat

dt,

Re z < 0 ,

=+

we find that

where H(x ) i s t h e H e a v i s i d e ' s f u n c t i o n One f i n d s e a s i l y t h a t

I n f a c t , u s i n g t h e r e s i d i u m c a l c u l u s ( o r (3.2.37)

w i t h z = -11,

one f i n d s

3.2. Singular Perturbations with Homogeneous Symbols

233

Therefore,

which l e a d s t o ( 3 . 2.39) when x The f o r m u l a e ( 3 . 2 . 3 8 ) .

> 0.

( 3 . 2 . 3 9 ) imply f o r R e z < 0

so t h a t

I n t h e same manner one f i n d s f o r R e z < 0

Let R e z > 0 and l e t z

=

r + <w i t h i n t e g e r r > 0 a n d R e 5 < 0 .

W e c a n w r i t e down:

Hence.

and i n o r d e r t o f i n d t h e k e r n e l of t h e o p e r a t o r s compute t h e i n v e r s e F o u r i e r t r a n s f o r m of f i n d s:

ED' , E D -

)

one h a s t o

15' / X c ( c S ' , ~ C n ) , a s w e l l . One

*

3 . Singular Perturbations on Smooth Manifolds without Boundary

234

Using t h e same argument a s p r e v i o u s l y , one f i n d s :

so t h a t

> 0 , then t h e l a s t formula y i e l d s :

If x

we can again rewrite t h e s i n g u l a r p e r t u r b a t i o n s

Combining ( 3 . 2 . 4 0 ) - ( 3 . 2 . 4 2 ) ,

a s a f a m i l y of i n t e g r o - d i f f e r e n t i a l

,ED,)

A:(ED'

o p e r a t o r s . The s a m e argument

l e a d s t o a s i m i l a r r e p r e s e n t a t i o n f o r X-Z ( s D ' , s D n ) . F i n a l l y , i f z = r > 0 i s i n t e g e r , t h e n one u s e s o n l y ( 3 . 2 . 4 1 ) , f o r g e t t i n g a r e p r e s e n t a t i o n of A T ( E D ' , E D

)

-

(3.2.42)

a s a f a m i l y of i n t e g r o - d i f f e r e n -

t i a l operators. Example 3 . 2 . 8 . Denote by S ( E , D ) t h e s i n g u l a r p e r t u r b a t i o n , whose symbol i s t h e f o l l o w i n g

e

matrix

where 0 E ( - n / 2 , n / 2 )

is given.

The s i n g u l a r p e r t u r b a t i o n S ( E , D ) w i t h t h e symbol ( 3 . 2 . 4 3 ) f o l l o w i n g S t o k e ' s problem i n R n (E

(3.2.44)

-2 2 i 8 e -A


where u : R n

,u> -f

)u

+ v p

= f,

= 0,

Cn, p : W n +

Ax i s t h e L a p l a c i a n , B

solves the

:

x

E xtn

x

E

IRn

C a r e t o be d e t e r m i n e d ,

: En+ C is

f

i s t h e g r a d i e n t and = d i v u

=

given,

au/ax.. l<j
'

One c h e c k s e a s i l y u s i n g t h e F o u r i e r t r a n s f o r m t h a t u = S (E,D I f . e x 2 -2i8 2 Since I+E e 151

(3.2.45)

z

O,V

(E,s)

ER+

x

nn,

if

e

E (-n/2,T/2),

it i s

3.3. Singular Perturbations with Variable Symbols q u i t e o b v i o u s t h a t t h e f o r m u l a (3.2.17) h o l d s f o r

235

r e p l a c e d by

E

Ee

-iB

.

I n d e e d , b o t h l e f t and r i g h t hand s i d e i n (3.2.17) a d m i t a n a n a l y t i c extension with r e s p ect t o E

E R

onto the half-plane

larg

€1

< ~/2.

Using t h e second formula (3.2.23) and t h e c l a s s i c a l R i e s z o p e r a t o r s Yk

(Example 3.2.4), one c a n r e p r e s e n t S e ( € , D ) as t h e c o m p o s i t i o n o f two

operators:

(3.2.46) where, a s For n = 3

(3.2.47)

N o t i c e t h a t f o r each 8 E (-n/2,n/2)one h a s :

(3.2.48) l i m

E - ~ S( E , D ) f ( x ) = e - 2 i e ( f ( x ) - ( Y , e

E++O

x

for

v

f

D

Y.)f)(x)), 7

E ( s ' ( R ~ ) ) "= s ~ ( R X ~ . . . X)S ' ( R " )

F u r t h e r m o r e , S ( E , D ) c a n be e x t e n d e d a s

e

(3.2.49) S B ( ~ , D x ):

x (H(s), E ( R n ) I n

+

(H(s-v),e (Rn)) n

w i t h v = (-2,0,-2),univormly w i t h r e s p e c t t o E E ( 0 . ~ ~ 1 ,v E~ < *, i . e . t h e norms of t h e o p e r a t o r s (3.2.49) a r e bounded by a c o n s t a n t which depends o n l y on

and n . I n d e e d , t h i s i s a n immediate consequence o f

E~

(3.2.43) and

Parceval's identity. Using a r e s u l t from [ M i k h . l , Z I , o n e c a n show t h a t (3.2.49) s t i l l h o l d s i n spaces spaces

H

( R n ) w i t h L -norms 1 < p < a, 5 s ),2,E( R n ) b e i n g t h e P ( R n ) ( s e e 3.12 f o r m o r e d e t a i l s ) .

(s)rP,E

H (s),E

3.3. S i n g u l a r p e r t u r b a t i o n s w i t h v a r i a b l e s y m b G W e c o n s i d e r h e r e c l a s s e s of symbols which are € u n c t i o n s of

( x , E , ~ ) and

i n t r o d u c e t h e c o r r e s p o n d i n g c l a s s e s of s i n g u l a r p e r t u r b a t i o n s which i n c l u d e s i n g u l a r l y perturbed d i f f e r e n t i a l o p e r a t o r s with v a r i a b l e c o e f f i c i e n t s ,

as w e l l . D e f i n i t i o n 3.3.1.

Let u be an open s u b s e t of cZass

S'

1,o a(x,E,c) E

R~

and Zet v

(u) i s d e f i n e d t o c o n s i s t

=

(vl,v2,v3)E R

3

.

The syrnboZ

o f a22 f u n c t i o n s

s a t i s f y i n g t h e c o n d i t i o n : f o r any compact K c u such t h a t and any couple of m Z t i - i n d i c e s a , t~h e r e e x i s t s a c o n s t a n t c

c

(UX ( o , E ~ ]x

Ka I3

.

3. Singular Perturbations on Smooth Manifolds without Bounda y

236

D e f i n i t i o n 3.3.2.

The symboZ a E

S'

1,o

(u) belongs t o t h e symbol c l a s s s'(u)

if t h e r e e x i s t s a

representation (3.3.2)

a ( x , ~ , t ; )= a ( x , ~ , < ) + r ( x , ~ , S ) 0

where a O ( x , ~ , E is ) homogeneous i n v +v2

1

(E-',s)

for

E

E ( O , c o l , 151 2 1 , of degree

i.e. - ( v +v )

a (x,t-lE,tS), 0

a0(x,E,S) = t

(3.3.3)

v

t 2 I,

v

E

E (o,EOl,

v 5 E mn, and where t h e remainder r ( x . E . 6 ) on each compact

K c

151

'

u s a t i s f i e s the

inequalities

for a l l x E K ,

E ( O , E ~ ] , E, E B~ and all m u l t i - i n d i c e s

E

a,B

P r o p o s i t i o n 3.3.3.

Let a E

Sv

1,o

(U),

b

E 'S

1,o

.

(U) Then

Proof. The s t a t e m e n t s of t h i s p r o p o s i t i o n are an immediate consequence of t h e chain rule. D e f i n i t i o n 3.3.4.

Let a E sV

op a

=

(u),( r e s p e c t i v e l y , a E sv (u)) . The s i n g u l a r p e r t u r b a t i o n

1,o a ( x , ~ , ~ is)

d e f i n e d as f o l l o v s

1,

231

3.3. Singular Perturbations with Variable Symbols

The s i n g u l a r p e r t u r b a t i o n a ( x , ~ , i~s) s a i d t o belong t o t h e c l a s s ( r e s p e c t i v e l y , t o t h e c l a s s op

op S;,,(U)

Sv(U)).

Theorem 3.3.5.

If

a

E

m

s ~ , ~ ( U ) ,t h e n a ( x , c , D ) : C o ( U ) + C"(Ui

a ( x , ~ , ~: )E ' ( u ) E"

-t

D'(u),

m 1 a(x,E,D) : Co(U)

+

V

E

( 0 , ~I . 0

E

and E"'a(x,E,D)

Cm(U)

and i t can be eztended a s

Moreover, : E'(U)

-t

D'(U)

E

(E

(O,En])

are e q u i c o n t i n u o u s . Proof. If a

rn

E sv

1,o

(U), u E C o ( U ) ,

then t h e function

Evla(x,E,D)u(x) = (2nl-n

e

i < x , S > '1

mn

E

a (x,E ,5)

( 5 )d5

c a n b e d i f f e r e n t i a t e d under t h e i n t e g r a l s i g n and t h e c o r r e s p o n d i n g i n t e g r a l s are a b s o l u t e l y and u n i f o r m l y c o n v e r g e n t w i t h respect t o x on any compact K c U , Further, l e t v

E

E

E

(O,E~]. m

The i n t e g r a t i o n by p a r t y i e l d s :

Co(U).

Using t h e l a s t i n e q u a l i t i e s , one shows t h a t t h e f u n c t i o n a l s

are w e l l d e f i n e d , i f u

E

V

E'(U),

E

E

( O , E ~ ] and, moreover, t h i s f a m i l y of

f u n c t i o n a l s i s equibounded i n t h e s e n s e o f D e f i n i t i o n 1.2.33.

In f a c t ,

w r i t i n g formally (3.3.12)

<E v

la ( x , E , D ) u , v > = ( 2 ~ ) I- ~ I

v(x)E"1a(x,E,S);(S)ei<x'5>

dsdx =

IRn lRn =

I ~ ( c (I ) mn mn

(2.rrl-n

and u s i n g ( 3 . 3 . 1 0 ) , defined f o r V u

E

Ev

1 i<x,S> a(x,E,<)v(x)e dx)dS

one g e t s t h e c o n c l u s i o n t h a t t h e l a s t i n t e g r a l i s w e l l

E'(U),

s i n c e t h e family o f f u n c t i o n s

i s r a p i d l y d e c r e a s i n g when

5

+

m

uniformly with r e s p e c t t o

E

E (O,E~],

3. Singular Perturbations on Smooth Manifolds without Bounda y

238

f o r any u E E ' ( U )

so t h a t t h e f a m i l y of f u n c t i o n a l s (3.3.11)

bounded w i t h respect t o

E

E (O,E

0

1.

i s uniformly

I

Remark 3.3.6.

(0) (0)

1

L e t a E S v ( " ) ( U ) where i n t e g e r v . t 0, j = 2,3, and l e t E a ( x , E , c ) be 3 polynomial i n ( ~ ~ 5 L)e .t a , ( x , E , S ) b e t h e p r i n c i p a l symbol o f a ,

a-a,

,

(0) ( 0 ) > v(l)+v(l) 1 2 v 1 +v2 I f a t e a c h s t e p o f t h i s p r o c e s s t h e remainder r k = (a-aoa ) E k- 1 (k) ( U ) , t h e n t h e symbol a ( x , E , S ) c a n b e r e p r e s e n t e d a s a f i n i t e sum:

where, of c o u r s e , u(O) > v(l) and moreover,

...-

Sv

a ( x , ~ , C )=

(3.3.13)

C ak(x,E,<) O
where ak have t h e v e c t o r i a l o r d e r v ( ~ ) a, r e homogeneous i n d e g r e e V : ~ ) + V ; ~ ) and, moreover, t h e o r d e r s v ( ~ ) ,0 5 k

=< N,

(E

-1

, 5 ) of

form a

s t r i c t l y d e c r e a s i n g sequence i n t h e f o l l o w i n g s e n s e :

The s p l i t t i n g (3.3.13)

i n t o a sum of homogeneous t e r m s ak, whose o r d e r s

v ( k ) s a t i s f y (3.3.14),

c a n be viewed a s a n a t u r a l g r a d u a t i o n of a ,

a n a l o g o u s t o (3.1.2). Of

c o u r s e , any symbol a

which i s w e l l d e f i n e d f o r

E

h a s a z e r o t e r m a.

Sv(U)

E (O,E~],

E

e x t e n d e d a s a homogeneous f u n c t i o n o f

of graduation,

151 2 1, x E U and can b e u n i q u e l y -1 , 5 ) E IR+x (lRn\{Oj) f o r V x E

(E

U

Example 3.3.7. The symbol a ( x , E , C ) c(x) : U

-f

C,

= E

2

j5j4+1512++c(x), w i t h b ( x ) : U

I t admits a n a t u r a l g r a d u a t i o n :

a (x,E,~) 0

+

Cn,

smooth f u n c t i o n s , b e l o n g s t o S V ( O ) ( U ) w i t h v ( O ) = (0,2,2).

2 = E

a

=

4

a o + a l + a 2 , where

IS/ +l5I2, al(x,E,5)

v ( O ) = (0,2,2), v ( l ) = (O,l,O), v")

= cb(x),C>, a2(x,€,S) = c ( x ) ,

= (O,O,O),

t h e c o n d i t i o n (3.3.14)

being

3.4. Continuity of Singular Perturbations

239

obviously s a t i s f i e d .

~ ~ / < / ~ + 1 < 1b ~e l o+n g~s ~ / c 1 ~

On t h e o t h e r hand, t h e symbol a ( x , ~ , E ; )= ~n ( R n ) , with v = ( 0 , 2 , 3 ) b u t does n o t belong t o S ( R ) t o Sv 1# O admit a n a t r u a l g r a d u a t i o n .

and d o e s n o t

3 . 4 . C o n t i n u i t y of s i n g u l a r p e r t u r b a t i o n s i n Sobolev s p a c e s of v e c t o r i a l o r d_ er _ The a i m of t h i s s e c t i o n i s t o p r o v e t h e f o l l o w i n g Theorem 3.4.1.

L e t a ( x , E , c ) E sV

1,O

the

( m n ) and Let

a ( x , s , c ) E s(IR:). Then f o r

v

s E R

3

foZlowingfamiLy of Linear mappings i s equicontinuous ( s e e D e f i n i t i o n 1 . 2 . 3 3 )

(3.4.1)

a ( x , E , D ) : H ( s ),E (7Rn)

-f

H(S-V)

,E

(Rn)

,

(o,Eol.

E

Proof. F i r s t assume t h a t s = v = 0 , s i n c e t h e argument i n t h i s c a s e i s p a r t i c u l a r l y simple.

The

assumption

a ( x , E , S ) E So 1no

(mn) n

S ( m n ) yields the

inequalities

/ a ( r l , ~ , S I) 5 CN-N,

(3.4.2)

v

N

since

naa(n,~,E)

D"a(x,E,S). x-trl x

= F

Therefore,

II

(3.4.3)

I

~ ( ~ , E , D ) (~0 I) , E

Since a(x.E.5)

= F

-1 a(rl.E.5) n+x

5

c ~ < ~ >I U-I ~ I ( oI)

and j e x p ( i x . n )

/

;E

G

+

5 1 one g e t s u s i n g ( 3 . 4 . 3 ) :

i s equivalent t o the inequality

Now c o n s i d e r any s and v . Then ( 3 . 4 . 1 )

where

V N.

L2(IRn),

5

KG i s a n i n t e g r a l o p e r a t o r w i t h t h e k e r n e l

and t h e c o n s t a n t C d o e s n o t depend on

E

and u.

Again t h e a s s u m p t i o n s on a ( x , E , S ) imply t h a t

:

3 . Singular Perturbations on Smooth Manifolds without Bounda y

240

The f o l l o w i n g i n e q u a l i t y h o l d s :

<E>T-T

(3.4.7)

4 21TI<E-T)>lTl

(3.4.7) h o l d s with (3.4.7)

with

Thus, (3.4.8)

N

T

and t h a t i m p l i e s t h a t

< 0 , one s w i t c h e s

5 and n and a p p l i e s

> 0.

-T

(3.4.7)

(3.4.6), /K(~,E,TI)

where C

Z 0. I f

T

2 4 2 <5-n>'

<5>'

I n d e e d , e v i d e n t l y one h a s :

E W.

t/ T

I

lead t o the inequality:

5 CN<S-n>-N,

depends o n l y on N ; h e r e N = N

-1

1

s

2

-v

2

I-1

s

3-v 3

I.

T h e r e f o r e , f o r V N one h a s :

where f ( 5 ) = C

N

<5>-N and f *

N

g s t a n d s f o r t h e c o n v o l u t i o n of f and g .

Applying t h e i n e q u a l i t y :

and t a k i n g N > n , one g e t s t h e c o n c l u s i o n t h a t t h e f a m i l y o f i n t e g r a l 2 n o p e r a t o r s w i t h k e r n e l s K ( C , E , ~ ) are c o n t i n u o u s i n L ( W ) and t h e i r norms

a r e u n i f o r m l y bounded w i t h r e s p e c t t o

E

E (O,E

0

1.

I

C o r o l l a r y 3.4.2.

Then t h e foZZowing f a m i l y o f l i n e a r mappings i s equicontinuous for each 3 s = (Sl,S2,S3) E R : a(X,E,D)

In f a c t , a ' ( x , E , D )

: II(s)

( S )P E

u n i f o r m l y bounded i n am(E,D)

,E(IRn)

: H

+

(Wn)

13

(s-u) , €

H(S-V) ,E

(2) , E (IR

E

(O,Eo1.

) and have t h e i r norms

E ( 0 ,3 ~a c c o r d i n g t o Theorem 3.3.1,

E

O

: H

n

(IR )

( s ) r E ( R n ) + H(S-V) ,E a s f o l l o w s from P r o p o s i t i o n 3.1.6.

while

uniformly with r e s p e c t t o

D e f i n i t i o n 3.4.3.

A singtilar pel-tilrbation

a(X,E,D)

v = ( v ~ P ~ ~ i ,f "f o~r )each

a(x,E,D)

: H

( S ) ,E

given

i s s a i d t o have v e c t o r i a l order E~

m

the family

( I R R n ) + H(S-V)

is equicontinuous and, moreover, f o r each

,E

(Wn),

(E

E

(o,Eol)

< v the mppings

E

E (O,E~I,

3.5. Pseudolocality of Singular Perturbations (Rn)

a(x,E,D) : H ( s )

H

+

(s-v)

o r have unbounded norms when

E

(E

24 1

E (O,E?I) e i t h e r are n o t continuous

,E

0.

+

C o r o l l a r y 3.4.4.

A f a m i l y o f p e r t u r b a t i o n s a ( x , E , D ) has order v E R

and a ( x , ~ , c )$?

S'

1 ro

( n n ), V

3

.

zf a ( x , E , c ) E s"

1,o

p < v.

(Rn)

Example 3.4.5. with A t h e Laplace o p e r a t o r , belongs

( i )A f a m i l y of p e r t u r b a t i o n s E'A'-A

( R n ) with V

t o S'

>

LI

and i t s o r d e r i s

(0,2,2)

1r o ( i i )A f a m i l y of p e r t u r b a t i o n s

a

1

E

= (0,2,2).

(x)D D with a ( x ) E C z ( R n ) kj k 1 kj

lsk,j
belongs t o S

Remark 3.4.6. Using t h e s a m e argument a s i n t h e p r o o f of Theorem 3.4.1,

one shows t h a t

m

(U) and g E C o ( U ) , t h e n H (u) 3 u + g O p a u E i f a ( x , E , E ) E S" 1,o ( s ) ,E H f U ) i s a f a m i l y of c o n t i n u o u s l i n e a r mappings, whose norms a r e ( s - v ) ,E u n i f o r m l y bounded w i t h r e s p e c t t o E E ( O , E ~ ] . I n o t h e r words, f o r each E~

> 0, s

E

R 3 t h e f o l l o w i n g f a m i l y o f l i n e a r mappings i s e q u i c o n t i n u o u s :

a(x,E,D) where

E +

gu E H

3.5.

H

1oc

H

:

loc

(u)

( s ) ,E

+

H ( s - " ), E

(U),

( U ) i s t h e f a m i l y of s p a c e s c f a l l f u n c t i o n s u s u c h t h a t

( s ) ,E

(U) f o r ( 5 ),E

v

g

E

m

Co(U).

The p s e u d o l o c a l i t y of s i n g u l a r p e r t u r b a t i o n s

W e s t a r t with t h e following

D e f i n i t i o n 3.5.1.

A f a m i l y of d i s t r i b u t i o n s

on an open s e t

v 5u

(O,E~I3

E + U(E,X)

i f f o r any g i v e n

E

E D' (u) i s s a i d t o be smooth

E ( 0 ,0 I~ it c o i n c i d e s idith some

) f o r any compact s e t f u n c t i o n u ( x ) E c ~ ( v and,moreover, (3.5.1)

max xEK

1

I D ~ ~ ( E , X )5

where t h e c o n s t a n t s

C

K,a

c ~ , ~ v ,a, v

KCC

do n o t depend on

E

KCC

v

one has:

v

-

E ( 0 , ~ 01

Definition 3.5.2.

The complement of t h e maximal open s e t on u h i c h a f a m i l y o f d i s t r i b u t i o n s E + U(E,X)

i s smooth, i s c a l l e d t h e s i n g u l a r support of u and denoted by

s i n g supp u ( E , x )

.

3. Singular Perturbations on Smooth Manifolds without Bounda y

2 42

Example 3.5.3. IR+ 3

The f a m i l y o f d i s t r i b u t i o n s :

singular support t h e o r i g i n x

=

E

+

Eaexp(-/xl

2

/E

2

)

E S ' (Xn) h a s as i t s

0 , V a E IR.

Our p u r p o s e i s t o prove t h e f o l l o w i n g Theorem 3.5.4.

Let ( o , E 3 E~ + I u(x,E) (3.5.2)

E

(u) and l e t

€ 1

(u). Then a ( x , E , ~ )E S" 1,o

s i n g s u p p ( ~ " a ( x , ~ , D ) u ( ~ , x5 ) ) s i n g supp u ( E , x ) .

Proof. W e associate w i t h E"la(x,E,D)

which i s a f a m i l y of

its kernel K(E,x,Y),

distributions

d e f i n e d by t h e r e l a t i o n

According t o Theorem 3.3.5,

the kernel-distribution m

f a m i l y of c o n t i n u o u s l i n e a r maps K : C o ( U ) W e are g o i n g t o show t h a t t h e f a m i l y (O,E

0

+

1

i n (3.5.3)

Cm(U) and

3

E +

K

:

defines a

E' (U)

+

D ' (u).

K ( E , x , ~ ) i s smooth o f f t h e

d i a g o n a l x = y i n UxU. Indeed, u s i n g (3.5.3)

and t h e d e f i n i t i o n of s i n g u l a r p e r t u r b a t i o n s ,

one f i n d s e a s i l y t h a t

Consequently,

The l a s t f o r m u l a f o r l a / > u +u +n c a n b e r e w r i t t e n as a c o n v e r g e n t i n t e g r a l 2 3

a n d , a s a consequence o f t h e u n i f o r m convergence of t h e i n t e g r a l on t h e r i g h t hand s i d e , one g e t s t h e c o n c l u s i o n t h a t ( x - ~ ) ~ K ( E , x , ~ E) C J ( U W ) i f

/ a / > v 2 +v3 +n+j. Moreover, a l l t h e d e r i v a t i v e s up t o t h e o r d e r j of (~-y)~K(~,xa ,y r e) u n i f o r m l y bounded w i t h r e s p e c t t o E E ( 0 . ~ ~ 1T. h a t p r o v e s t h e claim t h a t

E

+ K(E,x,~)

h a s as i t s s i n g u l a r support a c l o s e d set

243

3.6. Asymptotic Expansions of Symbols belonging t o t h e diagonal x = y i n UXU. I n o r d e r t o end t h e p r o o f of t h e theorem, w e need o n l y t o a p p l y t h e following Lemma 3.5.5.

Let

( O , E ~ ] 3 E + K ( ~ , x , y )E

D ' ( u x u ) . Suppose t h a t t h e famiZy of Zinear

maps m

m

(3.5.7)

K : Co(U)

+

C (U), K : E ' ( U )

is u n i f o r m l y bounded i n t h e diagonal in

E

+

D'(U)

E (O,c0], and, moreover, t h a t K is smooth off

UXU.

Then s i n g supp Ku c - s i n g supp u, V u E E ' ( U )

.

Proof of Lemma 3 . 5 . 5 . Let

v

and W b e open s u b s e t s of U s u c h t h a t W C C V. m

Let $ E

co(v),

Let

u E E ' ( U ) b e such t h a t u i s smooth i n V . Then Ku = K ( $ u ) + K ( ( l - $ ) u )

E -t

$

1 i n a neighborhood of t h e c l o s u r e of W. m

Since E

E +

$u i s smooth on U , and E

E (O,E~],

K(@)

$u E C (U) u n i f o r m l y w i t h r e s p e c t t o

+

0

i s smooth on U . On t h e o t h e r h a n d , w e c a n w r i t e

K ( ( l - $ ) u ) as an i n t e g r a l K((l-$)u) =

1

K(E,x,Y)

(I-$(y))u(y)dy.

U I f x E W , y E s u p p ( 1 - $ ( y ) ) , t h e n ( x , y ) b e l o n g s t o t h e complement o f some s u i t a b l e neighborhood of t h e d i a g o n a l i n U X U , t h a t i s ( x , y ) b e l o n g s t o a m

s u b s e t where K ( E , x , ~ ) i s smooth ( C

in (x,y) with a l l the derivatives

E ( 0 ,0 ~1 ) . Hence, K ( ( 1 - $ ) u ) i s smooth on V a s w e l l . S i n c e V w a s any open s u b s e t of U, one g e t s t h e c o n c l u s i o n

u n i f o r m l y bounded w i t h r e s p e c t t o

t h a t s i n g supp Ku C_ s i n g supp

E

U.

1

3 . 6 . Asymptotic Expansions of symbols A s it h a s been mentioned i n Remark 3 . 3 . 6 ,

admit a n a t u r a l g r a d u a t i o n

(3.3.13),

some p o l y n o m i a l i n (E,E)

symbols

(3.3.14).

One c a n c o n s i d e r f o r m a l i n f i n i t e sums o f symbols of t h e form ( 3 . 3 . 1 3 ) whose o r d e r s s a t i s f y ( 3 . 3 . 1 4 ) . One o f t h e b a s i c m o t i v a t i o n s t o c o n s i d e r such f o r m a l sums i s t h e f a c t t h a t t h e y a p p e a r i f one expands t h e i n v e r s e symbols t o e l l i p t i c polynomial symbols of t h e form ( 3 . 3 . 1 3 1 ,

(3.3.14)

into

a n a s y m p t o t i c series. On t h e o t h e r hand, it i s c o n v e n i e n t t o have a symbol f u n c t i o n f o r which a g i v e n f o r m a l sum of symbols c a n b e c o n s i d e r e d a s an

3. Singular Perturbations on Smooth Manifolds without Boundary

244

a p p r o p r i a t e l y i n t e r p r e t e d asymptotic expansion. Theorem 3 . 6 . 1 .

i)

L e t a . E syl0 ( u ) , v (1")

< v(j), j

7

v (11 ) J.

(3.6.1)

-a

0,1, and l e t

=

for j + ( 01

Then t h e r e e x i s t s a E s V

1.0

a(x,E,c) -

(3.6.2)

such t h a t (N) (U), V N t 0

C a . ( x , ~ , E ;E ) Sv 1r o O=<j
and moreover, f o r each p a i r of m u l t i - i n d i c e s a , @ and every compact s e t K c U

t h e f o l l o w i n g asymptotic r e l a t i o n s hold:

unifomnZy w i t h r e s p e c t t o (x,c;) E

K x

mn.

Proof. L e t + ( t )E

crn(K+) ,

Let K .

C K

K

c

o

E

...,

for t E

[o,tI, 4

z 1 for t t I.

b e a sequence of compact s e t s e x h a u s t i n g U, i . e .

3' 1 2 t h e union o f a l l K . c o i n c i d e s w i t h U. 3 We choose 6, so small t h a t

f o r V j t 1, V

E

E ( 0 . ~ ~ 1V, 5 E mn,

V x E Krr

v

a,B, r w i t h

l a i + l f I + r5 j .

T h i s i s p o s s i b l e s i n c e v!J)

+

-m.

I n f a c t , one h a s

Let

W e choose 6,

> 0 t o b e so s m a l l t h a t

I n t r o d u c i n g t h e numbers m. =

max t +lt
1

(1-1) 1

+ ( t I)

one c a n s a t i s f y t h e i n e q u a l i t y ( 3 . 6 . 6 ) by c h o o s i n g 6 . > 0 s u c h t h a t 7

3.6. Asymptotic Expansions of Symbols

245

Y. w i t h y , = ( v ~ J ) - v ~ J - ' ) ) - <' 0 .

6 . 4 ( 2 J C .m . ) 3

3

3

3

Moreover, w e r e q u i r e :

6.

1

+

j +

for

0

m.

Now we d e f i n e t h e symbol a ( x , ~ , E )t o be (3.6.7)

j

a(x,e,E)

-1

@ ( 6 . ~ ) a .( x , E , S ) ,

C

=

j>o

3

V (x,E,S)

E u

x

( 0 . ~ ~x 1R ~ .

One c h e c k s e a s i l y t h a t f o r a ( x , E , C ) d e f i n e d by ( 3 . 6 . 7 ) w i t h 6 . 3 s a t i s f y i n g (3.6.4), the r e l a t i o n s (3.6.2), (3.6.3) hold.

+

0,

+ m,

I n f a c t , i t i s enough t o check ( 3 . 6 . 3 ) s i n c e ( 3 . 6 . 2 ) i s a consequence of

( 3 . 6 . 3 ) . One g e t s u s i n g ( 3 . 6 . 5 ) w i t h j

for V

E

E (0,6N-1],

E

V x

Kr,

V

=

N and ( 3 . 6 . 4 ) :

5 E Rn , where t h e c o n s t a n t s C'

a,B,N,j

depend on t h e i r s u b s c r i p t s , and t h i s i s p r e c i s e l y ( 3 . 6 . 3 ) .

(3.6.8)

+

~ ( 1 ) 2

-m

for

Then t h e r e e x i s t s a E

Sv

j

--t

might

I

m.

(0) (U) such t h a t

1t o

(N)

(3.6.9)

a ( x , ~ , S -)

a.(x,E,S) 06j
E Sv

1t o

(U), V N t 0 ,

and moreover, for each p a i r o f m u l t i - i n d i c e s

uniformly w i t h r e s p e c t t o

E

a,B

and each compact K c u

E ( 0 . ~ ~ 1x , E K

Proof. The f u n c t i o n $

for

I a1 + / B I +r

E Cm(F+) b e i n g t h e same a s i n t h e p r o o f o f Theorem 3 . 6 . 1 ,

S j, x

E

Kr,

Kr,

r

=

sequence of compacts a s p r e v i o u s l y .

0,1,.

. ., b e i n g

t h e same e x h a u s t i n g U

3. Singular Perturbations on Smooth Manifolds without Boundary

246

S i n c e v ~ J )J-

for j +

-m

m,

one c h e c k s e a s i l y , u s i n g t h e L e i b n i t z

f o r m u l a and t h e same argument a s i n t h e p r o o f o f Theorem 3 . 6 . 1 ,

t h a t such

a c h o i c e of 6 . i s p o s s i b l e . A f t e r w a r d s , one d e f i n e s a ( x , ~ , S )t o b e : 1

I

and ( 3 . 6 . 9 ) , ( 3 . 6 . 1 0 ) c a n b e checked a s p r e v i o u s l y . [ v ] = v2+v3, \ v [

W e r e c a l l the notation:

=

vl+v2, V v

=

(vl,V2,v3) E R

3

.

Theorem 3.6.3.

Let a . E 3

(3.6.13)

"yl0

j), v

[v")]

iv

(I+') C

--,

'j),

for

j

j +

=

o , ~ ... , and l e t

m.

(0)

Then t h e r e e x i s t s a E a ( x , ~ , E -)

(3.6.14)

Sv

( U ) such t h a t

I:

a . ( x , ~ , F )E Svl,o

1 to

(N)

, V N

2 0

OSj
and, moreover, for each p a i r of m u l t i - i n d i c e s

a,B

and each compact

K c

u

t h e f o l l o w i n g asymptotic r e l a t i o n s h o l d : (j) (3.6.15)

E"

'

I: a , ) O<j
I-lalD6Da(a-

uniformly w i t h r e s p e c t t o

E

E

= O(lEC1

["(N) ] ) f o r / € E l

-f

m

( O , E ~ ] ,x E K .

Proof. __ With t h e same c h o i c e of t h e f u n c t i o n $ for j

-f

m

E

Cm(%

)

a s p r e v i o u s l y and 6 . C 0 , 1

such t h a t

f o r / a / + l B l + rS j , x E K r ,

K

1

c K

2

c

. _ .e x h a u s t i n g

U, one d e f i n e s a t o b e :

-

Xa. and c a l l Za. t h e f o r m a l symbol, i f a . have o r d e r s 3 I 3 s a t i s f y e i t h e r ( 3 . 6 . 1 ) o r ( 3 . 6 . 8 ) o r (3.6.13) and Ea, i s a s y m p t o t i c a l l y 3 c o n v e r g e n t t o a i n t h e s e n s e of (3.6.3) o r (3.6.10) o r ( 3 . 6 . 1 5 ) , r e s p e c t i v e l y . We shall w r i t e a

D e f i n i t i o n 3.6.4. (0)

The symbol a E sV

(0) (u) i (j)

s s a i d t o belong t o t h e c l a s s K~

(u) i f t h e r e

.

(UT) w i t h v (O) > v ( l ) t...> v ( J ) t . . , 1 u (1)1 c --, e r i s t symbols a . E s" 7 1.0 1 [ v ( J ) ] c -m f o r j + m, which a r e homogeneous of degree / v ( J ) I i n (E- , S ) ' g o

3.6. Asymptotic Expansions of SymOols for

E

x n , 151 t

E IR+, 5 E

I , i.e. ( j1

-1

a.(x,t 3

(3.6.18)

~ , t 5 )= t

241

+”( j )

v1

2

a . (x,E,t), 3

v

t t 1,

v

(E,<)

151

E

R+X

IR~,

2 1,

and such t h a t (N) (3.6.19)

(a

-

a,) E

C

S1)

(U),

V N 2 0.

O<j
With a .

u

-1 (E

x

,5)

t h e e x t e n s i o n of a . ( x , E , < ) from t h e s e t

( x , E , ~ )

70

R+

x

x

(mn\{

15 I

3

a s a homogeneous f u n c t i o n in v a r i a b l e

< 1))

R+ x ( I R ~ \ { o I o) f degree ( v ( J ) l , a ,

30

E

C m ( U x R+ x ( I R ~ \ { o I ) ) ,

t h e formal sum (3.6.20)

1 a. j t o 3’

(x,E,<)

i s c a l l e d t h e graded symbol and i s denoted b?j

ar(x,€,<).

D e f i n i t i o n 3.6.5.

A function a 0 ( x , € , t )

:

symbol c l a s s HV(u), v

u

=

x

R+

x

(Rn\{Ol)

(vl,v2,v3) E

i s homogeneous i n ( E - ~ , < )E R+

a0

f o r each p a i r o f m u l t i - i n d i c e s

where t h e c o n s t a n t s c

clBK

x

m3

+

c i s s a i d t o belong t o t h e

i f a O E c m ( ux R +

x

( n n \ 1 0 ) ) of degree l v l

(nnl{o})), =

vl+v2 and

a,B and each compact s u b s e t K c u h o l d :

depend only on t h e i r s u b s c r i p t s .

Remark 3.6.6. I t i s immediate t h a t t h e symbol a . i n D e f i n i t i o n 3.6.4 b e l o n g s t o 10 Hv(I) (U). I n f a c t , one h a s o n l y t o check t h e v a l i d i t y of (3.6.21) w i t h

v

=

v (I),

( E -1 , < )

which i s a consequence of (3.3.1) and t h e homogeneity of a , i n JO E R + x ( n n \ { 0 } ) of d e g r e e Iv(1)/.

Theorem 3.6.7.

L e t a E KV(’) (u) and l e t ar b e i t s graduate symbol. Then f o r V f E c:(u), t/

( x , ~E )

u

x

( R ~ \ { O } ) t h e following asymptotic f o m l a holds: -1

(3.6.22)

e-iE

iE-l<x,n>f(x)) < X ’ n > a ( x , c , D )( e

where a ( x , ~ , D )= O p a .

3. Singular Perturbations on Smooth Manifolds without Boundary

248

Moreover if K1 c u ,

K2

c (lRn\iO}) a r e any compact s u b s e t s and B is

any bounded s e t in c;(u), then ( 3 . 6 . 2 2 ) hoZds uniformly w i t h r e s p e c t t o E K~

(X,S)

x K~

and f E

B.

Proof. __

One h a s a c c o r d i n g t o t h e d e f i n i t i o n of a s i n g u l a r p e r t u r b a t i o n a ( x , E , D ) : -iE

-1

<x,r~>,(~

,E,D) (e

i E

-1

<x

'n>f(x)) =

-1 =

(2n)-n

.i<x,S-~

'>a(x,E,S)f(S-E

-1

q)dg =

mn =

(2n)-n J

e

i < x , E>

a (x,E

,S+E

-1

rl)

2 ( 5 )d5.

lRn

We a r e g o i n g t o u s e T a y l o r ' s f o r m u l a (3.6.23)

-1 a ( x , ~ , c + n) ~

where a ( a )

=

aaa

5

=

'a1 a

i'

1

c

=

a(")

-1 (X,E,E

n)cn

+ qN(x,E,5,q)

(0)

l 4 < N

D a and qN i s t h e r e m a i n d e r . S i n c e a E

5

Kv

(U) one

has def (3.6.24)

rN

=

(N)

(a-

z a,) OSj
E

S'

(u),

1 ,O

Therefore,

F u r t h e r , one h a s : (3.6.26)

if

E

a!") ( x , E , E - ' ~ ) 3

=

a!")

(X,E,E

-1 n)

=

E l " l - l v ( J ) ' a j(O a )( x , l , n ) ,

10

E (0,InjI. Using ( 3 . 6 . 2 5 ) ,

( 3 . 5 . 2 6 ) one g e t s t h e f o l l o w i n g a s y m p t o t i c r e l a t i o n

which means t h a t f o r V N t h e remainder

N

(X,E,E

-1

n)

d e f i n e d by ( 3 . 6 . 2 4 )

s a t i s f i e s (3.6.25). I n o r d e r t o f i n i s h t h e p r o o f w e have o n l y t o e s t i m a t e t h e t e r m

3.6. Asymptotic Expunsions of Symbols corresponding to 'the remainder q

249

N'

where 6 E ( 0 , l ) Denote

and let

depends only on its subscripts. where the constant C arK1tK2 Further for 2 ~ 1 5 12lrl and la1 (0) S" (U): 1to

=

N 2

via)

one gets, using one more

time the fact that a E la

(01)

(X,E,E

-1

-V

rl+65)j 5 c

(0)

E

Therefore, one finds for Q (f):

so that the last integral in the right hand side of (3.6.29) is

exponentially small when

E

-f

0:

(0)

3 . Singular Perturbations on Smooth Manifolds without Bounda y

250

S i n c e y depends o n l y on t h e d i a m e t e r of supp f , t h e i n t e g r a l (3.6.30) is e x p o n e n t i a l l y s m a l l u n i f o r m l y w i t h r e s p e c t t o f i n a g i v e n bounded s e t B

c C m ( U ) and w i t h r e s p e c t t o 0

T'

1

E K2 c U , and t h a t ends t h e p r o o f .

Remark 3.6.8.

E S v ( U ) t h e n t h e same argument as i n t h e p r o o f of Theorem 3.6.7.

If a

shows t h a t

-1

(3.6.31)

e-iE-'<xfn>

a ( x , E , D ) ( ei

= E-"'aO(X,l,T1)

f o r ri f R n , In1 2 1, x E K

C

<x*ri>f ( x ) )

E

+

o(E1-'v'),

U, f E B C C ; ( U ) ,

=

E +

0,

where a 0 ( x , E , C ) i s d e f i n e d

by (3.3.2), (3.3.3). One h a s t h e same a s y m p t o t i c formula (3.6.31) a l s o f o r ( u n i f o r m l y w i t h r e s p e c t t o q i n any g i v e n compact K2

C

V r- E IRn\{O]

IRn\{O}), b u t , o f

c o u r s e , a o ( x , E , C ) s h o u l d b e r e p l a c e d i n t h i s c a s e by i t s e x t e n s i o n t o

U

x

R+ X ( R n \ { O j ) as a homogeneous f u n c t i o n of (c-l,C) of d e g r e e

1\11

v +v

2'

1

3.7. Amplitudes, A d j o i n t s and P r o d u c t s of S i n g u l a r P e r t u r b a t i o n s W e s t a r t w i t h i n t r o d u c i n g c l a s s e s of s i n g u l a r p e r t u r b a t i o n s , which

w i l l l o o k a s more g e n e r a l , b u t i n f a c t , c o i n c i d e w i t h t h e c l a s s e s d e f i n e d p r e v i o u s l y i f some n a t u r a l c o n d i t i o n s a r e s a t i s f i e d . D e f i n i t i o n 3.7.1.

The f u n c t i o n a ( x , y , E , < ) E cm(u x u order v

=

u

e x i s t s a constant c

v

( O , E ~ ]x

nn)i s c a l l e d amplitude of

(vl,v2,v3) E m3 and i s said t o belong t o t h e c l a s s s v

i f f o r each compact K c

for

x

( x , y ) E K,

v

u and

x

> 0

(E,c)

The s i n g u l a r p e r t u r b a t i o n

(U

x

U)

there

such t h a t

E (o,cO1 A~

1 ,O

each t r i p l e of i n d i c e s a,B,y,

x

nn.

w i t h an amplitude a E S y , o ( U

x

U) i s d e f i n e d

a s t h e double i n t e g r a l

f o r V u E C:(U), in

<.

where t h e i n t e g r a t i o n i s done f i r s t i n y and afterwards

25 1

3.7. Amplitudes, Adjoints and Products m

m

The p r o o f o f Theorem 3 . 3 . 5 . shows t h a t A : Co(U) + C ( u ) , v E E ( o n E O 1 , "1 m m AE : C o ( U ) + C (U) a r e u n i f o r m l y b o u n d ed with r e s p e c t t o E E (O,soI. and t h a t E

its distributional

One can a s s o c i a t e w i t h t h e s i n g u l a r p e r t u r b a t i o n A kernel A(x,Y,E)

which c a n be f o r m a l l y d e f i n e d a s t h e f o l l o w i n g F o u r i e r

inteqral ~ ( x , y , E )= ( 2 7 1 1 - ~ i

(3.7.3)

mn

e

i<x-y

,5>

a (x,Y, E t S)dS.

A p r e c i s e d e f i n i t i o n o f t h e d i s t r i b u t i o n A ( x , y , ~ )c a n b e g i v e n , i f w e ,

i n s t a n c e , i n t e r p r e t e t h e r i g h t hand s i d e o f

for

(3.7.3) a s an o s c i l l a t o r y

i n t e g r a l i n the following sense. We may w r i t e ( s t i l l f o r m a l l y ) V N E Z+:

Now we u s e t h e L e i b n i t z f o r m u l a (see C h a p t e r 1 ) :

-

w i t h a s u i t a b l e cho c e o f t h e c o e f f i c i e n t s q

(3.7.6)

n,B'

( 3 . 7 . 5 ) a l l o w u s t o r e w r i t e A ( x , y , ~ )a s f o l l o w s :

Hence ( 3 . 7 . 4 ) , A(x,y,S

where

a r e w e l l defined continuous functions of

(x,y) E U

X

U, V

E

E ( 0 . ~ ~i f1 N

i s c h o s e n l a r g e enough.

Now,

(3.7.6),

( 3 . 7 . 7 ) can be t a k e n a s a d e f i n i t i o n of t h e d i s t r i b u t i o n

t h e d e r i v a t i v e s Dh i n ( 3 . 7 . 6 ) b e i n g i n t e r p r e t e d i n d i s t r i b u t i o n a l

A(x,Y,E),

sense. Example 3 . 7 . 2 . 2 2 2 -1 L e t a ( x , Y , E . t ) = (1+E q ( x , y ) c ) where q ( x , y )

E

m

C

(U

a E S"(U

X

U) w i t h v

=

(0,0,-2).

One f i n d s u s i n g

x

-'.

some c o n s t a n t qo > 0 t h e i n e q u a l i t i e s qo < q ( x , y ) < qo

U) s a t i s f i e s w i t h Obviously,

Example 3 . 2 . 2 .

that the

c o r r e s p o n d i n g d i s t r i b u t i o n a l k e r n e l A ( x , y , ~ )i s g i v e n by t h e f o r m u l a

3 . Singular Perturbations on Smooth Manifolds without Boundary

252

-2 where ho ( E , x ) i s g i v e n by ( 3 . 2 . 2 3 ) . m

The s i n g u l a r p e r t u r b a t i o n A

m

: Co(U)

+ C

(U) w i t h k e r n e l d i s t r i b u t i o n

(3.7.8), A u(x) =

I

A(x,y,~)u(y)dy,

U can not. b e r e d u c e d t o a s i n g u l a r p e r t u r b a t i o n of t h e form a ( x , ~ , D )w i t h symbol a ( x , e , S ) .

However, it c a n b e r e p r e s e n t e d a s a sum (3.7.9)

=

where a ( x , E , D )

a

N

(x,E,D)

+ RN,

,

V N > O

i s t h e s i n g u l a r p e r t u r b a t i o n w i t h t h e symbol

has i t s amplitude i n t h e c l a s s and t h e s i n g u l a r p e r t u r b a t i o n R (N1 N, E S" ( U X U) w i t h "(N) = (-N,O,-2-N). 1,o Moreover, t h e r e e x i s t s a s i n g u l a r p e r t u r b a t i o n a(x,E,D) w i t h t h e a r a d e d svmbol

and a s i n g u l a r p e r t u r b a t i o n R

with amplitude r ( x , y , E , S ) E S

f o r V u E E'(U) one h a s REuECm(U)and A

=a(x,E,D)+R

. Infact,one

R u =O(sm)

u s e s (3.6.7)

when E + O )

(--,

0 ,- m )

1, o

(i.e

such t h a t

i n o r d e r t o c o n s t r u c t a symbol

a ( x , E , C ) h a v i n g ( 3 . 7 . 1 0 ) a s i t s g r a d u a t e symbol. W e s h a l l show a l l t h i s l a t e r i n a more g e n e r a l s i t u a t i o n . T h e r e i s one i m p o r t a n t c l a s s o f s i n g u l a r p e r t u r b a t i o n s i n O p S " ( U which c a n be r e d u c e d t o t h e form a ( x , E , D ) E O p S " ( U 1 .

X

U)

These a r e p r o p e r l y

supported s i n g u l a r p e r t u r b a t i o n s . D e f i n i t i o n 3.7.3.

A singular perturbation

supported if

m

E"'A~

:

A

F o p sv (uxu), ( v = ( v 1 , ~ ,2u 3 ) ) , v +

c o ( u ) E ' ( u ) and its transpose -f

E

"AE

is c a l l e d properly m

: Co(Ul

-+

E ' (U),

( E E ( o , c 0 ] ) arebothequicontinuous. V1

One c a n show t h a t A E i s p r o p e r l y s u p p o r t e d i f f i t s k e r n e l A ( x , y , E ) h a s t h e A(. , ,E ) c u x u h a s compact p i - n p e r t y : t h e c l o s u r e of t h e u n i o n (iO<E$EO s ~ p p i n t e r s e c t i o n w i t h KxU and UXK f o r any compact K c U .

.

3.7. Amplitudes, Adjoints and Products intersection with

KXU

and UXK, f o r any compact s e t K c U .

I f AE i s p r o p e r l y s u p p o r t e d , t h e n A I n f a c t , AE

m : CO (U)

-t

C m ( U ) and A

9E

K t U be a compact. W e choose

i s contained i n K

253

x K1 w i t h K

1

: Cm(U)

: Cm(U) E

+

-t

D' (U).

Cm(U), V Let u

E

E

m

Co(U)

E ( 0 . ~ ~ 1 .

C m ( U ) and l e t

such t h a t supp A ( x , Y , E )

n

(U x K )

c U a n o t h e r compact and @ E 1 i n a

. T h e r e f o r e , one h a s : A u ( x ) = A E ( @ u()x ) , v x E K , 1 m so t h a t A u ( x ) E C ( i n t K ) , V E E ( O , E 1. S i n c e K i s any

neighborhood of K

V

E

E

(O,E~],

compact, it p r o v e s t h a t A

: Cm(U)

+

Cm(U), V

E

0 E (0.~~1.

Theorem 3 . 7 . 4 .

E op s"

Let

1 .o

(u

x

u ) be properly supported and l e t

a ( x , y , E , E ) be i t s

amp E t u d e . Then (3.7.11)

A

=

a(x,c,~),a

E op

S"

1 ,o (U)

'

where

and t h e forma2 symbo2 of a ( x , E , ~ )is given by t h e f o m 2 a

Proof. ___ Since A

: Cm(U) + Cm(U),

t h e symbol a ( x , E , g ) = e - i < x , S > A E(,i<x,S> ) i s a

w e l l d e f i n e d smooth f u n c t i o n of

(x,s) E U

x

Rn

depending smoothly on

E E ( 0 . ~ ~ 1 Fu . rther,

u(x)

=

IT-^

J

ei<x'5';(C)dC

Bn

and A

: C m ( U ) + C m ( U ) , one c a n a p p l y A

under t h e i n t e g r a l s i g n , so t h a t

E S" (U) and t h a t ( 3 . 7 . 1 3 ) h o l d s . 1 ,O (U x U) For d o i n g t h i s w e n o t i c e f i r s t t h a t f o r any a m p l i t u d e b E S' 1# O and any m u l t i - i n d i c e a t h e a m p l i t u d e s ( y - x ) " b ( x , y , ~ , c ) and D " b ( x , y , E , S ) 5 pa ( U x u ) w i t h urn = p-(O,lal,O). d e f i n e t h e same s i n g u l a r p e r t u r b a t i o n i n Op S and w e have o n l y t o show t h a t a

I n f a c t , one f i n d s f o r B

E,a

=

Op(y-x)"b(x,y,~,g):

3 . Singular Perturbations on Smooth Manifolds without Bounda y

254

Now w e w r i t e t h e f i n i t e T a y l o r e x p a n s i o n of a ( x , y , ~ , S )w i t h r e s p e c t t o y about t h e p o i n t x:

where

T h e r e f o r e one g e t s t h e c o n c l u s i o n t h a t a l o n g w i t h t h e a m p l i t u d e a ( x , y , € , t ) a l s o t h e amplitude

w i t h R g i v e n by ( 3 . 7 . 1 5 ) . d e f i n e s t h e same s i n g u l a r p e r t u r b a t i o n A N more o v e r ,

and,

for V N > 0. But t h a t means p r e c i s e l y t h a t t h e f o r m a l symbol of A ( 3 . 7 . 1 3 ) , so t h a t if a ( x , ~ , S )i s any symbol in S"

1 to

(U)

is g i v e n by

with t h e asymptotic

expansi o n

( s u c h a symbol e x i s t s a s a consequence o f Theorem cv1(A - a ( x , E , D ) )

f op s

(O'-mr-m)

1 r0

3.6.2),

then

uniformly with r e s p e c t t o E E ( 0 . ~ ~ 1 .

Theorem 3 . 7 . 5 .

If a ( x , c , D ) E op sv

1 ,O

(u

x

U)

is p o p e r l y supported, then its transpose

3.7. Amplitudes, Adjoints and Prodiicts

and i t s a d j o i n t a ( x , E , D ) * are again s i n g u l a r p e r t u r b a t i o n s i n

a ( x ,E , D )

s;,o(u

x

U). t

Moreover t h e i r formal symbols a , a

*

a r e g i v e n by t h e f o l l o w i n g

formulae:

(3.7.16) a*-X-

114 a!

D’DZa 5

( x ,E , )

*,

where a ( x , t ) * i s t h e (complezi conjugate o f a ( x , C ) . Proof. __ One can w r i t e :

=

(2.rr)-”

/,u(x) IR X

J

i

e

i<x-y

,c>

nn IRn 5 Y

a ( y , E , - 5 ) v ( y )d y d t d x

t so t h a t a ( x , ~ , D ) h a s a ( x , ~ , - E )a s one of i t s a m p l i t u d e s .

In t h e s a m e way one f i n d s

where a g a i n t h e u p p e r

*

means t h e (complex) c o n j u g a t e .

T h e r e f o r e , a ( x , E , D ) * h a s a s one of i t s a m p l i t u d e s t h e f u n c t i o n

a ( y , E , t )*. Applying Theorem 3 . 7 . 4 t o t h e a m p l i t u d e s a ( y , E , - C ) , a ( y , ~ , E , ) * one , gets

I

formulae ( 3 . 7 . 1 6 ) . Theorem 3 . 7 . 6 .

Let a(x,E,D) E op

’S

1,o

(u) and l e t

b(x,E,D)

E

Op S’

supported. Then (3.7.17)

a(x,E,D)

b(x,E,D)

E

Op S”+’(U 1r 0

x

U)

1n

o

(U) be p r o p e r l y

255

3. Singular Perturbations on Smooth Manifolds without Boundary

256

and, moreover, t h e forma2 symbol c (3.7.18)

c

besides c ,

-

1 l " l

Zc. 3'

=

l a l = j I:-- a!

-

Zc.

I

of a

b

Daa ( x , E ,C) D>

F

is given by the f o m l a ( x ,E , E , )

,

sv+~-(O,l,O) (U). 1.0

3

P r o_ of. _ We s h a l l f i r s t assume t h a t b ( x , E , F ) h a s compact s u p p o r t i n x b ( x , ~ , S )? 0 , V x E W K ,

E

E ( O , E ~ ] , 5 E l R n , where K

C

E

U, 1.e.

U i s some compact

set.

I n t h a t c a s e F u b i n i ' s theorem y i e l d s (3.7.19)

(a(x,E,D)

0

b ( x , E , D ) u ) ( x ) = (271

where (3.7.20)

c ( x , E , ~ )= (2n)-nJnei<X'TI'

a ( x , ~<+r?)b , (TI,€

R and, a s usual ; ( F ) Since x

-f

=

F

X+5

u(x), i)(n,~,c)

= F

X+r?

b(x,E,S

b ( x , E , S ) h a s compact s u p p o r t , c ( x , ~ , F )i n ( 3 . 7 . 2 0 ) i s w e l l

( X , E , F ) E S"+' (U) 1,o Now we a r e g o i n g t o p r o v e ( 3 . 7 . 1 8 ) . Using t h e f i n i t e T a y l o r e x p a n s i o n

d e f i n e d and one c h e c k s e a s i l y t h a t c

for a(x,E,F+q),

with

one c a n rewrite

(3.7.19) a s follwos:

where t h e f o l l o w i n g f o r m u l a h a s been u s e d :

3 . 7 . Amplitudes, Adjoints and Products

257

One c h e c k s e a s i l y , u s i n g t h e f a c t t h a t b ( x , E , S ) h a s compact s u p p o r t i n x , that cN(x,E,c)

=

( 2 n P J

ei<xr”

RN (x,E , 5 , n ) b (ri,E , 5 ) drl

mn b e l o n g s t o t h e symbol c l a s s Sv+U-(O,N+1,0) (U), V N 2 0 , and t h a t e n d s t h e 1r o p r o o f of ( 3 . 7 . 1 7 ) , ( 3 . 7 . 1 8 ) i n t h e c a s e , when x + b ( x , E , S ) h a s compact support. Now we d r o p t h e a s s u m p t i o n t h a t x

+

b ( x , E , S ) h a s compact s u p p o r t

a n d assume onLy t h a t b ( x , t , D ) i s p r o p e r l y s u p p o r t e d . A s a consequence of Theorem 3 . 7 . 5 t h e a d j o i n t b * ( x , E , D ) b e l o n g s t o Op S’

1 ,O

(U

X

U)

and i t s

a m p l i t u d e b* ( x , E , ~ h ) as t h e asymptotic expansion

Using t h e a m p l i t u d e b * ( x , c , c ) one c a n rewrite b ( x , ~ , D ) ua s f o l l o w s :

**

b ( x , ~ , D ) u ( x )= ( b ( x , ~ , D ) u(x) = =

( 2 n P kn;.e

i<x-y,S>

*

b ( y ,E,

5 ) *u ( y )dyd5.

Hence,

so t h a t c ( x , Y , E , < ) c(x,E,D)

=

a(x,c,D)

= a ( x , ~ , 5 ) b * ( y , ~ , E i) s* a n a m p l i t u d e o f 0

b(x,~,D).

Applying ( 3 . 7 . 1 3 ) and u s i n g ( 3 . 7 . 2 3 ) one f i n d s t h e f o l l o w i n g a s y m p t o t i c e x p a n s i o n f o r t h e symbol of c ( x , E , D ) :

3. Singular Perturbations on Smooth Manifolds without Bounda y

258

S i n c e e a c h t e r m i n t h e l a s t sum i s a n t i s y m m e t r i c w i t h r e s p e c t t o B,y when B+y > 0 , t h i s sum v a n i s h e s a n d w e a r e l e f t w i t h ( 3 . 7 . 1 8 ) i n t h i s c a s e , a s

I

well. Remark 3 . 7 . 1 .

A l l t h e r e s u l t s e s t a b l i s h e d h e r e f o r t h e S i n g u l a r P e r t u r b a t i o n s i n OP

a r e a l s o v a l i d f o r t h e classes Op s"(U) and Op K"(U)

sy,o(U)

(see D e f i n i t i o n 3 . 3 . 2

and 3 . 6 . 4 ) . 3 . 8 . The S t a t i o n a r y P h a s e , L a p l a c e and S a d d l e P o i n t Methods I n t h i s s e c t i o n we s h a l l i n v e s t i g a t e t h e a s y m p t o t i c b e h a v i o r o f i n t e g r a l s o f t h e form: I(p) =

(3.8.1)

J

f ( x ) ei p g ( x ) d x ,

p

E B + ,p

+

+a,

U where U C Bn i s bounded, f E Ci(U), g E C m ( U ) , I m g ( x ) :0 , We s h a l l s t a r t w i t h t h e c a s e n

=

v

-

x E U.

1, U = (x-,x+) c R.

Proposition 3.8.1.

Let g ' ( x ) # 0 , V x E

c n . T h e n f o r any non-negative i n t e g e r N and K

holds (3.8.2)

p

N

K

(d/dp) I ( p )

-f

0

when

P r o o f . An i n t e r g r a t i o n by p a r t

p

+

-.

yields:

~

A f t e r N p a r t i a l i n t e g r a t i o n s one g e t s : K

N

p

(d/dp) I ( p )

=

J

u

m

I

fN(x)eipg(X)dx, fN E Co(U).

D e f i n i t i o n 3.8.2.

A p o i n t x0 E g'(xo)

=

u i s s a i d t o be s t a t i o n a r y f o r t h e integraZ

(3.8.1)

i f

0.

The s t a t i o n a r y p o i n t x0 i s s a i d t o be r e g u l a r , i f g " ( x o ) # 0 . Theorem 3 . 8 . 3 .

Assume t h a t t h e r e i s only one s t a t i o n a r y p o i n t x0 f o r t h e i n t e g r a l ( 3 . 8 . 1 ) and t h a t it i s r e g u l a r . Then t h e f o l l o w i n g asymptotic formula h o l d s :

3.8. The S t a t m a r y Phase

259

and the coefficients a . ( f , g ) , j t 1, depend only on f ( x O ), g ( x o ) and their I derivatives at xo up to the order 2 j at most. The asymptotic formuZa ( 3 . 8 . 3 ) can be differentiated with respect to p

~

infinitely many times.

P r o o f . W e s h a l l assume t h a t g " ( x > 0. If g"(x < 0 , then f o r t h e 0 0 complex c o n j u g a t e I * ( p ) one h a s : g " ( x ) > 0. L e t $ E C" ( U ) b e such t h a t

0 0 < a / 2 , j , Z 0 f o r Ix-xOl 2 a , where t h e c o n s t a n t a i s 0 s u f f i c i e n t l y s m a l l and w i l l be chosen l a t e r on. R e w r i t i n g I ( p ) a s f o l l o w s :

j,

Z

1 f o r Ix-x

(3.8.5)

I(@)

I

1

=

ipg(x)dx f(x)$(x)e

+ i f ( x ) ( l - $ ( x ) ) ei p g ( x )dx

U

U

and n o t i c i n g t h a t t h e second i n t e g r a l o n r a p i d l y d e c r e a s i n g when p

+ m,

t h e r i g h t hand s i d e of

(3.8.5) is

a s a consequence of P r o p o s i t i o n 3.8.1,

g e t s t h e c o n c l u s i o n t h a t it s u f f i c e s t o show ( 3 . 8 . 3 ) ,

(3.8.4) f o r t h e

integral

I1(p)

=

J f (x)@(x)eipg(x)dx. U

The l a t t e r can b e r e w r i t t e n a s f o l l o w s : (3.8.6)

i

I(p) = eipg(xo)

i p h (x) f(x)$(x)e dx ,

R where (3.8.7)

h(x)

=

g(x)-g(x

0

A s a consequence of

(3.8.8)

y(x)

2

for x

+

x

0'

( 3 . 8 . 7 ) t h e change of v a r i a b l e s

( x - x O ) ( h ( x ) (x-x 1

=

3

= ~ g " ( x o ) ( x - x o )+ O ( / x - x o l ) ,

0

-2

1 ) 2 ,

x

=

x ( y ) , lx-xOl 5 a

i s a diffeomorphism of t h e c o r r e s p o n d i n g i n t e r v a l s , p r o v i d e d t h a t a i s

s u f f i c i e n t l y s m a l l . With such a c h o i c e o f a and u s i n g ( 3 . 8 . 8 1 , one c a n r e w r i t e I ( p ) , as follows: 1 (3.8.9)

I1(p)

=

e ipg

(xo) I R

ipg (xo) = e where (3.8.10)

q(y)

= f (X

2 f (x ( y ) ) $ (x ( y )) x ' ( y )eipy dy = 2

70 q ( y ) e i p Y d y ,

( y ) ) $ ( X ( y ) )x' ( y )+f (X ( - y ) ) V I ( x ( - y ) ) x ' ( - y )

.

one

3. Singular Perturbations on Smooth Manifolds without Bounda y

260 If

-

f(x(y))x'(y)

C b.y' jdo

is the Taylor expansion of f(x(y))x'(y) at y = 0 , then for q(y) given by ( 3 . 8 . 1 0 ) holds: (3.8.11)

q(y)

-

y2J, y + 0,

2 C b jto 2j

since q(y) is even and $(x(y)) : 1 in some small neighbourhood of zero. Notice that as a consequence of ( 3 . 8 . 7 ) ,

( 3 . 8 . 8 ) , the following formulae

hold: (3.8.12)

x'(y) ly=o

=

f. . r % " ( ~ O ) ) fbo , = f(xo)(~g"(xo))

!?-

Furthermore, denote

where

r

Y

is the following contour in

r

Y

I

= { z

1 z

E c

, Z

C':

:i = y+oe , u 2 0 1 ,

and the integration in ( 3 . 8 . 1 4 ) goes from

m

to the pointy

Rewriting P .( p ,y), 3

one can estimate it in the following fashion:

Using the functions ( 3 . 8 . 1 4 ) , integrating 2N+2 times by part in ( 3 . 8 . 1 3 ) and using the estimates ( 3 . 8 . 1 6 ) , one gets the following asymp-

totic formula for Q ( p ) : (3.8.17)

Q(p)

Z

=

15j 52N+2

(-1) j-1

(j-1)

9

1

m

(Y)P.(P,Y) 3

+ O(P

-N- 1

0

The Taylor expansion ( 3 . 8 . 1 1 ) for q(y) and ( 3 . 8 . 1 7 ) yield: (3.8.18)

Q(p) =

-

1 2 ( 2 j ) ! b 2jp2j + l ( P . 0 ) + O ( P - N - l ) OSjSN

I

P

-f

a,

),

P

-f

m.

3.8. The Stationary Phase

26 1

s i n c e q ( y ) :0 f o r y s u f f i c i e n t l y l a r g e . One e v a l u a t e s e a s i l y P 2 j + l ( p , 0 ) u s i n g ( 3 . 8 . 1 5 ) : - l e i ( ~ r / 4 )( 2 j + l )p - j - +

P2j+l(p,0) = - ( f ) T ( j + t ) ( ( 2 j ) ! )

(3.8.19)

so t h a t ( 3 . 8 . 1 8 ) ,

(3.8.19),

(3.8.9) y i e l d (3.8.3).

Using ( 3 . 8 . 1 2 ) and ( 3 . 8 . 1 9 ) w i t h j

=

0 , one g e t s t h e f o r m u l a ( 3 . 8 . 4 ) .

One c h e c k s e a s i l y t h e s t a t e m e n t c o n c e r n i n g t h e c o e f f i c i e n t s a . ( f , g ) , 3 j t 1 , and t h e f a c t t h a t ( 3 . 8 . 3 ) c a n b e d i f f e r e n t i a t e d i n f i n i t e l y many times with respect t o p .

I n d e e d , it s u f f i c e s t o show t h a t t h e a s y m p t o t i c

relation (3.8.20)

-

Z

~ ( p )

i ( ~ / 4 ()2 j + l ) p - j - + , r(j+?)bZje

~

j20

can b e t e r m by t e r m d i f f e r e n t i a t e d . One p r o c e e d s a s f o l l o w s . D i f f e r e n t i a t i n g k t i m e s ( 3 . 8 . 1 3 ) , one f i n d s : m

k (d/dp) Q ( P ) =

1

2 2 k ( i y 1 q(y)eiPY dy.

0 k A f t e r w a r d s , u s i n g t h e p r e v i o u s a r g u m e n t , one g e t s f o r ( d / d p ) Q ( p ) a n

a s y m p t o t i c e x p a n s i o n , which c o i n c i d e s w i t h t h e f o r m a l t e r m by t e r m d e r i v a t i v e o f o r d e r k w i t h r e s p e c t t o p o f t h e r i g h t hand s i d e i n ( 3 . 8 . 2 0 ) . The l a s t c l a i m f o l l o w s immediately i f one n o t i c e s t h a t 2 k (iy 1 q(y)

-

2i

k

b 2 j y2 ( J + k ) , f o r y + 0 ,

C

j20

k a n d , t h u s , f o r ( d / d p ) Q ( p ) , h o l d s t h e same k i n d o f a s y m p t o t i c e x p a n s i o n as (3.8.20) with c o e f f i c i e n t s c instead of b such t h a t c 21 21 21 k I = i b 2 ( j - k ) r 1 2 k. 21

=

0, j < k

and c

Now we d r o p t h e a s s u m p t i o n t h a t n = 1 and s h a l l c o n s i d e r t h e m u l t i d i m e n s i o n a l i n t e g r a l s ( 3 . 8 . 1 ) f o r any d i m e n s i o n . We s t a r t w i t h a s t a t e m e n t a n a l o g o u s t o P r o p o s i t i o n 3 . 8 . 1

in the

multidimensional case. Proposition 3.8.4.

L e t f E C:(U), V x E

E c r n ( U ) ,I m g ( x ) : 0 , V x E

U. Then for t h e i n t e g r a l

(3.8.21) Proof. ___

g

p N ( d / d p ) K I ( p )+ 0 Denote by

L

~ ( p given ) by

when

p

-f

a,

and assume t h a t Vg(x) # 0 , ( 3 . 8 . 1 ) holds:

V N 2 0, V K Z 0.

(x,a) t h e d i f f e r e n t i a l operator 9

3. Singular Perturbations on Smooth Manifolds without Boundary

262

and by L * ( x , a ) i t s f o r m a l a d j o i n t : 9

Hence, f o r any i n t e g e r K 2 0 , N 2 0 one f i n d s : K

( d / d p ) I(P) = ( i p ) - '

(3.8.24)

I U

K

f(x1(ig(x))

L ( x , a ) ei ? g ( x )dx g

K

( i p ) -1

K

eipg(x) L*(x,a) ( f ( x ) i g ( x ) dx = U g N+ 1 K K - (ip)-(N+l)i .ipg(x) ( L z ( x , a ) ) ( f ( x ) i g ( x ) )dx. =

U A s a consequence of

(3.8.24)

one g e t s ( 3 . 8 . 2 1 ) .

Definition 3.8.5.

A p o i n t xo E

u i . s~a i d t o be s t a t i o n a r y f o r t h e i n t e g r a l

Vg(x ) = 0. The s t a t i o n a q p o i n t xo

0

of t h e second d e r i v a t i v e s 2

2

D g(xo) =

d e t D g ( x ) # 0.

I

0

-

(3.8.1) i f

i s s a i d t o be r e g u l a r i f t h e m a t r i x is non-singular: K

I1

The i n v e s t i g a t i o n o f t h e a s y m p t o t i c b e h a v i o u r of t h e i n t e g r a l ( 3 . 8 . 1 ) for p +

c a n b e reduced t o t h e one d i m e n s i o n a l c a s e u s i n g t h e f o l l o w i n g

M o r s e ' s Lemma.

L e t g ( x ) , x E u b e r e a l valued and assume t h a t g ( x ) i s cm in some neighbourhood o f i t s r e g u l a r s t a t i o n a r y p o i n t xo E U. Then t h e r e e x i s t neighbourhoods Ox , Oo of t h e p o i n t s x

= xo

and y

=

0, r e s p e c t i v e Z 2 , and

0

a cm-diffeomorphism

h :

Ox + O o such t h a t 0

where u j ,

1 5 j 5 n,

are t h e eigenvalues of t h e m a t r i x

2

D g(x ) .

0

Proof o f Morse's Lemma Without r e s t r i c t i o n o f g e n e r a l i t y one c a n assume t h a t xo = 0, 2 D g ( 0 ) = d i a g ( v l , . . . , p n ) , s i n c e an o r t h o g o n a l t r a n s f o r m a t i o n r e d u c e s t h e X

g e n e r a l c a s e t o t h i s one. L e t 0

0

b e convex and s u f f i c i e n t l y s m a l l ,

3.8. The Stationay Phase

00

3 0. Since Vg(0)

=

0 , one h a s :

=

(

1/2 ) <s ( x )X I x> ,

where t h e m a t r i x S ( x )

=

I / S k j ( x ) I15k,jsn

263

i s d e f i n e d by t h e f o r m u l a e :

S ( x ) b e i n g , o f c o u r s e , symmetric, S ( O 1 = d i a g ( p l , ..., u n )

00’

function in

I d e n t i f y i n g t h e symmetric nxn m a t r i c e s w i t h R c o n s i d e r S ( x ) a s a Cm-map from

0,

m

matrix-

and C

n ( n + 1 ) / 2 one can

i n t o R n ( n + 1 ) / 2 . W e s e e k a smooth ( C m )

upper t r i a n g u l a r m a t r i x - f u n c t i o n Q ( X )

:

0O + f l R n ( n + 1 ) / 2 s u c h t h a t

where Q ( x ) * i s t h e c o n j u g a t e of Q ( x - ) . Denote F(Q,X)

=

Q*S(O)Q-S(X)

where Q i s any u p p e r t r i a n g u l a r m a t r i x and x E m

00 ’

One c a n c o n s i d e r F ( Q , x ) a s a C -map from R

n(n+1)/2

X

0,

into the

s p a c e o f a l l symmetric m a t r i c e s . One h a s F(Id,O)

=

and, given t h a t a l l

0

u,, I

1 5 j 6 n a r e d i f f e r e n t from z e r o , one f i n d s

easily that

QF ( Q r x ) / Q = I d , x = O = where J i s t h e m a t r i x which h a s t h e e n t r i e s : kj a k j = 1 , alm = 0 , V ( l , m )

# (k,j).

O b v i o u s l y , t h e m a t r i c e s J k j + J j kform a b a s i s i n t h e s p a c e o f a l l symmetric nxn m a t r i c e s , so t h a t

3 . Singular Perturbations on Smooth Manifolds without Boundary

264

t h e I m p l i c i t F u n c t i o n Theorem c a n b e a p p l i e d , which g u a r a n t e e s t h e e x i s tence of a (well-defined) C

m

upper t r i a n g u l a r m a t r i x Q ( x ) i n a

oo

s ~ i f i c i e n t l ys m a l l neighbourhood

of t h e o r i g i n , t h a t s a t i s f i e s t h e

conditions :

Obviously t h e map y

h(x)

=

def -

Q(x)x

i s a Cm diffeomorphism of in

0,

i n Rn o n t o some neighbourhood of t h e o r i g i n

which r e d u c e s g ( x ) t o t h e form

pln

Y’

(g

h)(y) = g ( O ) + $

0

2 C U.Y. I<j
’

Theholomorphversioiiofthislemmaistrue,too.Theproofis

Theorem 3.8

l e f t t o t h e reader.

&.

L e t u c R~ be bounded and l e t f t c ~ ( u ) ,g E cm(G), t h e f u n c t i o n g(x), x E

u, being

r e a l valued. Assume t h a t g ( x ) has p r e c i s e l y one

s t a t i o n a r y p o i n t xo in

5 and

t h a t xo is r e g u t a r .

Then for each i n t e g e r K b 1 one has: (3.8.26)

f ( x ) ei P 9 ( x ) d x

I(p) =

~

U - j-n/2

= ei p g ( x ~ ) c a.(f,g)p 0; j ‘K

’

where a . ( f , g ) 3

operators w i t h

m > K

=

+QK(p),

~ . ( x , a ) f ( x ) ~~ ~, (=x ~, a~being ), some l i n e a r d i f f e r e n t i a l 3 2j a t most, and whsre

m’

c - c o e f f i c i e n t s of order

o being some i n t e g e r s and

c K ( g ) > 0 being some constants,which may depend

on K and 9. The asymptotic f o m m * ~ a

can be d i f f e r e n 6 i a t e d wi+h m s p e c t t o

p

i n f i n i t e l y many t i m e s .

3.8. The Stationary Phase Proof. -

265

I t s u f f i c e s t o c o n s i d e r t h e c a s e when supp f b e l o n g s t o some s m a l l

0

neighbourhood

o f t h e p o i n t x o , s i n c e t h e p a r t i t i o n o f u n i t y and

XQ

P r o p o s i t i o n 3.8.4

can be used i n o r d e r t o reduce t h e g e n e r a l s i t u a t i o n

t o t h i s case. Assuming t h a t

0

i s so s m a l l t h a t Morse's Lemma c a n b e

XO

a p p l i e d , w e can rewrite I ( p ) a s follows:

Using r e p e a t e d l y t h e one d i m e n s i o n a l s t a t i o n a r y phase method n t i m e s , one g e t s (3.8.27) f o r I ( p ) and t h e c o r r e s p o n d i n g f o r m u l a e f o r i t s derivatives

.

I

W e are g o i n g t o g e t e x p l i c i t

formulae f o r t h e c o e f f i c i e n t s a . ( f , g ) i n 1

(3.8.27). Denote by Q

(3) the differential operator 4 (xo)

2 -1 . 2 i s t h e i n v e r s e m a t r i x f o r t h e m a t r i x D g ( x ) of t h e where D g ( x o ) 0 second d e r i v a t i v e s of g ( x ) a t x o , and i n t r o d u c e

2

u 9 ( x0 1

where

i s t h e s i g n a t u r e of D g ( x

).

0

Theorem 3.8.7.

Under t h e assumptions o f Theorem 3.8.6. t h e foZZowing asymptotic formula holds for

where

Q

g i v e n by (3.8.1):

I(p)

9 (xo)

(a),

h (x,xo), y 9

a r e given r e s p e c t i v e z y by (3.8.28), (3.8.29),

(3.8.30), where (3.8.32)

\

and where f o r

=

p

n/2 + k

-

[2k/3]

>= 1 holds:

w i t h some i n t e g e r

M k ' O

3. Singular Perturbations on Smooth Manifolds without Boundary

266 Proof. -

Denote iph (x,xo) v(p,x0,x) = f(x)e g

and

For a symmetric regular matrix A one has the following formula

where

- (2T)-n/21det Al-f e(in/4) sign A 'n,A

-

sign A being the signature of A. Using Parceval's identity and the last formula, one can rewrite I ( p ) as follows: 1

Expanding the exponential under the integral sign into Taylor's series, one gets:

and that is precisely (3.8.31). Since h ( x , x ~ )has at x = x0 a zero of order 3 at least, the degree 4

of the polynomial

is [2j/?] at most. Writing I ( p ) in the form

using the fact that the coefficients of an asymptotic expansion are uniquely d e f i n e d and that the degree of q . ( p ) is at most [2j/3 , one gets the 7 p-' with p 5 n/Z+k-[Z(k-1)/31

conclusion that all the terms containing have to be included in the sum

c

-1

L-

Sj'P),

O<j
(3.8.33).

I

3.8. The Stationary Phase

267

Remark 3 . 8 .-8.. Taking o n l y t h e f i r s t t e r m i n t h e e x p a n s i o n ( 3 . 8 . 3 1 ) , one g e t s t h e f o l l o w i n g a s y m p t o t i c formula f o r I ( p )

when p

+ m.

Remark 3 . 8 . 9 . I f g ( x ) h a s a f i n i t e number o f c r i t i c a l p o i n t s x l , ...,x regular, then t h e localization principle

cl

which all a r e

( p a r t i t i o n o f u n i t y and

Proposition 3.8.4)

l e a d s t o t h e following asymptotic formula f o r I ( p )

(3.8.35)

X

I(p) =

I . (p) + O ( P - ~ ) , for p

:

+ m,

l<j
0'

I

I n a p p l i c a t i o n s f and g may a l s o depend ( c o n t i n u o u s l y ) o f some m

p a r a m e t e r w b e l o n g i n g t o a compact C -manifold w i t h o u t boundary ( f o r

R

instance t h e u n i t sphere

i n Wn ) ; b e s i d e s f may a l s o h a v e some weak

(polynomial l i k e ) dependence upon p E IR+. More p r e c i s e l y , w e consi.der now i n t e g r a l s o f t h e form i p g ( x ,w ) dx f(x,w,p)e U where, a s p r e v i o u s l y , U i s bounded, U C IRn,

(3.8.36)

I(p,w)

1

=

o E M , M b e i n g a compact

m

riemannian C -manifold of f i n i t e dimension 9, and p E IR+, a s p r e v i o u s l y . The f u n c t i o n f , c a l l e d a l s o t h e a m p l i t u d e , and t h e f u n c t i o n g , c a l . l e d a l s o t h e p h a s e , are supposed t o s a t i s f y t h e f o l l o w i n g c o n d i t i o n s . 1'

f E Cm(U

Z 0 For e a c h

X

M X R+), g

E M

(wo,po)

x

E Cm(U

x M),

a n d , moreover, g i s r e a l v a l u e d .

R+ , h o l d s : supp f ( x , w 0 , p 0 )

i s some compact which d o e s n o t depend on

C K C U,

where K

(wo,po).

3" For e a c h t r i p l e of m u l t i - i n d i c e s a , B , y h o l d s :

where m i s some g i v e n number and t h e c o n s t a n t s C

Cr,B,Y

depend o n l y on t h e i r

subscripts. 4 " For each g i v e n

c r i t i c a l point x

iz

=

E M t h e f u n c t i o n g(x,w)

x (w) E U, 0

:

U

i . e . t h e equation

-t

R h a s p r e c i s e l y one

3 . Singular Perturbations on Smooth Manifolds without Boundary

268 (3.8.38)

Vxg(x,o)

=

0

h a s p r e c i s e l y one z e r o x = x o ( w ) ,

v

E

w

M.

2

5" The e i g e n v a l u e s p I, (a), 1 6 j 6 n , of t h e m a t r i x Dx g ( xo ( w ) ,w) the inequalities:

(3.8.39)

inf

0, V j = 1 , 2 ,..., n

( p . ( w ) ( 2 po

wEM

satisfy

'

Introduce the d i f f e r e n t i a l operator

2 -1 2 t h e i n v e r s e m a t r i x of D g ( x o ( w ), w ) , w i t h D g ( x o ( w ),w)

and d e n o t e , a s

p r e v i o u s l y by y = y ( w ) t h e c o n s t a n t ( 3 . 8 . 3 0 ) w i t h xo = xo(w) t h e o n l y n n c i r t i c a l p o i n t of g i n U. Denote by h (x,w) t h e f u n c t i o n ( 3 . 8 . 2 9 ) where 9 xo = x ( 0 ) i s t h e c r i t i c a l p o i n t of g i n U. 0

Theroem 3.8.10.

Assume t h e c o n d i t i o n s l o k 2 1

holds:

where

%

= n/2tk-[Zk/3]-m

I%(P,~)

I

t o be f u l f i l l e d . Then for each i n t e g e r

5 O

and

5 Ck, V

( 0 , ~E)

M

x

IR,

w i t h some c o n s t a n t Ck which depends only on i t s s u b s c r i p t . The a s y m p t o t i c expunsion

D + -

can be d i f f e r e n t i a t e d i n f i n i t e l y many t i m e s w i t h r e s p e c t t o and holds uniformly w i t h r e s p e c t t o

E

w

(w,p)

E M

x

M.

P r o_ o f . A s a consequence of t h e c o n d i t i o n s 4 " , 5 O , t h e c r i t i c a l p o i n t _ x ( w ) of g i s r e g u l a r , V w E M a n d , moreover, t h e s i g n a t u r e a ( x ( w ) ,w) O 2 ¶ O of D g ( x ( w ) , w ) x o

d o e s n o t depend on w

E

M.

The i m p l i c i t f u n c t i o n s theorem

R+

3.8. The Stationary Phase m

i m p l i e s t h a t xo(w) E C ( M ) .

r(

,

)

=

{w

I

w

w ) t h e length of t h e P P' and w and 6 > 0 t o be chosen l a t e r o n . L e t

P c M b e one o f s u c h domains and l e t w l ,

Ro. I f x ( w ) f

K, V w E Q

0

2O,

distance

E M , r ( w , w ) < 6) w i t h r ( w

geodesic j o i n i n g w

0

m

M b e i n g equipped w i t h a C -riemannian

and compact, one c a n c o v e r M by a f i n i t e number o f domains

R(wp,6)

R

269

0

...,wcl

b e l o c a l c o o r d i n a t e s on

w i t h K t h e compact d e f i n e d i n t h e c o n d i t i o n

t h e n f o r any i n t e g e r k t 0 and any p a i r o f m u l t i - i n d i c e s one h a s :

I D U~ D ~PI ( ~ , L I ) I

(3.8.43)

5

ck,a,B

P

i n f a c t , using again the operator

-k

,v

L (w,x,a ¶

t h e f a c t t h a t V g(x,w) # 0 , V (x,w) E

0

)

x

R+.

d e f i n e d by ( 3 . 8 . 2 2 ) and

c x co one

same p r o c e d u r e as i n P r o p o s i t i o n 3 . 8 . 4 . Now, l e t R

E no

(W,P)

g e t s (3.8.43) u s i n g t h e

and ( 3 . 8 . 3 7 ) .

b e such a domain t h a t t h e s e t Uo

=

x (R ) h a s a non-

0

0

empty i n t e r s e c t i o n w i t h K i n t h e c o n d i t i o n 2 " . Without r e s t r i c t i o n o f g e n e r a l i t y one c a n assume t h a t wo = 0 , x 0 ( w 0 ) = 0 , g ( x0 ( w 0 ) , w o ) = 0 . I f 6 i n t h e d e f i n i t i o n o f Ro i s s u f f i c i e n t l y small, t h e n one c a n a g a i n u s e Morse's Lemma i n o r d e r t o r e d u c e by an a p p r o p r i a t e d i f f e o m o r p h i s m x = h(y,w) the function g(x,w) t o t h e q u a d r a t i c function:

with

u1

Uo and T , = * I . 3 we u s e t h e p a r t i t i o n of u n i t y ~i ( X I + $ ( x )

2

-1

-

The i n t e g r a l I ( p , w ) 1

where f l ( y , w , p )

=

2

1 in

fi,

0 for 1 E 1 f o r x E U with U2 c U o , U 2 c U 1 . D e n o t e b y I . ( p , w ) , j = 1 , 2 x !$ 2 1 t h e c o r r e s p o n d i n g c o n t r i b u t i o n s i n t o I (p , w ) , so t h a t i (p,W) = I 1 ( p ,W) + I 2( P , w ) NOW

VJ

can be r e w r i t t e n as follows:

f ( h ( y , w ), w , p ) ($1 o h ) ( y , w ) . d e t h ' y ( y , w ) .

The f u n c t i o n f l h a s compact s u p p o r t i n y E R n , and s a t i s f i e s ( 3 . 8 . 3 7 ) i n h((U1

x

Go))

x

Ro

x

R + . Applying t h e s t a t i o n a r y p h a s e method

r e p e a t e d l y i n v a r i a b l e s y1,y2, . . . , y n ,

one g e t s f o r I 1 ( p , w )

asymptotic

e x p a n s i o n s o f t h e form ( 3 . 8 . 3 1 ) and t h e ( u n i f o r m i n w , p ) e s t i m a t e f o r

the remainder of the form (3.8.33) w i t h

\

= n/2tk-[2k/3]-m,

since t h e

amplitude f s a t i s f i e s (3.8.37). The same argument a s p r e v i o u s l y i n t h e p r o o f of Theorem 3.8.3 y i e l d s t h e c o r r e s p o n d i n g a s y m p t o t i c formulae f o r t h e d e r i v a t i v e s o f I ( p , w ) .

3 . Singular Perturbations on Smooth Manifolds without Boundaty

270

C o r o l l a r y 3.8.11.

Assume t h a t f,g s a t i s f y t h e c o n d i t i o n s o f Theorem 3.8.10 and assume, furthermore, t h a t f belongs t o some bounded s e t B i n ci(U

x

M

x

R + ) . Then

is uniform w i t h r e s p e c t t o f E

t h e e s t i m a t e of t h e remainder in (3.8.42)

B.

To c o n c l u d e t h i s p a r a g r a p h w e s h a l l c o n s i d e r t h e a s y m p t o t i c b e h a v i o u r

of I ( p ) when p i s a complex p a r a m e t e r , p

+

-.

Theorem 3.8.12.

The assumptions o f Theorem 3.8.6 (3.8.44)

being f u l f i l l e d , assume a d d i t i o n a l l y t h a t

g ( x o ) > g ( x ) , V x E v\{xo}.

Then t h e asymptotic expansions ( 3 . 8 . 2 6 1 ,

and t h e corresponding

(3.8.31)

e s t i m a t e s f o r t h e remainders are s t i l l v a l i d when such t h a t I m p

5 0,

/p

2 1,

p

is a complez parameter,

uniformly w i t h r e s p e c t t o a r g

p.

If (3.8.45)

g(xo) < g(x)

v

x

E v\{xol,

then t h e s t a t e m e n t s ab ve a r e t r u e f o r I m p L

o,

Ip

1

2 1.

Proof. W e s h a l l assume t h a t ( 3 . 8 . 4 4 ) h o l d s , t h e case when one h a s ( 3 . 8 . 4 5 ) __

c a n b e t r e a t e d i n t h e c o m p l e t e l y analoguous way. i t s u f f i c e s t o p r o v e t h i s theorem f o r n

Evidently,

=

1 , s i n c e Morse’s

Lemma r e d u c e s t h e m u l t i d i m e n s i o n a l case t o n = 1. I n one d i m e n s i o n a l

s i t u a t i o n i t i s enough t o show t h a t ( 3 . 8 . 1 6 ) h o l d s f o r complex p s u c h t h a t I m p S 0 . For d o i n g t h a t one i n t e g r a t e s i n ( 3 . 8 . 1 4 ) a l o n g t h e r a y

r

Y

= {z

I

z E

c,

i0 z=y+oe , a 2 0, 8

where y = a r g p . Then f o r z E

r

Y

one f i n d s

and t h a t y i e l d s ( 3 . 8 . 1 6 ) .

I

Remark 3.8.13. If

( 3 . 8 . 4 4 ) h o l d s , t h e n one h a s f o r p +

m

=

n/4-y/2},

3.8. The Stationary Phase

27 1

uniformly with r e s p e c t t o a r g p . The p r o c e d u r e above f o r computing a s y m p t o t i c s o f t h e form ( 3 . 8 . 4 6 ) i s c a l l e d t h e L a p l a c e method. I f f , g a r e a n a l y t i c and U i s a m a n i f o l d i n Cn of r e a l dimension n ,

t h e n t h e p r o c e d u r e above h a s t h e name: t h e mountain p a s s method ( o r t h e s a d d l e p o i n t method). W e s h a l l s t a t e t h e main r e s u l t i n t h i s case which w i l l b e u s e d l a t e r on for d e s c r i b i n g a s y m p t o t i c b e h a v i o u r of s i n g u l a r l y p e r t u r b e d Poisson type o p e r a t o r s .

E C n , b e holomorphic a t t h e p o i n t z o , which i s supposed t o

Let g ( z ) , z

b e a r e g u l a r c r i t i c a l p o i n t f o r g ( z ) . L e t U be a s u f f i c i e n t l y s m a l l neighbourhood o f z o , U

=

{z

1

Iz-zo/

<

6) and l e t

G~ =

u n

{Re(ig(z)-ig(zo))>

-ul,

LG

u n

{Re(ig(z)-ig(z ) )

-GI,

=

0

=

where u > 0 i s s u f f i c i e n t l y s m a l l . The r e l a t i v e homology g r o u p H ( Gu , L u )

i s isomorphic t o Z . Denote

by y t h e g e n e r a t i n g c y c l e o f t h i s t r o u p . I f g ( z ) =

C 22 t h e n one l Z j 6 n I'

can d e f i n e Y a s f o l l o w s

(3.8.47)

y

=

=...=

{yl

yn

=

01 n u

Theorem 3.8.14.

L e t f ( z ) , g ( z ) be holomorphic a t z o E y, z o being a reguZar s t a t i o n a r y p o i n t f o r g ( z ) and l e t J f ( z ) eiPg(')dz, Y y being t h e generating c y c l e d e f i n e d above. ~ ( p )=

Then one has f o r (3.8.48)

I(p)

p

-

-f

-:

p -n/2

eipg(zo)

I: a . p - 1 . jto

'

The main term i n t h e asymptotic expansion ( 3 . 8 . 4 8 ) has t h e form: (3.8.49)

I(p)

-

f ( z o ) ( 2 n / p ) " l 2 ( d e t DZq(zO))-' 2

rJhere ( d e t D2g(z z

o

))

eipg(zo),

' I 2i s d e f i n e d by t h e o r i e n t a t i o n o f

y.

3 . Singular Perturbations on Smooth Manifolds without Bounda y

212 Proof. -

I n a s u f f i c i e n t l y s m a l l neighbourhood of zo one c a n u s e a holomor-

p h i c d i f f e o m o r p h i s m z = h ( w ) (holomorph v e r s i o n o f Morse's Lemma) which r e d u c e s g ( z ) t o a sum of squares: (g

0

h ) (w) = g ( z o ) +

b e s i d e s , one h a s 2

h'(0)

I: 1sj
=

2 w. J'

-1

(det DZg(zO))'.

N

L e t y b e t h e image of y u n d e r t h e holomorphic diffeomorphism h . The Cauchy theorem y i e l d s :

/ (f /f ( 2 ) e i p g ( z ) d z = N

Y

o

i p (g

h) (w)h'(w)e

0

h ) (w)dw

Y

/ (f~h)(w)h'(w)eip(goh)(W)d+ w / ( f ~ h ) ( w ) h ' ( W ) . i p ( g o h ) ( W ) dw, YO Y1 where yo i s t h e c y c l e d e f i n e d by ( 3 . 8 . 4 7 ) and R e i ( g , h ) (w) = -u f o r w E yl, =

so t h a t t h e i n t e g r a l o v e r y1 h a s t h e o r d e r

when p

-t m

and i s , t h u s , e x p o n e n t i a l l y s m a l l compared t o t h e i n t e g r a l o v e r

y o . Now t o t h e i n t e g r a l o v e r yo t h e L a p l a c e method c a n b e a p p l i e d , which l e a d s t o t h e asymptotic formula ( 3 . 8 . 4 9 ) .

I

Example 3.8.15. Let h

E (O,hO] be

(3.8.50)

a s m a l l parameter. Consider t h e following i n t e g r a l

Ph ( x ,t ) = (Znh)-'J

where ( x , t ) E R

2

e

3 ih-' ( ( x - t ) <+ ( t / 6 ) 5 )

d<

IR

.

The d i s t r i b u t i o n ( 3 . 8 . 5 0 )

i s t h e s o l u t i o n of t h e Cauchy problem:

l i m P (x,t) = 6(x),

t+o

where 6 ( x ) i s D i r a c ' s m a s s a t t h e o r i g i n . We a r e g o i n g t o f i n d t h e a s y m p t o t i c b e h a v i o u r o f P h ( x , t ) when h Introduce

-f

0.

3.8. The Stationary Phase

Assume f i r s t t h a t a = x / t > 1. Then 5

+

213

g ( x , t ; C ) d o e s n o t have s t a t i o n a r y

p o i n t s on R and t h e p a r t i a l i n t e g r a t i o n i n t3.8.50)

yields:

u n i f o r m l y w i t h r e s p e c t t o ( x , t ) i n any compact s e t K b e l o n g i n g t o t h e halfplane x / t

> 1.

let a

Now,

=

x / t < 1. Then

5

+

g ( x , t ; c ) h a s two s t a t i o n a r y p o i n t s :

which are r e g u l a r , s i n c e a c: 1.

,f

L e t f ( 5 ) E C:(lR)

:

R

+

R b e such t h a t f ( 5 ) : 1 f o r

5 E [<-(a)-1,5+(a)+l]. Then a g a i n t h e p a r t i a l i n t e g r a t i o n y i e l d s : 0 0 3 i h - l ( ( x - t ) S - ( t / 6 ) < )d5 = O(hN ) , ( 2 a h ) - l / (1-f ( 5 ) ) e R V N 2 0,

v

t # 0,

so t h a t one h a s i n t h i s c a s e :

(3.8.51)

P h ( x , t ) = (2nh)

-1

3 ih-' ( ( x - t ) 5+ ( t / 6 ) 5 ) d S

J f(5)e R

Applying ( 3 . 8 . 3 1 ) t o t h e i n t e g r a l o n

+

O(hm).

t h e r i g h t hand s i d e o f

one g e t s t h e f u l l a s y m p t o t i c e x p a n s i o n f o r P

h

( x , t ) when h

+

(3.8.51)

0.

W e w r i t e here only t h e f i r s t t e r m of t h i s asymptotics:

Ph(x,t) = 2 ) (ah)-'(tl-'(l-a)

(3.8.52)

-'cos

(n/4-Z3j2 ( 3 h ) - l t ( l - a ) 3 / 2 ) ( 1 + 0 ( h )) h + O

where a = x / t < 1; thus one h a s : s i n g s u p p Ph = { x S t j . One c a n a l s o e x p r e s s P ( x , t ) i n t e r m s of A i r y ' s f u n c t i o n ( s e e , f o r h 1 ) . Namely, one f i n d s e a s i l y :

i n s t a n c e [Was, 1

P ( x , t ) = c h-5/6t-1/3

h

0

( t - x ) ' l 6 A i ( c h-2/3t-1/3 ( t - x ) ) 1

,

2-5/3rr - 1/ 2 , c = - 61/3 = 0 1 Using t h e s a d d l e p o i n t method ( o r t h e a s y m p t o t i c e x p a n s i o n f o r A i r y ' s

where c

f u n c t i o n ) one c a n show, t h a t f o r a = x / t > 1 t h e f u n c t i o n Ph d e c r e a s e s a s e-h-ly(xft) NOW,

w i t h some y ( x , t ) > 0 , when h + 0 .

assume t h a t s t i l l a

=

x / t < 1 and t h a t

,

3 . Singular Perturbations on Smooth Manifolds without Bounda y

214

(1-a)

-

,

h'

f o r h + 0,

where u < 2 / 3 .

5

A f t e r t h e change of v a r i a b l e

L

= ( l - a ) 2 ~t ,h e i n t e g r a l

(3.8.50) takes

t h e form: P ( x , t ) = (Zah)-'(l-a)

h

f 1 ei t h - ' ( I - a ) 3 / 2 R

(-q+

( 1 / 6 ) n 3 ) dll

where p =

h-1(l-a)3/2

30/2-1 O(h

=

for h

m

-f

+

0.

Hence, one c a n a p p l y a g a i n t h e s t a t i o n a r y p h a s e method f o r computing t h e asymptotics f o r P ( x , t ) with p as a l a r g e parameter. h T h e r e f o r e , t h e a s y m p t o t i c s above are v a l i d f o r P ( x , t ) i n t h e r e g i o n h x / t < 1-Ch' f o r any c o n s t a n t s C > 0 , o E ( 0 , 2 / 3 ) . N o t i c e t h a t i n t h e r e g i o n 1-a

=

1 +u

O(h

)

w i t h any g i v e n

0

> 0 t h e main

t e r m i n t h e a s y m p t o t i c e x p a n s i o n f o r P ( x , t ) h a s t h e form: h Ph(X,t)

-c

t

-1/3h-2/3

where

m

@ - 2 / 3 cos

= 3-2/321/3n-1

e de

0 Example 3.8.16. Again l e t h

For t

E

IR,

E

(0,h

x E

0

1

\=

b e a s m a l l parameter. Consider t h e i n t e g r a l :

{x

1

x

E IR,

x/h

E

Z}

t h e function E ( x , t ) is the h

s o l u t i o n o f t h e Cauchy problem:

LE

at

h

( x , t ) + (2h)

l i m E (x,t)

=

-1

(Eh(x+h,t)-E(x-h,t) = 0 ,

6h(x),

t-to where 6 (0) = h-', 6h(x) = 0 f o r x h The p h a s e f u n c t i o n

h a s two s t a t i o n a r y p o i n t s

E %..{0}

3.8. The Stationary Phase

+

(3.8.54) if

Eg(a) =

a

arccos a ,

_+

275

= x/t

la1 C I , moreover, f o r la1 < 1 t h e s t a t i o n a r y p o i n t s ( 3 . 8 . 5 4 ) are r e g u l a r . m

Hence, f o r la1 > 1 , i . e .

1x1 > I t / , on: has: E ( x , t ) = O ( h ) , h h u n i f o r m l y w i t h r e s p e c t t o ( x , t ) i n any compact b e l o n g i n g t o t h e s e t IIxj

'

-f

0

It\}.

Assume now t h a t 1x1 < Itl. L e t f ( < ) E C m ( R ) be s u c h t h a t 0

supp f c ( - 2 , 2 ) ,

E

f E 1 for 5

[-3/2,3/2].

Then one h a s :

One c a n c o n s i d e r t h e i n t e g r a l

2 m a s a n e x t e n s i o n o f E ( x , t ) t o R up t o a n e r r o r O(h ) , f o r h + 0 . h Applying t h e s t a t i o n a r y p h a s e method ( f o r m u l a ( 3 . 8 . 3 1 ) ) t o t h e i n t e g r a l ( 3 . 8 . 5 6 ) one g e t s t h e f u l l a s u m p t o t i c e x p a n s i o n f o r Eh ( x , t ) when h + 0 . Again w e w r i t e h e r e o n l y t h e f i r s t t e r m i n t h i s e x p a n s i o n : (3.8.57)

Eh ( x , t )

where la1 < 1 , a

=

-1 2 -d 2*(nh) ' / t / - f ( l - a ) cos(h-'t(aarccosa-Jl-a

x/t;

t h u s , s i n g s u p p Eh =

Now assume t h a t (1-a ) -ha

with

0

{\XI

2

)+n/4),

5 Itl}.

E (2/5,2/3).

Then ( 3 . 8 . 5 3 ) c a n b e r e w r i t t e n a s f o l l o w s :

where $ ( S )

= 5-(1/6)5

3

-sin

5

5

= O(5 )

A f t e r t h e change o f v a r i a b l e

Since p

=

( 1 - a ) 3/2h-1

each i n t e g r a l

+ m

(h

+

5

=

when

5

(1-a)

t n,

+ 0.

( 3 . 8 . 5 8 ) becomes

0 ) i s a l a r g e p a r a m e t e r , one c a n a p p l y t o

3. Singular Perturbations on Smooth Manifolds without Boundary

216

t h e S t a t i o n a r y Phase Method. Hence, w i t h 1-a = Ch',

2/5 < u < 2/3, one f i n d s :

and moreover,

T h e r e f o r e , t h e main t e r m i n t h e a s y m p t o t i c b e h a v i o u r of E ( x , t ) i n t h e h c a s e c o n s i d e r e d i s g i v e n by t h e f o r m u l a :

-

The a p p l i c a t i o n o f t h e S t a t i o n a r y Phase Method t o t h e i n t e g r a l o n 3/2 h-l + the large ( 3 . 8 . 5 9 ) w i t h P = (1-a)

t h e r i g h t hand s i d e o f

parameter l e a d s t o t h e following asymptotic formula for E ( x , t ) : h

where c . > 0 , j = 1 , 2 are any g i v e n c o n s t a n t s 7 y = m i n I 1 - 3 ~ / 2 , 5u/2-11 > 0.

< c 2 , 2/5 < a < 2/3 and

C

Example 3.8 . 1 7 . Consider t h e i n t e g r a l

0 i s an i n t e g e r . where h E ( 0 , h ] i s a s m a l l p a r a m e t e r and p 0 The f u n c t i o n ( 3 . 8 . 6 1 ) i s t h e s o l u t i o n of t h e Cauchy problem:

lim

t++o

G

(x,t) = 6(x).

Using t h e change of v a r i a b l e 5

where

1 -

prh +

(x/t-1)

2p- 1

,€ one f i n d s :

3.8. The Stationary Phase

277

2p (3.8.63)

a

= t(x/t-l)

2p- 1

For p = 1 u s i n g Cauchy‘s theorem one f i n d s e a s i l y : -

( 2 n t h ) - f e - ( x - t ) 2/ ( 2 t h )

-

Gl,h(~,t)

W e a r e g o i n g t o compute t h e f i r s t t e r m i n t h e a s y m p t o t i c e x p a n s i o n of G

Pth

( x , t ) when h

0 , u s i n g t h e S a d d l e P o i n t Method.

-f

The polynomial

g (5) P

=

i ~ - ( 2 p ) - ~ 5 ’ P , gp(5) :

c

1

+

c

1

has t w o regular stationary points n i_ ni -__ 2 (2p-1) Z(Zp-1) (3.8.64) = e , C 1 = -e

c0

w i t h t h e same imaginary p a r t , I m 5 .

I

g

P

(Lo)

= g

P

(ill

=

=

sin(;

~i 2(2p-1)

)

> 0 , where

i ( 1 - i ) e 2 ( 2 p - 1 ) , R e g (5.)< 0 , 2P P I

and f o r any o t h e r s t a t i o n a r y p o i n t 5

one h a s :

I t i s e a s i l y seen t h a t

max R e g ( 5 ) = R e g ( < , ) , P I

j = 0,l.

<EY

I n d e e d , t h e s t a t i o n a r y p o i n t s of t h e f u n c t i o n R e g ( 5 ) : y + R a r e t h e 2p- 1 P p o i n t s , where R e 5 = 0, i . e . the points C j , j = 0 , l i n (3.8.64) o n l y , so t h a t one h a s

The Cauchy theorem y i e l d s :

Applying t h e S a d d l e P o i n t Method t o t h e i n t e g r a l on t h e r i g h t hand

3 . Singular Perturbations on Smooth Manifolds without Boundary

278

s i d e o f ( 3 . 8 . 6 5 ) a t each s t a t i o n a r y p o i n t < , , j = 0 , 1 , one g e t s t h e f u l l 3 asymptotic expansion f o r G ( x , t ) when h + O . Wewritedown e x p l i c i t l y the P,h f i r s t t e r m of t h i s e x p a n s i o n : I

-1 2p-1 -f (3.8.66)G ( x , t ) = 2(2nh)-+(xt-l-l) a R e { J ; ; - - e a g P ( S O ) ] ( l + O ( h a - l ) )= 9 ( C 0 ) P,h P _ _P_ 1 2p- 1 -f -c h - l a = (2nth) 2 ( x / t - l ) (2p-1) w p ( c o s ( b h-'a+a ) ( l + O ( h a - l ) ) ~

P

P

Note t h a t one h a s : s i n g supp G

P*h

= ix = t}.

Example 3 . 8 . 1 8 . C o n s i d e r t h e Cauchy problem

d K (x,t) +h-l(%(x,t)-%(x-h,t)) dt

= 0, x

E \,

t

E

W+

h

(3.8.67) 1i m

t++o where

%(x,t)

= Ah(X)

"i, and 6h ( x ) a r e d e f i n e d i n Example 3.8.16. Using t h e F o u r i e r t r a n s f o r m i n x E "i, , one f i n d s e a s i l y :

(3.8.68)

% ( x , t ) = (2nh)-'

J

e

h

-1

(ix<-t(l-e

-i<

))dC.

/
R e w r i t i n g t h e l a s t i n t e g r a l i n t h e form:

one g e t s :

W e s h a l l f i n d t h e a s y m p t o t i c b e h a v i o u r of K ( x , t ) f o r x > 0 , x E h 0.

when h

-f

The f u n c t i o n (3.8.71 h a s one s t a t i o n a r y p o i n t

c0

= i In

a.

%,

t > 0,

3.8. The Stationary Phase

279

Denote (3.8-72)

Y = i~

1

5 E c

1

,

Im g(a,i)

One f i n d s e a s i l y t h a t t h e c u r v e y i n

=

I m g(a,c 1,

c1

<

(Re

0

71.

i s g i v e n by t h e e q u a t i o n

Furthermore, denoting

Y,-

=

Ii

I

c

E

1

c ,c

= fn+in,

rl

E IR+},

the integrals (2nh)-l J

e

t h - I ( i a c - ( l-e-ii

Yf e x i s t and c o i n c i d e . Hence, t h e Cauchy theorem y i e l d s : (3.8.73)

%(x,t)

= (2nh)

-1

J e

th-l(iac-(l-e-i'

))dc

Y A s t r a i g h t f o r w a r d computation

shows t h a t

Applying t o t h e l a s t i n t e g r a l t h e s a d d l e p o i n t method one g e t s t h e f u l l a s y m p t o t i c e x p a n s i o n f o r % ( x , t ) , x > 0 , when h

-f

0. W e w r i t e only the

f i r s t term i n t h i s e x p a n s i o n : (3.8.74)

Kh(x,t) =

(2nh)

-

(27ihx/t)

-1/2

(x/t)

-1/2

e

-(x/h+1/2)

(3.8.75)

Y,(x,t)

-h

ln(x/t))

-1

w i t h some a

Now assume t h a t ( t - x ) - h a i n t e g r a l (3.8.68)

-h-'(t-x+x

(t-x)

E

,

-

x > 0, t > 0, h

(1/3,1/2).

-f

0.

Rewriting t h e

i n t h e form =

(21rh)-'(a-l)

<

e

t h - I (a-1 )

(is- ( 1/2) t; * ) +th-lJ, ( (a-1 5 ) as

"a

one g e t s ( a s p r e c i o u s l y i n Example 3 . 8 . 1 6 )

t h e conclusion,

that in the

case c o n s i d e r e d t h e main c o n t r i b u t i o n i n t h e a s y m p t o t i c b e h a v i o u r o f %(x,t) for x - t

-

h'

,

u E

(1/3,1/2),

i s g i v e n by t h e i n t e g r a l

3 . Singular Perturbations on Smooth Manifolds without Boundary

280

i n t h e p r e v i o u s example The f u n c t i o n

where

$(5)

=

(4/3)sin 5

-

(1/6) s i n 25 - ( 1 - c o s

5)2

i s t h e s o l u t i o n of t h e Cauchy problem

i s t h e s h i f t o p e r a t o r 8 u ( x ) = u ( x + h ). h One f i n d s i n t h e same way a s p r e v i o u s l y t h a t f o r

(x-t)-ha,

t h e main t e r m i n t h e a s y m p t o t i c b e h a v i o u r o f K

( x , t ) when h

where Bh

c o i n c i d e s w i t h t h e r i g h t hand s i d e o f $(c)

=

U E (2/5,2/3), +

2rh (3.8.66) with p = 2 , since

~ - ( 1 / 4 ) < ~ + 0 (
+

0

0.

Example 3.8.19. Consider t h e f u n c t i o n

where

E

E ( 0 . ~ ~i s1 a small p a r a m e t e r .

The f u n c t i o n P ( x ) i s t h e s o l u t i o n o f t h e boundary v a l u e problem: E

2 (1-E A ) P ( x )

=

0,

2

x E W + = {x

1

x E W 2 ,x 2 > 01

(3.8.77)

P ( x ) = 6 ( x 1 ) , l i m P ( x ) = 0. x ++a ++o 2 2 We s h a l l a p p l y t h e s a d d l e p o i n t method f o r computing t h e a s y m p t o t i c lim

x

e x p a n s i o n o f P ( x ) f o r E + 0 , and a g i v e n x E R:. The f u n c t i o n

3.8. The Stationary Phase

28 1

c a n be e x t e n d e d a n a l y t i c a l l y i n t o t h e complex p l a n e 5 c u t s along t h e r a y s 1

1-

{5

=

[

5

=

in, n

domain where

=

1

{C

< -1).

5 = in,

E

C1 w i t h two

=

1+ U 1-, t h e

ri > 1) arid

W e s h a l l d e n o t e by C 1 \ l ,

1

is analytic.

The f u n c t i o n

g(a,<)

(3.8.78)

1 h a s i n C \{I}

2 1/2 ia<-(1+< )

.

one s t a t i o n a r y p o i n t

5 (a) 0

(3.8.79)

=

=

ia(l+a2 ) -4

2 1/2 and g ( a , s ( a ) ) = - ( l + a

0

.

The s e t

i s c o n n e c t e d , c o n t a i n s t h e r e a l a x e s and i t s boundary 20'

C+an 5

=

+ m ,

0 a s i t s asymptote when

<

+-, 5

>

has t h e l i n e

0 and t h e l i n e <-an

=

0 when

5 < 0.

A c t u a l l y , a f t e r an e l e m e n t a r y Computation, one f i n d s t h a t f o r a # 0

w h i l e f o r a = 0 one h a s : Uc = C(')\{l].

0

Denote

One f i n d s e a s i l y t h a t f o r a # 0 t h e c o n t o u r

r

i s a p i e c e o f hyper-

bola defined as follows:

while f o r a = 0 t h e contour The c u r v e I' when 5

+

-, n

ro

c o i n c i d e s w i t h t h e r e a l a x e s R.

l i e s i n D+ and h a s t h e l i n e at-q

> 0 and t h e l i n e a<+q = 0 when 5

d i s t a n c e from a p o i n t 5

E r

+

=

-,

0 as i t s asymptote rl < 0 .

t o aD+ goes t o i n f i n i t y when 5

Hence, t h e +

-.

Thus,

3 . Singular Perturbations on Smooth Manifolds without Bounda y

282

t h e Cauchy t h e o r e m y i e l d s : 2 1/z

-1 (3.8.82)

PE(x) = ( Z n ~ ) - l J eE

(ix1'-x2(1+5

)

'dc,

x

E R 2+ .

ra Since g ( a , < ) = R e g(c(,L), V Re g ( O r , < ) on

r

< c ra,

e a c h s t a t i o n a r y p o i n t of

i s also a s t a t i o n a r y p o i n t of g ( a , < ) . Hence

by ( 3 . 8 . 7 9 ) i s t h e o n l y s t a t i o n a r y p o i n t of Re g ( a , < ) on ly Re g(a,<) i n (3.8.79)

+

--

when i

+

-, < E Ta

<

r .

( a ) defined 0 Since obvious-

one g e t s t h e c o n c l u s i o n , t h a t i 0 ( a )

i s t h e maximal p o i n t f o r Re g ( a , < ) on

r

a'

so t h a t

3.8. The Stationary Phase

283

x E which f o r n

=

q =i x

>

0},

2 c o i n c i d e s w i t h P ( x ) i n t h e p r e v i o u s example.

The f u n c t i o n P n,E

( x ) i s t h e s o l u t i o n of t h e boundary v a l u e problem:

2

x E Ry

(1-E A)P (xi = 0 , nrE

W e a r e g o i n g t o a p p l y t h e S a d d l e P o i n t Method f o r computina

t h e asymptotic expansion of P

( x ) f o r ~ 1 x I - l+ 0 .

n.E

The f u n c t i o n (3.8.89)

g (x,5 )

is t h e a n a l y t i c a l extension of g ( x , S ' ) ,

which c o n t a i n s Rn-'

domain i n Cn-'

1/ 2

i < x ' , 5 '>-xn( 1+<<' , 5 '>I

=

5 ' E Rn-' i n t o some c o n n e c t e d

and t h e o n l y s t a t i o n a r y p o i n t 5 '

s'

5 ; = ixvlxl-' of g ( x , 5 ' ) . One f i n d s : g ( x . 5 ' )

=

-1~1.

Again i n t r o d u c e

D+

=

rx

=

{C'

(3.8.90)

I

5' E

I

5 ' E D,:

C

n- 1

,

Re g ( x , < ' ) < 01,

Im g(x,<')

=

I m g(x,L*) =

i n t h e p r e v i o u s Example 3.8.19 one can show t h a t Rn- 1 so t h a t t h e Cauchy theorem y i e l d s :

AS

-1 (3.8.91)

=

n,E

(2.rrE)l-n

eE

rX

01. i s homological t o

.

( i < x ' , c ' > - x n ( 1 + < ~ ' , < * > )dg ~ /' *

rX

Again, t h e s a m e argument a s i n Example 3 . 8 . 1 9 l e a d s t o t h e c o n c l u s i o n -1 . i s t h e o n l y maximal p o i n t of t h e f u n c t i o n Re q ( x , L ' ) on t h a t 5; = i x ' / x j

rx,

t h a t i s Re g(x,<') < R e g ( x , < ' ) , V 5 E rx\{c;}.

I n o r d e r t o a p p l y t o t h e i n t e g r a l ( 3 . 8 . 9 1 ) t h e S a d d l e P o i n t Method, one h a s t o compute (3.8.92)

A 9 (x)

=

2 -det D 5 , g , g ( x , 5 t ) \

<'=
3. Singular Perturbations on Smooth Manifolds without Bounda y

284 One finds: (3.8.93)

A

9

where y' = x

t (x) = Ixln-' detl)Id+y' y'll

-1

x'.

Denoting t

D(E,Y') = det( ( I d + ~ y y'I '

1

2 2 one finds: D(o,~')z 1, (a/aE)D(E,Y')lE=O = l y~l , ( a 2 / a E )D(E,~'): 0, so that D(1,y') = 1 + ly'l

2

and (3.8.94)

A

9

(x)

=

X:~]X]~.

Now we can write the first term in the asymptotic expansion of

Again, as in the previous example one finds that for

/XI/€

-+

0 the

(x) is nothing else but the first term in the asymptotic expansion for P n,E Poisson kernel for the Laplace operator in En with the Dirichlet boundary condition at x

where

=

0, i.e.

is the area of the unit sphere in IRn.

The two asymptotics (3.8.95),

(3.8.96) can be written as one

asymptotic formula:

for ( ( x / / E + E / ~ x l )+ - ~m .

Let A be symmetric positive definite nxn matrix. Denote by P (X) n,A,E 2 E
the Poisson kernel for the operator 1 -

a

7

=

a/ax., with the Dirichlet boundary condition at x 7

...,

= 0.

A similar argument leads to the following asymptotic formula:

3.8. The Stationary Phase

for

-1

(IxI/E+E/IxI)

-f

m,

where

01

= A

-1

,

285

Ix1f = .

i s t h e s o l u t i o n o f t h e Cauchy problem:

l i m E (x,t) = 0, l i m

t+o

a/at

E

(x,t) = 6(x)

t-to

One h a s : (3.8.99)

E ( x , t ) E 0 f o r 1x1 > t .

I n d e e d , assume, f o r i n s t a n c e , t h a t x > t . Using t h e a n a l y t i c e x t e n s i o n of

<E> i n t o t h e domain C1\{l} from Example 3 . 8 . 1 9 a n d t h e Cauchy

theorem, one c a n s h i f t t h e i n t e g r a t i o n on t h e r i g h t hand s i d e o f ( 3 . 8 . 9 7 ) 1 and i s homological t o t h e o v e r t o any c o n t o u r r , which l i e s i n C \(l} r e a l a x i s IR.

It is e a s i l y seen t h a t f o r x > t t h e contour

r

c o n s i s t i n g of

I

two s i d e s o f t h e c u t 1 = ( 5 5 E C1, 5 = i n , ri > 1 1 run i n t h e o p p o s i t e lj2 d i r e c t i o n s , where ( l + g L ) h a s t h e imaginary p a r t of o p p o s i t e s i g n s , i s

*

homological t o IR. One c h e c k s e a s i l y , t h a t t h e c o r r e s p o n d i n g i n t e g r a l o v e r i s i d e n t i c a l l y z e r o . I f x < - t , t h e n t h e c o r r e s p o n d i n g homological c o n t o u r

c o n s i s t s o f t h e Cut 1

= {<

1

5 E

C1,

5

=

in,

ri < -1)

r u n t w i c e from t w o

d i f f e r e n t s i d e s i n two o p p o s i t e d i r e c t i o n s . Now assume t h a t 1x1 < t . W e a p p l y t h e s t a t i o n a r y p h a s e method f o r computing t h e f i r s t t e r m i n t h e a s y m p t o t i c e x p a n s i o n of E ( x , t ) when

E

-f

One f i n d s i n t h a t case f o r E ( x , t ) :

N o t i c e t h a t t h e s i n g u l a r s u p p o r t of t h e f a m i l y of d i s t r i b u t i o n s ( 3 . 8 . 9 7 ) c o i n c i d e s w i t h t h e c l o s u r e o f t h e cone V+

=

((x,t)

I

(x,t) E R x

IR+

,

1x1 < t } , as a consequence o f t h e a s y m p t o t i c f o r m u l a ( 3 . 8 . 1 0 0 ) . A s i m i l a r a s y m p t o t i c formula h o l d s a l s o i n t h e c a s e when x

E lRn , n > 1.

The p r e v i o u s argument y i e l d s ( 3 . 8 . 9 9 ) i n t h e m u l t i d i m e n s i o n a l case x E

mn,

as w e l l .

0.

r

286

3. Singular Perturbations on Smooth Manifolds without Boundary

Example 3.8.22 W e s h a l l c o n s i d e r h e r e a one p a r a m e t e r f a m i l y of f i n i t e d i f f e r e n c e

a p p r o x i m a t i o n s ( i n s p a c e v a r i a b l e ) of a h y p e r b o l i c f i r s t o r d e r d i f f e r e n t i a l o p e r a t o r , t h e mesh-size h b e i n g t h e s m a l l p a r a m e t e r . The s i n g u l a r s u p p o r t and t h e a s y m p t o t i c b e h a v i o u r

(for h

-t

0 ) of t h e

Fundamental S o l u t i o n o f t h e Cauchy problem f o r t h e s e a p p r o x i m a t i o n s , w i l l b e i n v e s t i g a t e d under t h e a s s u m p t i o n t h a t t h e y s a t i s f y t h e von Neumann's s t a b i l i t y condition. L e t a ( s ) : C1 + C1 b e a n a n a l y t i c f u n c t i o n of

s =

n+iT,

which s a t i s f i e s

the conditions: in q E R ,

( i )a ( s ) i s Z v - p e r i o d i c

( i i )a ( 0 ) = 0 , a ' ( 0 ) = i w w i t h s o m e w 6 R ,

E R.

( i i i )Re a ( q ) 5 0 , V r' Let

(3.8.101) a ( s )

iws+ C

=

k2p

a s k

k

b e t h e T a y l o r e x p a n s i o n f o r a ( s ) i n a neighbourhood of z e r o . D e f i n i t i o n 1.

1 " . The f u n c t i o n a ( s ) i s s a i d t o b e p - p a r a b o l i c if

i n t h e sense of Petrovsky

(i)- ( i i i )a r e f u l f i l l e d and, moreover, i f

(3.8.102)

R e a ( q ) < 0 , V I)

p = 2 r i s even and R e a

P

E

37,

r' # 0 (mod 2 n ) ,

< 0.

2". The f u n c t i o n a ( s ) i s s a i d t o b e ( p , q ) - p a r a b o l i c i n t h e s e n s e of S h i l o v if ( i ) - ( i i i )(,3 . 8 . 1 0 2 ) are f u l l f i l l e d and t h e c o e f f i c i e n t s ak i n

(3.8.101)

s a t i s f y the conditions: ( 3 . 8 . 1 0 3 ) a # 0 , R e ak = 0 , p i k < q , R e a < 0 , q = 2 r i s even. P q

3 " . The f u n c t i o n a ( s ) i s s a i d t o b e p - h y p e r b o l i c i f ( i ) - ( i i ia) re f u l f i l l e d a n t ( 3 . 8 . 1 0 4 ) Re a ( r l )

Let

5'

R

Z

0 , V II

E IR,

a

P

# 0.

b e a one p a r a m e t e r f a m i l y o f meshes i n I R , c o n t a i n i n g t h e

o r i g i n , h b e i n g t h e mesh s i z e . Denote by Ah t h e f i n i t e d i f f e r e n c e o p e r a t o r a s s o c i a t e d w i t h t h e f u n c t i o n a ( s ) above a c c o r d i n g t o t h e f o r m u l a (3.8.105)

= h-la(hDx),

D~ = - i a / a x

3.8. The Stationary Phase

287

t h e f u n c t i o n a ( n ) b e i n g a l s o c a l l e d t h e symbol o f A Since a ( n ) i s 2n-periodic, a(n)

z c

=

h

.

it c a n b e expanded i n t o a F o u r i e r series,

eikq,

kEZ so t h a t t h e o p e r a t o r Ah i s a l s o w e l l - d e f i n e d on t h e m e s h - f u n c t i o n s U(X)

:

R

h

C1 w i t h compact s u p p o r t by t h e f o r m u l a :

-f

(Ah u ) (x) L e t E ( x , t ) : IRh

h

=

1 h-lCku(x+kh), kEZ

R+ b e t h e s o l u t i o n o f t h e Cauchy problem:

X

h

(3.8.106)

x E R h

(D + i A ) E h ( x , t ) = 0 ,

( x , t ) E Rh x

lim

x E \,

t

E

t++o

h

(x,O) = 6 h ( x ) ,

where, a s p r e v i o u s l y , 6 ( x ) = 0 , V x # 0 , d h ( 0 ) h

The o p e r a t o r D

+inh t

=

R+

h

-1

,

D

=-ia/at.

t i s a f i n i t e d i f f e r e n c e approximation ( i n space

v a r i a b l e x ) of t h e h y p e r b o l i c o p e r a t o r Dt-uDX,

t h e approximation e r r o r

being 0(hP-'). Definition 2. The o p e r a t o r D t + i A h

i s s a i d t o be p-parabolic

or ( p , q ) - p a r a b o l i c o r p-

h y p e r b o l i c if t h e symbol a ( n ) of Ah i s p - p a r a b o l i c ,

or ( p , q ) - p a r a b o l i c o r

p-hyperbolic. Definition 3.

E R x %+ i s s a i d t o b e r e g u l a r f o r E ( x , t ) i f t h e r e (x h O'tO) e x i s t c o n s t a n t s Ckl such t h a t

A point

k l

IDtDx,hEh(~,t)I

5 Ckl,

f o r e a c h map ( x , t ) : (O,ho] + R h

V k,l,

V h E (O,hol,

X

R+ w i t h ( x , t )

+

(x0,t0) for h

-+

0;

here D = - i ( B - l ) / h , where 0 i s t h e s h i f t o p e r a t o r 8' u ( x ) = u ( x + h ) . x,h h h h The c l o s u r e o f t h e complement i n R X of t h e s e t of a l l r e g u l a r p o i n t s of E ( x , t ) i s c a l l e d t h e s i n g u l a r s u p p o r t o f Eh and i s d e n o t e d h s i n g supp E h' C l a i m 1.

I f t h e approximation D t + i A h

i s ( p , q )- p a r a b o l i c , t h e n

( 3 . 8 . 1 0 7 ) s i n g supp Eh = { ( x , t ) E R

X

z+1

x + u t = 0)

3 . Singular Perturbations on Smooth Manifolds without Boundary

288

Proof of Claim 1. Using the discrete Fourier transform in space variable x E l€i,,one gets for E (x,t) the formula h

Analyticity of a(<) and 27-periodicity yield

as a consequence of Cauchy's theorem.

We shall choose Then for

<

=

sufficiently small and such that sgn T =sgn(x+wt)

T

q+iT with l q l 5 6, using the formula

one gets the conclusion that for 1171 5 6,

] T I 5 TO,

sgn

T

=

sgn(x+ut) with

sufficiently small, the following inequality holds: (3.8.110) Re(ix<+ta(<))

v

for

17

E [-6,61,

<-\TI

Ix+wt//2

V T E [ - T ~ , T ~ ] , sgn T = sgn(x+wt).

since Re a(n)

s -c6, v

q E [-rr,nI,

J r i J2

s

with some positive

constant C6 (this is a consequence of (p,q)-parabolicity), one can choose so small that Re a(<) S -(1/2)Cs,

T~

1.r

I

v

ri with

6 2

6 T ~ sgn , T = sgn(x+wt) , so that combining

lql

$ 7,

t/ T

with

t with (3.8.110) one will

have : (3.8.111) Re(ix<+ta(<)) 5 for

v

11 E

[-x,al,

V T

E

-IT\

/x+wtl/2

t-TO,~Ol,

sgn

T =

sgn(x At) with

T~

sufficiently

small. Taking T =

T~

sgn(x+wt) in (3.8.109) and using (3.8.1111, one gets

the following estimate for Eh (x,t): -To I X+Wt I / (2h) /Eh(x,t)) 5 h-'e k 1 For the derivatives Dx,hDtEh(~,t)one gets, using the same argument, the estimates:

k

1

h-(k+l+l)e-~0/~+~t)/)2h)

3.8. The Stationary Phase Now l e t x t w t

0, t > 0. As a consequence o f

=

289 (p,q)-parabolicity,

the

main c o n t r i b u t i o n i n t o t h e a s y m p t o t i c b e h a v i o u r o f t h e i n t e g r a l (3.8.108)

i s g i v e n by a neighbourhood o f z e r o R e a ( q ) 5 -C6,

in1

5 6 s i n c e f o r 1q1 2 6 one h a s :

w i t h some p o s i t i v e c o n s t a n t C 6 , depending on 6 . Hence,

E ( x , t ) = (2rrh)

-1

h

J

h - l ( i x q + t a ( q ) ) d r lt O ( e - t h - l C

6).

lrlIS6

One h a s i n t h e case c o n s i d e r e d : ixq+ta(q)

t a rl Pb ( q ) , P

=

E C"([-6,61), b ( 0 )

where b ( n )

R e a

P


P

# O

= 1.

I n t r o d u c i n g a new v a r i a b l e

5 w i t h zl/'

=

lip,

q(b(rl))

rl =

t h e main b r a n c h ( z 1 /P

Q ( 5 ) , rl'(0)

l Z Z1

=

1,

= 1) of t h i s multivalued f u n c t i o n ,

one c a n r e w r i t e t h e l a s t a s y m p t o t i c f o r m u l a f o r Eh ( x , t ) w i t h x+wt = 0 and 6 so s m a l l t h a t 5 f-t rl t o b e a b i h o l o m o r p h i c d i f f e o m o r p h i s m , i n t h e following fashion:

'I

Y6 where t h e c u r v e y6 i n a s m a l l complex neighbourhood o f t h e o r i g i n i s t h e image of [-S,S] under t h e b i h o l o m o r p h i c d i f f e o m o r p h i s m rl

5.

++

N o t i c e t h a t R e ( a g P ) < 0 , b' g E y6\iO}. I f 6 i s s u f f i c i e n t l y s m a l l P t h e n one w i l l have R e ( a iP)< -C6 < 0 on t h e segments P 1* = {Z, E C 5 = Re < ( ? 6 ) + i . r , C I I m <(?6)\}. Using t h e Cauchy theorem,

\TI

1

one c a n d i f f o r m t h e c u r v e y 6 i n t o t h e c o n t o u r which i s t h e u n i o n of t h e i n t e r v a l [Re 5 ( - 6 ) ,

Re

<(&)I

C

R

and t h e c o r r e s p o n d i n g p i e c e s o f t h e

segments 1

2-

Obviously, t h e i n t e g r a l s o v e r t h e s e p i e c e s of 1+ g i v e e x p o n e n t i a l l y -

s m a l l terms when h

+

0 , so t h a t t h e main c o n t r i b u t i o n i n t o a s y m p t o t i c

b e h a v i o u r of E ( x , t ) i n t h e case c o n s i d e r e d ( x + o t = 0 , t > 0 ) i s g i v e n by h t h e i n t e g r a l o v e r t h e i c t e r v a l I = [Re 5 ( - 6 ) , R e 5 ( 6 ) 1 C W , -1 t a h-lSP - t h C6 E ( x , t ) = (2nh)-' I q ' ( 5 ) e dS+O(e ) = h I

t a h-lEP =

(2nh)-' J e I

d5+0((h/t)(2/p)-1)

=

3. Singular Perturbations on Smooth Manifolds without Boundary

290

R

where

a c

P

=

cp

dg # 0,

J e

(an)-'

w

t h e l a t t e r i n t e g r a l b e i n g c o n v e r g e n t s i n c e Re a

P

5 0, a

P

f C, p = 2 r

2

2.

T h a t e n d s t h e p r o o f of Claim 1. C l a i m 2.

If t h e a p p r o x i m a t i o n D t + i A h

for (x,t) E \ x

i s p-hyperbolic,

then E ( x , t ) is w e l l defined h

R and

( 3 . 8 . 1 1 2 ) s i n g s u p p Eh

=

{(x,t) E R

2

1

3 rl E R s . ~ .a ' ( n )

= -ix/t}.

N o t i c e t h a t t h e n u l l - c h a r a c t e r i s t i c x+wt = 0 i s c o n t a i n e d i n s i n g supp E h' since a ' ( 0 ) = i w . Proof of C l a i m 2 . since a ( n )

=

iw(n), w

:

[-n,n]

-Z

IR b e i n g 2 n - p e r i o d i c r e a l v a l u e d a n a l y t i c

f u n c t i o n , one can r e w r i t e ( 3 . 8 . 1 0 8 ) a s f o l l o w s

and a p p l y t o t h e i n t e g r a l on t h e r i g h t - h a n d s i d e of if w'(n)

3.8.1,

(3.8.113) P r o p o s i t i o n

# - x / t , V n € [-ii,n]

S i n c e w ( n ) i s a n a n a l y t i c f u n c t i o n , t h e r e may b e o n l y a f i n i t e number of 1 5 j 5 r , where w " ( g . ) = 0 . F o r t h e same r e a s o n f o r p o i n t s E . E [-n,.rr], 3 3 e a c h g i v e n v a l u e of x / t t h e r e may b e o n l y f i n i t e l y many p o i n t s n . ( x / t )

where w ' ( n )

= -x/t.

s t a t i o n a r y p o i n t ri with 5 .

3'

7 L e t x / t b e such t h a t t h e r e e x i s t s a t l e a s t one

f o r x r l + t w ( n ) which i s r e g u l a r , i . e . d o e s n o t c o i n c i d e

1 5 j 5 r d e f i n e d above a s z e r o s of w " ( n ) .

Then t h e c o r r e s p o n d i n g

c o n t r i b u t i o n i n t o t h e a s y m p t o t i c b e h a v i o u r of E ( x , t ) f o r h j. 0 from t h i s h s t a t i o n a r y p o i n t c a n b e computed by u s i n g t h e S t a t i o n a r y Phase method, so t h a t t h e l i n e s x t t w ' ( n ) = O b e l o n g t o s i n g s u p p Eh if n s ( x / t )

ft 1 5 , ,..., 5,).

I f min a ' ( n ) < - x O / t O < max a ' ( n ) t h e n f o r any ( x , t ) s u c h t h a t Ix / t - x / t l 0

0

< 6 with 6 s u f f i c i e n t l y s m a l l t h e r e a r e s t i l l s t a t i o n a r y p o i n t s

n s ( x / t ) , s o l u t i o n s o f t h e e q u a t i o n a'(n)

=

-x/t,

so t h a t i n f a c t t h e r e a r e

continuum l i n e s x t t w ' ( q s ( x / t ) ) = 0 i n R2 b e l o n g i n g t o s i n g supp Eh,

ns(x/t)

3.8. The S t a i m a y Phase

291

b e i n g a r e g u l a r s t a t i o n a r y p o i n t , and t h e r e a r e o n l y f i n i t e l y many l i n e s i n t h e s e t on t h e r i g h t hand s i d e o f

( 3 . 8 . 1 1 2 ) f o r which t h e r e may b e no

r e g u l a r s t a t i o n a r y p o i n t s f o r t h e i n t e g r a l d e f i n i n g Eh, c h a r a c t e r i s t i c x+wt = 0 b e i n g one o f them, u n l e s s p

=

the null-

2 . Hence ( 3 . 8 . 1 1 2 )

i s proved. W e state here

(without a f u l l proof) t h e following r e s u l t concerning

t h e a s y m p t o t i c b e h a v i o u r of E ( x , t ) ( f o r h + 0) i n s o m e s t r i p h a ( h ) 5 x+wt [ < a 2 ( h ) w i t h a . ( h ) -+ 0 f o r h -f 0 , s a t i s f y i n g a p p r o p r i a t e c o n d i t i o n s 1 3

I

Claim 3 . 1 " . The a p p r o x i m a t i o n D + i A h b e i n g p - p a r a b o l i c , t formula h o l d s : (3.8.114) E ( x , t ) = h h

-1/2 -l/G (p-1))

t

t h e following asymptotic

I x+wt 1 - (p-2) / (p-1)

V ( x , t ) such t h a t h~x+otl-P(P-l)+h-l/x+wtl (p+l)'(p-l) Aj,

Bjr

-+

0 when h

+

0; here

j = 1 , 2 a r e some c o n s t a n t s , R e A . > 0 , j = 1 , 2 , which depend o n l y 1 i n (3.8.101). P

on w and t h e c o e f f i c i e n t a 2'.

+inh b e i n g p - h y p e r b o l i c , assume a d d i t i o n a l l y t h a t t # 0 (mod Z I T ) . Then t h e f o l l o w i n g a s y m p t o t i c f o r m u l a e

The a p p r o x i m a t i o n D

# 0, V n

a'(n)-iw hold:

a ) f o r p even

V ( x , t ) s u c h t h a t h l x + w t /- o / ( P - l ) + h - l A,

I'

B., 1

j

=

Ix+wt/

1 , 2 , are some non-vanishing

which depend o n l y on w and a

P

(p+l)'(p-l)

+ 0 when h

constants, I m A . 1 i n (3.8.101).

=

-f

0; h e r e

0 , j = 1,2,

b ) f o r p odd, one h a s t h e same f o r m u l a ( 3 . 8 . 1 1 4 ) a s i n t h e p - p a r a b o l i c case f o r sgn(x+wt) t h e p-parabolic

=

-sgn I m ap, a n d , t h e same f o r m u l a ( 3 . 8 . 1 1 5 ) a s i n

c a s e w i t h even p , when s n g ( x + w t ) = sgn I m a

P

.

The f o l l o w i n g argument g i v e s an i d e a how t o p r o v e t h e f o r m u l a e ( 3 . 8 . 1 1 4 ) , (3.8.115).

3 . Singular Perturbations on Smooth Manifolds without Bounda y

292

Introducing the new variable n = /(x/t)+o(l’(p-l)E,

one rewrites the

integral (3.8.108) in the form where the phase function in the exponent is as follows:

where $ ( n ) = a(q)-iwn-a np

P

=

for rl

O(np+l),

-f

0.

One can consider O(x,t,E) as a small perturbation of the phase function

when a =

I (x/t)+wIP-l

is small, the remainder being of order

+1 p+l $ ( a S ) = O(ap 5 ) , when a

One has the natural conditions on large parameter when h

+

+

0.

1 (X/t)+ W /

:

h-ll (X/t)+W lp’(p-l) has to be a

0 in order to apply saddle point or stationay

phase method to the corresponding integral with the phase function iS+a SP, while h-ll(x/t)+wl(p+l)/(p-l) has to be a small parameter when

P

h

+

0, for being able to use the Taylor expansion for the function

exp(ih-’t$(aE)) as in Example 3.8.16. Thus, the computation of the main term in the asymptotic expansion of E (x,t) for (x,t) such that -1

h 1 (x/t)+wl p/(p-l) + m, h-ll (x/t)+wl t p + l ) / ( p - l ) + O the case of the polynomial phase function iS+a 5’.

P

(h+O) is reduced to The use of the saddle

point (as in Example 3.8.17) or stationary phase method (in the corresponding p-hyperbolic case), leads to the formulae (3.8.114), (3.8.115) above. Example 3.8.23. 1 Let 5 E C , 151 < 1 and let:

where 5

*

is the complex conjugate of 5.

Introduce s;(c;e)Je=l = r,

o

<

r 5 I.

With $(x,t,<,w) = xw+(t/r)arg S(r;,eiw) consider the following integral

3.8. The Stationary Phase

293

It i s e a s i l y seen t h a t E ( 5 ; x . t ) is t h e s o l u t ion of t h e following f i n i t e h d i f f e r e n c e i n i t i a l v a l u e problem:

E (5;x,t) h where R

1

8

U ( t )

E

= {kh, k

h o p e r a t o r on

1

6h(x),

=

Z},

lR1

T ,+

= {mT,

m E

8 u ( x ) = u ( x + h ), and 6

%,

i s the s h i f t h 1 i s t h e s h i f t o p e r a t o r on IR T,+

Z+>,

h

T = rh,

8

‘

= U(t+T).

The problem ( 3 . 8 . 1 1 7 ) i s a n i m p l i c i t d i f f e r e n c e a p p r o x i m a t i o n o f t h e f o l l o w i n g d i f f e r e n t i a l Cauchy problem: (D -D ) E ( x , t ) t x

=

l i m E(x,t) t++o

6(x)

=

0,

x E IR, t E Rt

where 6 ( x ) i s t h e Dirac 6 - f u n c t i o n . Obviously, E ( x , t )

=

6 ( x + t ) , so t h a t s i n g supp

E

=

{x+t=O}. W e a r e

g o i n g t o i n v e s t i g a t e t h e s i n g u l a r s u p p o r t o f t h e mesh-function Obviously, f o r 5

=

0

(1- =

l ) , one h a s :

E (O;x,t)

=

Eh ( 5 ; x . t ) .

6h ( x t t ) , so t h a t

h s i n g supp E ( O ; x , t ) = s i n g supp E ( x , t ) . F u r t h e r , s i n c e h iw

IS(<;e

)I

=

1,

v

w E [-n,nI,

so t h a t it i s e a s i l y s e e n t h a t t h e s t a t i o n a r y p o i n t s of t h e p h a s e f u n c t i o n $ a r e s o l u t i o n s of t h e e q u a t i o n :

/eiw-5/’

= a -1 j 1 - 5 l 2 ,

a

=

-x/t.

The l a s t e q u a t i o n h a s p r e c i s e l y two z e r o s w . ( < ; a ) E [ - n , n ] , 3 -1 c1 E [ X - ( c ) ,X+(C) ] , w i t h

provided t h a t

A s t r a i g h t f o r w a r d computation shows t h a t b o t h s t a t i o n a r y p o i n t s

w . ( ~ ; x , t ) ,j = 1 , 2 , are r e g u l a r , p r o v i d e d t h a t 5 # 0 . Applying t h e 3 s t a t i o n a r y p h a s e method t o t h e i n t e g r a l ( 3 . 8 . 1 1 6 ) one g e t s a f u l l a s y m p t o t i c e x p a n s i o n f o r Eh ( < ; x , t ) ,

3. Singular Perturbations on Smooth Manifolds without Bounda y

294

f o r each ( x , t ) E

B

X

R + such t h a t ( - t / x )

E

[x-(<),x+(<)].

T h e r e f o r e , one h a s

T h i s phenomemon o f s p r e a d i n g o u t of t h e s i n g u l a r s u p p o r t o c c u r s f o r a l l h y p e r b o l i c f i n i t e d i f f e r e n c e a p p r o x i m a t i o n s ( i . e . such a p p r o x i m a t i o n s iw t h a t jS(<,e ) E 1 ) of h y p e r b o l i c d i f f e r e n t i a l o p e r a t o r s .

I

One h a s

a n d , moreover, one c h e c k s e a s i l y t h a t

where

F u r t h e r , w i t h wo

=

-2 a r g

<

one f i n d s :

Using t h e s a m e argument a s i n Example 3.8.18 one f i n d s t h e f i r s t t e r m i n t h e a s y m p t o t i c e x p a n s i o n f o r E ( x , t ) i n a neighbourhood of t h e z e r o h 2 -1 -f a, (x+t)3h-1 + 0 for h c h a r a c t e r i s t i c x + t = 0 such t h a t ( x + t ) h

+

0.

W e w r i t e it h e r e down:

N o t i c e t h a t t h e f i r s t e x p o n e n t i a l term on t h e r i g h t hand s i d e of t h e l a s t formula (which i s t h e c o n t r i b u t i o n from w ( < ; a ) ) c o n v e r g e s t o 6 ( x + t ) a s 1 h + 0 and i t s b e h a v i o u r i s l i k e t h e one of t h e fundamental s o l u t i o n f o r S c h r o d i n g e r ' s e q u a t i o n . The second e x p o n e n t i a l t e r m (which i s t h e

3.9. The Fourier Integral Singular Perturbations c o n t r i b u t i o n from w when h

+

2

295

(<;a)) c o n v e r g e s t o z e r o i n t h e d i s t r i b u t i o n s e n s e ,

0.

3.9.

The F o u r i e r I n t e g r a l S i n g u l a r P e r t u r b a t i o n s n. L e t U . E IR , j = 1,2. 3

'

Definition 3.9.1.

The f u n c t i o n $ ( x , y , h , S ) , ( x , y , h , S ) E u1 x U2 x IR x I R ~i s s a i d t o be a phase f u n c t i o n i f t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d : l o $ i s reaZ valued, I $ ( X , Y , E - ' , < ) E S : ~ 6 ° " ' 2"

f o r each compact s e t K c u1

151 Let

(U1 x U2)

there e x i s t s a constant c

K

l o , 2" t h e f u n c t i o n ei'

A s a consequence o f

when

x U2

such t h a t

is rapidlyoscillating

-t m .

x

E

m

C

0

(IRn)

x

and

E 1 for

151

5 R w i t h some R <

m.

D e f i n i t i o n 3.9.2.

L e t $ ( x , y , X , S ) be a phase f u n c t i o n and l e t a ( x , y , e , < ) E S;,,(U, an amplitude (see D e f i n i t i o n 3 . 7 . 1 ) . The operator

x

u 2 ) be

i s c a l l e d t h e Fourier i n t e g r a l s i n g u l a r p e r t u r b a t i o n (F.I.S.P.) provided t h a t t h e r i g h t hand s i d e i n ( 3 . 9 . 2 ) i s p o i n t w i s e convergent. I f a ( x , y , ~ , S )E

su(U1

X

( s e e D e f i n i t i o n 3.3.2) t h e n , as u s u a l ,

U2)

a O ( x , y , ~ , S )s t a n d s f o r t h e c o r r e s p o n d i n g p r i n c i p a l symbol. Theorem 3 . 9 . 3 .

Let AE be a FISP. Then @

@

: cm(U )

0

2

+

Cm(ul)

f o r each given

E

> 0.

P r o_ o f . W e s h a l l p r o v e somewhat more. Namely, t h a t f o r e a c h E > 0 and e a c h _ p a i r o f i n t e g e r s K > 0 , 1 > 0 s u c h t h a t u +u + n + l < K o n e h a s :

A :

:

c1 0 ( u 2 )-t cK ( u l ) .

integral

2 3 L e t f i r s t K = 0 . Then u +u + n i l < 0 , so t h a t t h e 2 3

3. Singular Perturbations on Smooth Manifolds without Boundary

296

i s a b s o l u t e l y (and u n i f o r m l y w i t h r e s p e c t to x E U 1 ) c o n v e r g e n t , a n d , moreover, one c a n d i f f e r e n t i a t e it 1 t i m e s u n d e r t h e i n t e g r a l s i g n , t h e

c o r r e s p o n d i n g i n t e g r a l s b e i n g a b s o l u t e l y and u n i f o r m l y i n x E U 1 converg e n t , t o o . That proves t h e c l a i m f o r K = 0. W e u s e t h e i n d u c t i o n f o r p r o v i n g i t f o r any i n t e g e r K

0. L e t K = 1 .

Denote

so t h a t a f t e r one p a r t i a l i n t e g r a t i o n i n t h e r i g h t hand s i d e of

(3.9.2),

one g e t s : dy dS =

where L

* is b

t h e ( f o r r a l l y ) a d j o i n t o p e r a t o r of

consequence o f belong t o

sV - ( O , 1t o

Lb,

and where, a s a

( 3 . 9 . 1 ) , t h e a m p l i t u d e s of t h e o p e r a t o r s AE 0 5 j 5 n, $,I' 1, O ) (U1 x U 2 ) . 3

Example 3 . 9 . 4 . Consider t h e following F o u r i e r I n t e g r a l S i n g u l a r P e r t u r b a t i o n : (3.9.61

(E:(t)u) ( x )

=

(2i)-'((E~,+[t)u(x)-(EE (t)u)( X I ) , n,-

where

One c h e c k s e a s i l y t h a t v ( E , t , x )

=

( E z ( t ) u ) ( x ) i s t h e s o l u t i o n of t h e

f o l l o w i n g s i n g u l a r l y p e r t u r b e d Cauchy problem: (E

2

(3.9.8) lim

t++o

a 2 / a t L.- €

2

A + i ) v ( E , t , x=) 0, t > 0 , x n

a -v t++o at

v ( E , ~ , x ) = 0, l i m

E R ~ ,

( ~ , t , x )= 6 ( x ) .

One h a s t h e f o l l o w i n g formula f o r t h e d i s t r i b u t i o n a l k e r n e l s E n ( € , t ; x - y ) of t h e f a m i l y o f o p e r a t o r s

E + EE(t):

3.9. The Fourier Integral Singular Perturbations

(3.9.9)

E (E,t;x) = F -1

E<Ec>-l sin(t

X'S

Introducing

to

E R , 15,/

= E

-1

-

, 5

=

297

E -1 < E s > ) .

(Co,5) E R

n+l

-

, x

=

(xo,x) E Rn+',

one can rewrite (3.9.9) as follows:

Let n = 2m+2, where rn 2 0 is integer. One has in that case (see [ G - Sh, I ] ) : (3.9. where C12m+2

6 /:(I

is the area of the unit sphere in R

t) is the &-function on the sphere

(3.9.12)

(6(l;l-t),$)

=

t

t

=

{y 1

and the distribution

x E iR2m+3, 1y1

= t}, i.e.

.

J $(:)do_, S

S

2m+3

V Ji E C m ( ~ R 2 m + 3 ) X

Using (3.9.10), (3.9.11), one finds:

Furthermore, one has

where, as usual, s

=

max {s,O}.

Hence, (3.9.131, (3.9.14) yields: -1

(3.9.15) E 2 m + 2 ( ~ , t ;=~) and for m = 0 one finds:

Therefore the FISP (3.9.6), (3.9.7) with n equivalent representation:

=

2m+2 admits the following

3 . Singular Perturbations on Smooth Manifolds without Boundary

298 and f o r n

2 m + l one c a n u s e t h e c l a s s i c a l d e s c e n t method f o r g e t t i n g a

=

s i m i l a r r e p r e s e n t a t i o n f o r ( E : (~ t )~u )~( x ) ( s e e , [Cour.,

1

I.

One f i n d s t h e f o l l o w i n g f o r m u l a f o r t h e c o r r e s p o n d i n g k e r n e l d i s t r i b u t i o n :

where J ( p ) i s t h e B e s s e l f u n c t i o n of o r d e r z e r o and H ( T ) , T C R , i s 0 H e a v i s i d e ' s f u n c t i o n (see, f o r i n s t a n c e , [Grad.-Ryz., 11 f o r t h e B e s s e l f u n c t i o n s . N

Indeed, l e t a g a i n 5 =

(5,,5) E R

2m+2

-

, x

= (xO,x) C R

Using t h e d e s c e n t method, one f i n d s f o r

t h e following formula

Again a p p l y i n g ( 3 . 9 . 1 0 ) , one g e t s :

since

1

1

e-ipe(1-82)-4d8

=

71

Jo(p),

V p 6 IR

-1 (see [ G r a d . - R y z . ,

1

1).

I n p a r t i c u l a r , one h a s :

L e t B b e symmetric p o s i t i v e d e f i n i t e nxn m a t r i x . The s o l u t i o n of t h e s i n g u l a r l y p e r t u r b e d Cauchy problem: (E

2 2

at-E

lim

two

2


v = 0,

a

x' x

0,

>+l)V

lim

t++O

a tv

=

t > 0,

u(x)

X

6

mn

2m+2

.

3.9. The Fourier Integral Singular Perturbations

299

c a n b e w r i t t e n a g a i n a s a sum o f S i n g u l a r F o u r i e r I n t e g r a l O p e r a t o r s :

and one c h e c k s e a s i l y u s i n g t h e s u b s t i t u t i o n y = B-*x,

that the

d i s t r i b u t i o n a l k e r n e l E n , B ( ~ , t ; x - y ) of EE ( t ) f o r n = 2m+2 i s g i v e n by n,B t h e formula

The cone V = { ( p , 5 ) E R

n+l

,

*

=

u2}

i s n a t u r a l l y connected with t h e

d i f f e r e n t i a l o p e r a t o r above. L e t V b e t h e d u a l cone of V , n +1 2 V* = { ( t , x ) E R . = t 1. A s a consequence o f t h e l a s t f o r m u l a f o r E 2 m + 2 , B ( ~one , t ;g~ e t )s ,t h e c o n c l u s i o n t h a t i t s s i n g u l a r s u p p o r t , s i n g supp E

2m+2,B whose boundary i s

c o i n c i d e s with t h e set

*

v

Dv*

=

( t , x ) E R n + ' , 5 t

.

We a r e g o i n g t o r e s t r i c t s l i g h t l y t h e c l a s s of p h a s e f u n c t i o n s and s h a l l c o n s i d e r , from now on t h e p h a s e f u n c t i o n s $ ( x , y , A . S ) which s a t i s f y t h e following Condition 3.9.5.

The phase f u n c t i o n s @ ( x , y , X , E ) are p o s i t i v e l y homogeneous f u n c t i o n s of degree 1 i n v a r i a b l e s ( h . 5 ) f o r h

2

2

2 f.

For such p h a s e f u n c t i o n s ( 3 . 9 . 1 ) can b e s t a t e d i n t h e form

O f c o u r s e , i n t h i s case @ ( X , Y , E

-1

,<)

€ .S(lfo'l)(Ul

x

U2).

On t h e o t h e r hand, w e s h a l l e x t e n d t h e cl.ass of t h e o p e r a t o r s c o n s i d e r e d i n t h e f o l l o w i n g way. We s h a l l d e n o t e by K t h e c l a s s of i n t e g r a l o p e r a t o r s (3.9.18)

(Ku) ( E , x )

=

1

K(E,x,y)u(y)dy

Rn such t h a t ~ ( E , x , y )E C ~ ( U x u2) u n i f o r m l y w i t h r e s p e c t t o E E ( o , E o ] 1 W e s h a l l use t h e n o t ati o n ( 4 ) f o r t h e f a m i l i e s of c o n t i n u o u s _1,0 l i n e a r mappings A C m ( U 1 + C ( U 1 ) , V E E (O,E 1, which c a n be I$: 0 2 0 r e p r e s e n t e d i n t h e form ( 3 . 9 . 2 ) w i t h t h e phase f u n c t i o n $ s a t i s f y i n g

s"

2

1,

3 . Singular Perturbations on Smooth Manifolds without Boundary

300

Condition 3.9.5 and with an amplitude a E Sv ( u x u), a being defined up to 1,o an element i n K . We shall need later on the following Corollary 3.9.6. Let A

A

$,I

a2

'j

@

E S y , o ( $ ) . Then A

$

can be represented in the form (3.9.5) with

C SV-e2($) where e 2 = ( 0 , 1 , 0 ) . Furthermore, the amplitudes ia a and 1,O 61 4, with a = a/aC., define the same singular Fourier Integral Operator 3

'j

according to the formula (3.9.2). As a consequence of Schwartz's Kernel Theorem (see [Sch, l]), one

can associate with A -V E

'A(E,x,Y) E D ' ( U ,

(3.9.19)

: Cm(U )

0 X

+ Cm(U

0 2 U 1 for each 2

E

1

a family of distributions E ( 0 . ~ ~such 1 that

e " l ( ~ u ) (E,x) = , @

v

u E c;(u2).

We are going to localize the singularities of the kernel distribution A(E,x,Y) associated with E"'A

4-

With @(x,y,h,E) positively homogeneous of degree 1 in ( h , c ) for h

2

+ ] < I2

t

phase function, satisfying (3.9.17), introduce the set

and denote

Theorem 3.9.7.

The singular support of the family E + A(E,X,YI of distributional kernels defined by (3.9.19) is contained in the set Q defined by (3.9.201, 4 (3.9.21): (3.9.22)

5 Q+. sing supp (A,~,x,y)

Proof. As

a consequence of Corollary 3.9.6, it suffices to consider the case,

when v + n < 0 , v +u +n < 0. Then E + A(~,x,y)is a family of continuous 2 2 3 in (x,y) E u l x u2 functions uniformly with respect to E E ( O , E I . More0 over, one has for A(~,x,y)the formula: (3.9.23)

A(~,x,y)=

I mn

E'

-1 a(x,y,~,<)e~'(~'~'~ ")d<.

1

3.9. The Fourier Integral Singular Perturbations W e show t h a t u n i f o r m l y w i t h r e s p e c t t o

E

A(€,.,.)

u C C (U1

Indeed, l e t

V (x,y) E

a.

E

30 1

one h a s :

C ( U 1x U 2 ) \ Q $ ) .

X

X 2 + / E l 2 2 1 one h a s V r$ # 0 , 5

Then f o r

U2)\Q4.

Using t h e o p e r a t o r ( 3 . 8 . 2 2 ) w i t h $ i n s t e a d o f g , one g e t s

t h e following representation f o r A ( ~ , x , y ) : A(E,x,~) = Jn

E"

1a l ( x , y , t , S ) e i r $ ( X ' Y ' E

-1

")dS,

V ( x , y ) Ea

R with

E

so t h a t , i n f a c t , A ( E , . , . )

C

1

(u)u n i f o r m l y

with respect t o

E

E ( 0 , ~1. 0

Using r e p e a t e d l y t h e o p e r a t o r ( 3 . 8 . 2 2 ) a n d t h e p a r t i a l i n t e g r a t i o n , one g e t s t h e c o n c l u s i o n t h a t A ( € , . , . ) to

E C

(u)u n i f o r m l y

with r e s p e c t

8

E (O,Eo1.

E

m

Example 3 . 9 . 8 . Let n

=

n2 = n , U1 = U2 and $ ( x , y , A , S ) = < x - y , F > , so t h a t t h e c o r r e s -

ponding S i n g u l a r F o u r i e r I n t e g r a l O p e r a t o r i s j u s t a s i n g u l a r p e r t u r -

sv ( U ) i n t r o d u c e d i n D e f i n i t i o n 3 . 3 . 1 . 1,o One f i n d s e a s i l y i n t h i s case

b a t i o n of t h e c l a s s

A s a consequence, one g e t s a g a i n t h e p s e u d o l o c a l i t y p r o p e r t y o f

singular perturbations: s i n g supp(E"la(x,E,D)u)

5

s i n g supp u .

Example 3 . 9 . 9 . Let n

=

(3.9.24)

n2

=

n + l , U1 = U2

$(x,y,t,X,E)

=

=

{(x,t) E R

n+ 1

},

and l e t

< x - y , C > + ( t - T ) ( A 2+)t ,

where B i s symmetric p o s i t i v e d e f i n i t e nxn m a t r i x . One h a s :

Q (X,C) r$

=

{(X,t,y,T) E U

X

2 U, X-y+(t-T) ( A +)-*BC = 0).

Hence i n t r o d u c i n g rl = B t 5 , one f i n d s e a s i l y :

3. Singular Perturbations on Smooth Manifolds without Boundary

302

The phase f u n c t i o n ( 3 . 9 . 2 4 ) a p p e a r s w h i l e s o l v i n g t h e Cauchy problem f o r t h e s i n g u l a r l y p e r t u r b e d wave o p e r a t o r : 2 2 2 a t - € CBa X '

ax>+]

(see Example 3 . 9 . 4 ) , so t h a t t h e s i n g u l a r i t i e s o f t h e c o r r e s p o n d i n g

fundamental s o l u t i o n

a r e c o n t a i n e d i n Q $ . A c t u a l l y , f o r n = 2mc3, t h e

s i n g u l a r s u p p o r t of t h e fundamental s o l u t i o n i s e x a c t l y t h e whole Q$, a s Example 3.9.4

shows.

Now w e s h a l l i n t r o d u c e t h e c l a s s of

hyperbolic singular perturbations

and i n v e s t i g a t e t h e s i n g u l a r i t i e s of t h e k e r n e l d i s t r i b u t i o n s a s s o c i a t e d w i t h t h e s i n g u l a r F o u r i e r i n t e g r a l o p e r a t o r s which s o l v e t h e Cauchy problem f o r h y p e r b o l i c s i n g u l a r p e r t u r b a t i o n s . D e f i n i t i o n 3.9.10.

With a ( x , t , E , c , < ) E S v ( ~ y + l,) R:+'=

R

~

x

XR

t,+'

O p a i s s a i d t o be a

s t r i c t l y hyperboZic s i n g u l a r p e r t u r b a t i o n i f v 2 2 0 , v3 2 0 a r e i n t e g e r , a ( x , t , c , S , < ) i s polynomial i n 5

of degree v 2 + v 3 and t h e principa2 symbol

a ( x , t , ~ , S , < )s a t i s f i e s t h e c o n d i t i o n : 0 The zeros < . ( x , t , ~ , S ) ,1 5 j 6 v2+v3,0f t h e equation 1

(3.9.25)

a0(x,t,E,S, 5 . )

I

=

0

n+ 1

are r e a l and d i s t i n c t , v ( x , t ) E R +

,v

(E,c)

E R + x ( B ~ l\ o } ) .

H y p e r b o l i c s i n g u l a r p e r t u r b a t i o n s whose symbols a r e polynomial i n ( E , t ) , a r e o f s p e c i a l i n t e r e s t . W e s h a l l c o n s i d e r t h e Cauchy problem f o r t h e c o r r e s p o n d i n g h y p e r b o l i c d i f f e r e n t i a l s i n g u l a r p e r t u r b a t i o n s whose symbols have c o n s t a n t c o e f f i c i e n t s and c o i n c i d e w i t h t h e i r p r i n c i p a l symbols. L e t ao(t,S,<)

E P

b e s t r i c t l y h y p e r b o l i c . Without r e s t r i c t i o n of

g e n e r a l i t y , one c a n assume t h a t v 1 = 0 . I f v 3 = 0 , t h e n a. on

E

d o e s n o t depend

and becomes j u s t t h e c l a s s i c a l s t r i c t l y h y p e r b o l i c polynomial w i t h

3.9. The Fourier lntegral Singular Perturbations

303

c o n s t a n t c o e f f i c i e n t s ( s e e [ P e t r . 1 1 ) . T h e r e f o r e , w e s h a l l assume t h a t

v3 > 0 , v2 2 0. C o n s i d e r t h e Cauchy problem:

n+l

aO(E,Dx,Dt)u = 0 ,

( x , t ) E R+

(3.9.26)

where 6 . = 0 , 1 5 j < m, bm = 1 i s t h e Kronecker symbol and v E C m ( R n ) 1m 0 Let <.(E,<), 1 5 j 5 m = v 2 + v 3 be t h e z e r o s of t h e e q u a t i o n

.

1

(3.9.27)

aO(E,S,<.(E,S)) = 0. 3

homogeneous f u n c t i o n s o f ( E -1 , S ) o f d e g r e e 1 . 0 F u r t h e r m o r e , d e n o t i n g by < . ( c ) , 1 6 j 6 v2 t h e z e r o s of t h e r e d u c e d

Obviously,

<.(E,S) a r e 1

3

equation: (3.9.28)

0 <,(c)

0

0

ao(S,<.(S)) 3

=

0,

b e i n g homogeneous i n

6 o f d e g r e e 1 , one h a s t h e a s y m p t o t i c f o r m u l a e :

J

(3.9.29)

< . ( E , S ) = <0. ( E ) + O ( E 2~ )S, ~ f o r 3

3

and f o l l o w i n g i n e q u a l i t i e s f o r C , ( E , ~ ) 3

(3.9.30)

l?j(~,c)I

1 5 j 5 v2

5 C/SI,

where C > 0 i s some c o n s t a n t .

<,(E,E)

For t h e o t h e r v3 z e r o s

3

of

(3.9.27) t h e following i n e q u a l i t i e s

hold (3.9.31)

C-lE-I

5

IC.(E,S)

3

I

5 C(E

-2+1~/2)t,

v2

<

j 5 v 3'

where a g a i n C i s some c o n s t a n t . The s o l u t i o n of (3.9.32) where

u ( E , x , ~ )=

( 3 . 9 . 2 6 ) c a n b e w r i t t e n a s f o l l o w s ( s e e [Gel-Sh,

I

I

R

~

C

15jsv R ~ +v 2 3

a.

11):

3. Singular Perturbations on Smooth Manifolds without Boundary

304

One h a s f o r e a c h g i v e n t > 0

I: (A v ) ( E , x , ~ ) , l s j s v +v 'j 2 3 i s t h e S i n g u l a r F o u r i e r I n t e g r a l O p e r a t o r w i t h t h e p h a s e $ . and where A I' 1 t h e amplitude a , , i n t r o d u c e d above. 1 One f i n d s : =

U(E,X,t)

Q

(h,E;t)

E Rn

= { (x,y)

x Rn

,

(x-y)t

-1

=

}.

-vn<j(l,'l)

$3

rl=A-l 5 : Rn\{O}

L e t K , be t h e c l o s u r e o f t h e r a n g e of V 5 . ( 1 , n ) I '13 K = u K Denote j' (x,y,t)

K* =

Rn

x

Rn

x

R+

,

(x-y)/t

E

-f

nn

and l e t

K}.

so t h a t K*

U

=

(t).

Q

t>O,j

'j

A s a consequence of Theorem 3 . 9 . 7 t h e s i n g u l a r s u p p o r t of t h e k e r n e l

d i s t r i b u t i o n a s s o c i a t e d w i t h t h e s o l u t i o n u ( E , x , ~ )o f

(3.9.26),

is

For c l a s s i c a l h y p e r b o l i c o p e r a t o r s ( v 3 = O ) , t h e s e t K*

c o n t a i n e d i n K*.

i s n o t h i n g e l s e b u t t h e d u a l c o n e of r a y s f o r t h e c h a r a c t e r i s t i c cone of

(see f o r i n s t a n c e [Cour I]).

t h e o p e r a t o r a,(E)

I n t h i s c a s e dim K* = 2n,

i

w h i l e i n t h e g e n e r a l case of h y p e r b o l i c s i n g u l a r p e r t u r b a t i o n s onemay h a v e : dim K

*

=

2 n + l , a s it was t h e case i n Example 3 . 9 . 4 .

Using ( 3 . 9 . 3 2 ) one g e t s a f t e r t h e change o f v a r i a b l e s E C

+

5 the

f o l l o w i n g formula f o r t h e d i s t r i b u t i o n a l k e r n e l o f t h e o p e r a t o r , t h a t s o l v e s t h e Cauchy problem ( 3 . 9 . 2 6 ) (3.9.341

:

-1 im-1(2n)-nE1-n/n c a . (l,c)eiE ' j ' t ~ x ~ y ~ l ~ ' ) d ~ , R 1 c j s v +v 2 3 a r e g i v e n by ( 3 . 9 . 3 3 ) .

E(E,x-y,t)

=

where a . and 4 . 7 7 Let E . ( ( x - y ) / t ) be a s t a t i o n a r y p o i n t f o r 1 (3.9.35)

V5cj(l,S.) 1

=

(x-y)/t,

0.

1' l - e .

1 5 j 5 m.

Definition 3.9.11.

The p o i n t ( x , y , t ) E

K*

i s s a i d t o be non-focaZ i f t h e corresponding

s t a t i o n a r y p o i n t s 5 . ( ( x - y ) / t ) d e f i n e d by ( 3 . 9 . 3 5 ) are r e g u l a r , i . e . t h e 5

3.9. The Fourier Integral Singular Perturbations

matrices

D

2

305

5 ( 1 , c . ) are non-singular.

5 ,

I

Applying t h e s t a t i o n a r y p h a s e method t o t h e i n t e g r a l ( 3 . 9 . 3 4 ) a t e a c h n o n - f o c a l p o i n t ( x , y , t ) a n d assuming a 6 l i t i o n a l l y t h a t t h e c o r r e s ponding s t a t i o n a r y p o i n t s S . ( ( x - y ) / t ) a r e w e l l - d e f i n e d by ( 3 . 9 . 3 5 ) , one

I

g e t s t h e following assymptotic formula f o r E ( E , x - Y , ~ ) a t each such p o i n t (x,y,t): (3.9.36)

E(~,x-y,t)

N

v

+I,

-1 (2Tr)

i

-n/2

t

-1/2

E

1-n/2

~

q . ( ( x - y ) / t )e 1 6 j 6 v +v 7 2 3

itE-lAj

((x-y)/t)

A . ( 2 ), z E R n , i s t h e Legendre t r a n s f o r m of t h e f u n c t i o n 5 , ( 1 , c ) , 5 E 3 1 i . e . A.(z) and < . ( l , E ) a r e r e l a t e d by t h e Lagrange t r a n s f o r m a t i o n 1 3

where

(3.9.37)

A.(z) = < Z , S > + < . ( l , S ) , 3

1

v65 3 (1,5)

mn,

= -2,

and where

2

X . ( 5 . ) b e i n g t h e s i g n a t u r e o f t h e m a t r i x D E E < , ( 1, 5 , ) . I

1 ( 3 . 9 . 3 6 ) , E ( ~ , x - y , t )a t e a c h n o n - f o c a l p o i n t i s

3

A s a consequence o f

r a p i d l y o s c i l l a t i n g when For e a c h ( x , y , t ) multi-indice

c1

K

*

E +

0.

one f i n d s , a p p l y i n g P r o p o s i t i o n 3 . 8 . 4 ,

t h a t f o r each

t h e following asymptotic formula h o l d s :

2 2 For t h e wave o p e r a t o r a ( E . S . ~ ) = 5 -151 one h a s c1 = 151, c 2 = -1c1, 0 $ l ( t , x , y , l , S ) = < x - y , C > + t / < I , $ 2 ( t , x , y , 1 , 5 ) = < x - y , C > - t / E and t h e

s t a t i o n a r y p o i n t s a r e n o t w e l l - d e f i n e d by ( 3 . 9 . 3 5 ) so t h a t ( 3 . 9 . 3 6 ) c a n not hold. Notice t h a t i n t h i s case, i n f a c t , $

E

s(orlto)(UxU).

For t h e s i n g u l a r l y p e r t u r b e d wave o p e r a t o r w i t h t h e symbol ao(E,S,S)

c2

= E 2 5 2 -E 2 [ 5 1 2 + 1 ( s e e Example 3 . 9 . 4 )

= - E - ~ < E ~ >and

<

=

€-kc>,

t h e c o r r e s p o n d i n g s t a t i o n a r y p o i n t s a r e w e l l - d e f i n e d by

( 3 . 9 . 3 5 ) p r o v i d e d t h a t Ix-yl

and

one h a s :

< t . One f i n d s e a s i l y i n t h i s c a s e

3 . Singular Perturbations on Smooth Manifolds without Boundary

306

2 2 -2 I D s S 5 j ( l , c3. ) I = ( l - l x - y ] t

Hence, one has K* (x,y,t)

E

K

*

=

{ ( x , y , t ) , Ix-yI

l+n/2

< t} i n t h i s c a s e , e a c h p o i n t

i s n o n - f o c a l and t h e f o r m u l a e ( 3 . 9 . 3 6 ) - ( 3 . 9 . 3 8 ) a r e v a l i d

f o r t h e d i s t r i b u t i o n a l k e r n e l E ( ~ , x - y , t )of t h e corresponding o p e r a t o r s o l v i n g t h e Cauchy problem ( 3 . 9 . 2 6 ) f o r t h e S i n g u l a r l y P e r t u r b e d Wave 2 2 2 O p e r a t o r E at-€ A + l ( s e e ( 3 . 8 . 1 0 0 ) where t h e f i r s t t e r m i n t h e a s y m p t o t i c e x p a n s i o n f o r E ( E , x , t ) i s e x p l i c i t e l y w r i t t e n down). N o t i c e t h a t i n t h e c a s e of t h e s i n g u l a r l y p e r t u r b e d wave o p e r a t o r

a t- & 2 A x +1 -

2 2

t h e c o r r e s p o n d i n g p h a s e f u n c t i o n s a r e $ ( t , x , y , ~ , < )= 1 < x - y , S > + t ~' < E S > , G 2 ( t , x , y , & , S ) = < x - y , S > - - t ~ - ~ < & and c > $ j € S ( l ' o r l ) ( U x U) E

but

$jtr

S (o.l,o)

(UX

U).

Hence, it i s n a t u r a l t o i n t r o d u c e t h e f o l l o w i n g D e f i n i t i o n 3.9.12.

A Fourier I n t e g r a l S i n g u l a r Perturbation A'

$

i s s a i d t o be a Proper Fourier

I n t e g r a l S i n g u l a r Perturbation (PFISP) i f i n ( 3 . 9 . 2 ) t h e corresponding phase -1

function $ ( x , Y , E , t ) s a t i s f i e s t h e conditions: 1'

$

b u t $(X,Y,E 2"

i s real-valued, $ ( x , Y , E -1 , c ) E S ( l ' o ' l ) ( u l -1

, S ) ?! S(o,l,o)

(x,y,h,t!

variabzes ( A , 5 ) f o r

(U1

x

x

u2),

U2).

i s p o s i t i v e l y homogeneous o f degree 1 w i t h r e s p e c t t o

1 A I 2+ 15 I

2

+ and, moreover, $ s a t i s f i e s

vx,x,S$ f 0 , vy,h,<$ # 0 ,

v

(x,y) €

u1

x

u2, v

the condition

( h , t ) E IRn+l,

2 h + ( < I 2 = 1.

I f $ s a t i s f i e s l o , 2" t h e n i t i s c a l l e d a proper phase f u n c t i o n s

family. Remark 3.9.13. I f $ ( x , y , A , t ) i s a p r o p e r phase f u n c t i o n t h e n t h e f o l l o w i n g i n e q u a l i t i e s

3.10. Diffeornorphisms and Singular Perturbations

v

( x , y , C ) E U1

U2

X

x

R n and when 5 +

The F I S P ' s i n Example 3.9.4 D e f i n i t i o n 3.9.14.

(Vl.0,V3)

m,

V (x,y,X) E U 1 x U

307

2

x IRR\{O}.

are p r o p e r . n+l

with a ( x , t , E , c , c ) E S (R+ 1 , R y t l = nn x R ~ , + , op a is s a i d t o be a Proper S t r i c t l y Hyperbolic S i n g u l a r P e r t u r b a t i o n i f v3 > 0 is i n t e g e r a ( x , t , E , S , c ) i s polynomial i n 5 of degree v3 and t h e p r i n c i p a l symbol a o ( x , t , ~ , S , c )s a t i s f i e s t h e c o n d i t i o n :

The z e r o s 5 . ( x , t , c ) , 1 5 j 5 v3, of t h e equation: 7

ao(x,t,l,S,c)

=

0

a r e r e a l and d i s t i n c t , V ( x , t , < ) E

I < ,3 ( x . t . 6 ) I

b 6 > 0,

I R ~x IR

x

R;, and, moreover,

t.+ 1 5 j 5 v3, V ( x , t , c ) , where 6 i s some c o n s t a n t .

Remark 3.9.15. L e t a ( E , D ,Dt) b e a Proper S t r i c t l y Hyperbolic S i n g u l a r P e r t u r b a t i o n o x whose symbol a O ( E , E , L ) E S ( 0 ' 0 ' v 3 ) (R:+l ) i s polynomial i n (5,C.) and -1 homogeneous i n ( E ,c,c) of d e g r e e 0 . L e t E ( ~ , x - y , t ) b e t h e k e r n e l d i s t r i b u t i o n of t h e o p e r a t o r E ( t ) t h a t s o l v e s t h e c o r r e s p o n d i n g Cauchy Problem ( 3 . 9 . 2 6 ) ( w i t h v2 = 0 ) . * n c R X R n x I7 b e d e f i n e d , a s p r e v i o u s l y i n t h e c a s e v 2 b 0 , so x Y t,+ * t h a t s i n g supp E ( ~ , x - y , t ) E K Then f o r P r o p e r H y p e r b o l i c S i n g u l a r

Let K

.

P e r t u r b a t i o n s w i t h polynomial symbols a ( E , E , < ) always h a s dim K formula ( 3 . 9 . 3 4 )

0

*

= 2n+l

( w i t h v2

E S ( 0 f 0 ' v 3 ) ( m y + ' ) one

(see a l s o Example 3 . 9 . 4 ) . =

In t h i s case i n t h e

0 ) f o r E ( ~ , x - y , t ) a l l $ j , 1 5 j Iv 3 , a r e

p r o p e r phase f u n c t i o n s a n d , i t c a n b e shown t h a t t h e measure of t h e s e t of a l l f o c a l p o i n t s i n K

i s z e r o , so t h a t t h e s t a t i o n a r y p h a s e method c a n

be a p p l i e d , t h e a s y m p t o t i c f o r m u l a ( 3 . 9 . 3 6 ) b e i n g v a l i d a . e . i n K

*

.

3 . 1 0 . Diffeomorphisms of S i n g u l a r P e r t L r b a t i o n s on m a n i f o l d s W e s t a r t w i t h some a s y m p t o t i c f o r m u l a f o r S i n g u l a r P e r t u r b a t i o n s

which p l a y s t h e r o l e comparable w i t h L e i b n i t z ' f o r m u l a f o r D i f f e r e n t i a l Operators. Theorem 3 . 1 0 . 1 .

Let

f

E c;(u), g E C m ( U ) , g being real-vaZued and dg ( x ) # 0 , V x E supp f .

L e t a ( x , E , ~ )E o p s;,,(u), N 2 o one has:

a ( x , E , ~ ):

m

c o ( u ) + c m ( u ) . Then f o r each i n t e g e r

3. Singular Perturbations on Smooth Manifolds without Bounda y

308

where h (x,Y) g

= g ( x ) - g ( y l -

and where t h e remainder s a t i s f i e s on any compact s e t K c

c

depending onZy on

N,K

(3.10.2)

N

u (with a constant

and KJ the foZZowing e s t i m a t e

/ R N ( x , ~ , p )5 / C

N.K

E

-U1

p

-N+[N/2]+V2

<EP>

v3

.

Proof. Without r e s t r i c t i o n of g e n e r a l i t y , one can assume t h a t u z + v 3 < -n,

as t h e

c a s e v2+u3 2 -n can b e reduced t o t h i s one u s i n g t h e i n t e g r a t i o n by p a r t . Denote

With v2+u3 < -n,

Q ( x , ~ , p )c a n b e w r i t t e n a s a d o u b l e i n t e g r a l and a f t e r

t h e change o f v a r i a b l e s p 5 (3.10.4)

+

5 one g e t s f o r

1 1

= (2.ir)-"pn

Q(x,E,P)

R

Q(x,E,P)

t h e f o l l o w i n g formula:

q ( x , y , ~ , p , 6 ) e ~ WdS, ~ ~ ( ~ ' ~ ~ ~ )

U

where

The p o i n t (3.10.6)

M(x) = ( x , V g ( x ) ) E lRn

Y

X

lRn

5

i s t h e o n l y s t a t i o n a r y p o i n t o f t h e phase f u n c t i o n il,

: R n x IRn y 6 + R . W e s h a l l u s e a p a r t i t i o n of u n i t y i n 6 - v a r i a b l e and a f t e r w a r d s a p p l y

Theorem 3 . 8 . 1 0 i n o r d e r t o f i n d a n a s y m p t o t i c e x p a n s i o n f o r Q ( x , E , P ) by ( 3 . 1 0 . 4 ) .

( 3 . 1 0 . 5 ) when p

+

F i r s t w e show t h a t t h e c o n t r i b u t i o n i n t o Q ( x , E , P ) o v e r t h e areas for p

-f

-,

{[c/

< 6, y

given

-.

E U} and

> R, y

from t h e i n t e g r a t i o n

E Ul i s of o r d e r

-m

O(p

)

p r o v i d e d t h a t 6 i s s u f f i c i e n t l y s m a l l and R i s s u f f i c i e n t l y

l a r g e . S i n c e dg # 0 on supp f , one c a n choose 6 ' > 0 , R ' > 0 , such t h a t 6 ' 5 IVg(y)l 5 R ' ,

V y C supp f . L e t 6 < 6 ' .

x 1 ( 5 ) : 0 f o r 151 > 6 , x , ( S )

!

1 for

and l e t x , ( S )

151 < 6 / 2 . Denote

E C;(lRn)

,

3.10. Diffeomorphisms and Singular Perturbations

Q 1 ( x , ~ , p )=

J 1 mn u

309

qx,eip'dydS.

L e t K C U b e any g i v e n compact. W e s h a l l show t h a t f o r any m u l t i i n d i c e s

a,B and any i n t e g e r N 2 0 h o l d s :

Denote

Given t h e c h o i c e o f t h e c u t - o f f

x l ( S ) , one o b v i o u s l y h a s :

function

IVg(y)-SI 2 c o > 0 , V ( x , y , S ) E K

X

X

supp

xl.

Therefore, the

c o e f f i c i e n t s o f t h e o p e r a t o r L ( y , C , a ) a r e smooth and bounded on Y U X supp x . F u r t h e r m o r e , s i n c e a ( x , E , S ) E sv (U) one o b v i o u s l y h a s f o r 1 1,o any a : -V

(3.10.9)

lD;q(x,y,E,P,S)\

5 C

a,K

E

l



V2

<EPS>

v3

,

V ( x , y , ~ , p , S )E K x

X

(O,E

0

1

x

l R + x lRn,

depends o n l y on a and K .

where t h e c o n s t a n t C U,K

Using t h e f o r m u l a

L ( Y , s , a y ) ei P J l

= ipeiP',

and d e n o t i n g by Lt(y,S,a t

L (y,c,a

Y

)

=

Y

t h e f o r m a l a d j o i n t o p e r a t o r of L ,

-
~(Y,s)>,

one g e t s , a f t e r one i n t e g r a t i o n by p a r t f o r Q ( x , E , P )

1

AS a consequence of

estimate :

t h e following formula:

( 3 . 1 0 . 9 ) , one g e t s f o r Q l ( x , & , p ) t h e f o l l o w i n g

3 . Singular Perturbations on Smooth Manifolds without Boundary

310

A f t e r N i n t e g r a t i o n s b y p a r t u s i n g t h e o p e r a t o r L ( y , S , a ) , one f i n d s Y -W

I

IQ~(X,E,P)

=

3, IpV2-N<Ep>v

'N,K€

'd (x,E,P) E K

X

( 0 . ~ ~X 1R + , p 2 1.

D i f f e r e n t i a t i n g Q ( x , E , ~ )w i t h r e s p e c t t o x and p, one g e t s t h e same k i n d

1

of i n t e g r a l s , so t h a t ( 3 . 1 0 . 7 )

for p

-f

-,

u n i f o r m l y w i t h r e s p e c t t o x E K C C U and

Now, l e t R > R ' 5 IVyg(y)

x2

0 for

i s proved. N o t i c e , t h a t

151

5 R,

x2

Z

1,

V y 6 supp

E

f and l e t

vl

X,E,P) Q1( E [ O , E 1. E

=

O(p

-m

)

0

x,(S)

E Cm(lRn)

,

1 for 151 2 2 R , b e t h e c o r r e s p o n d i n g c u t - o f f

f u n c t i o n . Denote by Q ~ ( X , E , P )t h e f u n c t i o n

Q2(x,~,p) =

IqX2eiP'dydc, Rn U

t h e d o u b l e i n t e g r a l i n t h e r i g h t hand s i d e b e i n g c o n v e r g e n t s i n c e v2+v3

-n.

Using a g a i n t h e o p e r a t o r L ( y , S , a ) i n t r o d u c e d by ( 3 . 1 0 . 8 ) , Y Q2(x,€,p) i n t h e form:

S i n c e Vg(y)-C

# 0,

V (y,C) E

V

supp

x2

(y,S)

1 .( y , < )

1

E

X

c s(o,-l,o) l,o (U)

x

supp

x2,

one h a s i n f a c t ,

one c a n r e w r i t e

lv g(v)-cj

w i t h s o m e p o s i t i v e c o n s t a n t C , so t h a t

Y

5

c,

a n d , t h e r e f o r e , Q2(x,€,p) c a n b e e s t i m a t e d , a s

follows:

-vl v2-1 I Q ~ ( x , E , P )5~ CE

P

R e i t e r a t i n g t h e p a r t i a l i n t e g r a t i o n , one g e t f o r Q 2 ( x , ~ , p ) t h e s a m e

estimate a s f o r Q ( x , E , P ) . D e r i v a t i v e s of Q2 w i t h respect t o x and p b e i n g 1 e x p r e s s e d by i n t e g r a l s o f t h e same k i n d , a s Q, i t s e l f , one g e t s f i n a l l y ,

f o r V (x,E,P) E K

x

( 0 , 1~ X R + , p t 1, and any i n t e g e r N t 0.

N o t i c e t h a t E V ' D a D B i ( x , E , P ) = O(p X P 2 r e s p e c t t o x E K C C U, and E E [ O , E 1 . 0

Let

x3(S)

=

1 - x 1 ( S ) - x 2 ( S ) and l e t

-a

)

for p

+

m,

uniformly with

3.10. Diffeomorphisms and Singular Perturbations

31 1

I n o r d e r t o b e a b l e t o a p p l y t h e s t a t i o n a r y p h a s e method t o t h e i n t e g r a l ( 3 . 1 0 . 1 1 ) one h a s t o check t h a t t h e o n l y s t a t i o n a r y p o i n t M(x) = ( x , V g ( x ) ) , M(x) E supp qx3, of t h e p h a s e f u n c t i o n $,

i s regular.

Indeed one f i n d s :

so t h a t a t t h e s t a t i o n a r y p o i n t M(x) = ( x , V g ( x ) ) , one f i n d s :

I n a neighbourhood of t h e s t a t i o n a r y p o i n t M(x) one h a s :

I n t r o d u c i n g t h e v a r i a b l e s z = y-x, 2 =

1

0

n

=

S - V g ( x ) - f g x x ( x ) ( y - X I , one f i n d s t h a t

, -Id

-Id,

0

1

so t h a t

Applying t o ( 3 . 1 0 . 1 1 ) t h e s t a t i o n a r y phase method and t a k i n g i n t o a c c o u n t t h e estimates f o r Q . ( x , E , P ) , j 1 e x p a n s i o n f o r Q ( x , E , P ) when p + m.

=

1 , 2 , one g e t s a n a s y m p t o t i c

I t r e m a i n s t o compute t h e c o e f f i c i e n t s of t h i s e x p a n s i o n . For d o i n g

t h a t , w e n o t i c e f i r s t t h a t f o r g ( x ) = < x , n > w i t h I? E lRn and f o r any i n t e g e r N > 0 one h a s :

where t h e remainder i s g i v e n by t h e formula

a given vector

3. Singular Perturbations on Smooth Manifolds without Boundary

312

m

w i t h some 8 E ( 0 , l ) and E l E C O ( U ) . ( U ), one g e t s f o r R N f t h e f o l l o w i n g and a E Sv 1 ,o

E C;(U)

S i n c e Daf e st i m a t e :

-V

(3.10.13)

IRN(x,E,p,rl,Dx)f(x)I 5 C N ( f l ) E

w i t h some c o n s t a n t C ( f l ) > 0 N S e t t i n g n = Vg(x), f l (x)

=

V

-N

V <EP>

P'

3

f ( x ) ei p h 9 ( x ' y ) and a p p l y i n g t h e f o r m u l a

(3.10.12) t o t h e i d e n t i t y e-ipg ( x ) a( x , E , D X ) ( ei W f )

~

ip iphg(x,y)) -ipg (x) f(y)e a ( y , E , D 1 (e Y=x Y

I

= e

one g e t s t h e f o r m u l a ( 3 . 1 0 . 1 ) . The e s t i m a t e ( 3 . 1 0 . 2 ) f o r t h e r e m a i n d e r R N ( x , E , p ) f o l l o w s from

I

grows a t most l i k e ( 3 . 1 0 . 1 3 ) and t h e f a c t t h a t t h e c o n s t a n t C ( f e i p h l N Y=x P "/21 f o r p + m, g i v e n t h a t t h e f u n c t i o n h ( x , y ) h a s a z e r o o f o r d e r 2 when y-x

+

9

I

0.

C o r o l l a r y 3.10.2. Taking N = 1 i n ( 3 . 1 0 . 1 ) , ( 3 . 1 0 . 2 ) , one g e t s t h e f o l l o w i n g aSymptOtiC formula : ( 3 . 1 0 . 1 4 ) e - i p g ( x )a ( x , E , D

) (f

( x ) ei p g ( x ) )

=

f ( x ) a(x,E,pvxg(x))

-

2 2 - i { + ( p / 2 ) T r (DE5a( x ,E , 5 ) Dxxg ( x ) ) }

5

-v

+

O(E

1p"2-2<Ep>v

C o r o l l a r y 3.10.3. -1 . Taking p = E i n (3.10.1),

3

).

(3.10.21,

one f i n d s

iE-lg(X) ( 3 . 1 0 . 1 5 ) e-iE - l g ( x ) a ( x , E , D x ) ( f ( x ) e ) =

N-[N/2]-V

+

O(E

-V2

1

)

u n i f o r m l y w i t h r e s p e c t t o x i n any compact s e t K C C U and w i t h r e s p e c t t o

3.10. Diffeornorphisnzs and Singular Perturbations

313

m

f in any bounded set in

Co(U).

= v, In particular, if a E Kv(U) and ar = C a . ord a . = v(1), v(') j z o I' 1 is its graduate symbol, then the following asymptotic formula holds:

where, as usual a(a)= aaa, the asymptotic relation (3.10.16) being uniform

5

with respect to x I n any compact set K c c

U,

with respect to f in any

m

given bounded set in C ( U ) . 0 We are going to apply Theorem 3.10.1 in order to show how the symbols of singular perturbations are transformed under the coordinates diffeomorphisms. This will enable us to introduce singular perturbations on smooth manifolds without boundary. Theorem 3.10.1 will also be applied for giving a different proof of the formulae for the symbols of product of properly supported singular perturbations.

We shall need the following useful auxiliary result. Theorem 3.10.4. (j)

(u), j b 0 w i t h v ( 1 ) 1,o L e t ( O , E ~ I 3 E + a(x,~,c)E cm(ux

Let a E

1

E Sv

=

(j) (vl,v2 ,v3), ViJ) c

::

R~ ) be continuous w i t h r e s p e c t t o

E ( 0 . ~ ~and 1 assume t h a t t h e r e e x i s t c o n s t a n t s

c

a. 5

-a

andp

for

=

j

+

m.

u ( a , B ) such

that

f o r any x E U,

E

E (o,c01,

5 E R ~ .

Further, assume t h a t t h e r e e x i s t u ( N )

J-

-m

for

N +

and such t h a t on every

compact s u b s e t K cc u one has:

uniformly w i t h r e s p e c t t o x on any compact s u b s e t K

a U and

6

E ( 0 . ~ ~ 1

3. Singular Perturbations on Smooth Manifolds without Boundary

314 Proof. -

(01

According t o Theorem 3 . 6 . 2 ,

t h e r e e x i s t s a symbol b ( x , ~ , S )C Sv

1# O

(U) such

that

(N) (b(x,E,S) -

Z O<=j
and such t h a t ( 3 . 1 0 . 1 9 )

a . ( x , ~ , S ) C)

'

S"

1 to

(U)

h o l d s w i t h b i n s t e a d of a . m

I t r e m a i n s t o show t h a t r = ( a - b ) E sy,o(U), where

v

m

=

(v1,-m,v3). The i n e q u a l i t i e s ( 3 . 1 0 . 1 8 ) imply:

(3.10.20)

Ir(x,E,S)

I

-V =

< 'N,KE

V

l < E ; > - N < ~ S >', V x E

K C C U, V E

E (O,E~],

V N Z O . W e have t o check t h a t (3.10.20) h o l d s f o r t h e d e r i v a t i v e s B w D D r ( x , E , S ) . F o r d o i n g t h a t , o n e may u s e t h e Kolmoqorov i n e q u a l i t y

x s

z

SUP

/ a l = l xEK 1

2 IDa43(x)( 5 C s u p I43(x)( Z 1Da@(x)I, xEK2 1452

where K , a r e compact s e t s , K CC i n t K 2' 7 1 O b v i o u s l y , i n o r d e r t o show t h e v a l i d i t y of t h i s l a s t i n e q u a l i t y i t s u f f i c e s t o c o n s i d e r t h e one d i m e n s i o n a l case. Using t h e T a y l o r formula i n t h e form:

43 ' ( x )

= 'clp

26

6

( x + 6 )-43 (x-6) 1 + -(a" ( y + 6 )- $ ' I 4

and c h o o s i n g 6 = ( 2 s u p I @ ( x )I/sup

I

(y-6) )

,

I

I$"(x) ) ' ,

one g e t s t h e Kolmogorov

i n e q u a l i t y above. Indeed, a p p l y i n g t h i s l a s t i n e q u a l i t y t o t h e f u n c t i o n s @c(X,E,T))

with K

r(X,&,c+rl)

1 = K X { O } , K 2 = K 6 X Irl g e t s , using (3.10.17), (3.10.20)

/ 1111

5 61, K6 = { x

I

d i s t ( x , K ) 6 6}, one

3.10. Diffeornorphisrns and Singular Perturbations Now w e g i v e a d i f f e r e n t p r o o f of Theorem 3.7.6,

315

which i s b a s e d on t h e

s t a t i o n a r y phase method. Theorem 3.10.5.

L e t a ( x , ~ , ~: )c;(u)

+

c;(u),

a(x,~,<E )

m

b(x,E,D) :

c;(u)

+

C o ( u ) , b ( x , E , c ) E S'

1 .o

Sv

(u), and

1 PO ( U ) be p r o p e r l y supported

s i n g u l a r p e r t u r b a t i o n s . Then t h e i r product c ( x , E , D )

=

a ( x , ~ , D 0) b ( x , ~ , D ) ,

i s a properZy supported singuZar p e r t u r b a t i o n , whose symbol E Sv+' ( u ) has t h e f o l l o w i n g asymptotic expansion: 1,o

c(x,E,S)

t h e asymptotic r e l a t i o n (3.10.21) being understood in t h e f o l l o w i n g sense

u n i f o m l y w i t h r e s p e c t t o x i n any compact s u b s e t K C C to

E

E

U and w i t h r e s p e c t

(O,Eo1.

Proof. Let u

m

E C o ( U ) . S i n c e a ( x , ~ , D ) :C;(U)

Applying (3.10.1) w i t h p

=

151,

+

C E ( U ) , one h a s

g ( x ) = <x,w> w

=


i n order t o

I

i l ~<x,w>

compute t h e a s y m p t o t i c e x p a n s i o n o f a ( x , 6 , D ) ( b ( x , E , < ) e -f

m,

one g e t s t h e formulae

c ( x , E , ~ )of

)

when

(3.10.21), (3.10.20) f o r t h e symbol

the singular perturbation c = a

o

b . Now Theorem 3.10.4 i m p l i e s

E S"+'(U). I 1P O For some classes of s i n g u l a r p e r t u r b a t i o n s t h e c o m p o s i t i o n f o r m u l a C(X,E,S)

N

(3.10.21) i s v a l i d w i t h a remainder which i s of o r d e r O ( E < E S > - ~ )when 6 ' 0 ,

5'm.

D e f i n i t i o n 3.10.6.

A singular perturbation A o p J~

1 ,o

(u) i f

v

E op

= (vl,0,v3) and

( u ) i s s a i d t o b e in t h e c l a s s 1,o i t s symbol i s a cm-function of t h e form

Sv

a ( x , s , E ( ) , which s a t i s f i e s t h e i n e q u a l i t i e s

3 . Singular Perturbations on Smooth Manifolds without Boundary

316

and f o r V ( x , E , ~ ) E K

x

( 0 . ~ ~x 1iRn

where K is any compact s e t in

U.

Theorem 3.10.7.

L e t AE

=

Op a E Jv ( u ) , BE = Op b E Sy,o(u) be properly supported. Then 1r 0 C = AE o BE is a p r o p e r l y supported singuZar p e r t u r b a t i o n

t h e i r composition

i n OP Sv+' ( U l whose symbol c(x,E,~)has t h e f o l l o w i n g asymptotic expansion: 1,O

where t h e asymptotic r e l a t i o n (3.10.24) i s i n t e r p r e t e d in t h e f o l l o w i n g sense: f o r each N

= O,l,

... t h e

remainder

moreover,

Proof. ~

m

m

Cn

Let u E C (K). Since A

: CO(U)

0

-f

C

(U)

, one has

so that (3.10.1), (3.10.2) yield

+ (271)-n

where r

N

i rN(x,E,5)ei<X"> mn

u ( 5 )dS ,

satisfies the inequality IrN(X,E,S)I 5

N-(V +Ul) 1

cN , K ~

<5'

V +U

V

+!.I

v

-N

, V x E K ,

2<E5>

E E

(0,E0It v 5 E R n ,

I

as a consequence of (3.10.2), (3.10.23). Corollary 3.10.8. If a(x,~,D)E Op J"

1 to

a(x,E,D)

o

(U)

with v = (v1,0,v2),b ( x , E , D )

E

Op

b(x,E,D) = Op(a(x,E,~)b(x,E,~)) + ERE

(u) sV 1r o

then

3.20. Diffeornovhisrns and Singular Perturbations where R

E Op sl,o(U) (Y) with y = v+~1-(0,0,1), y

317

< v+u

The following result shows how the symbols of singular perturbations are transformed after a diffeomorphic change of variables.

,

lRn be bounded domains and let x : U Y diffeomorphism. Let AE = Op a, a ( x , ~ , 5 )E Sv ( u ) and let Let U

C

IR:

V

C

-

m

+

V be a C

1#O

Theorem 3.10.9.

The operator AE

m

:

Co(V)

-f

Cm(V) defined by (3.10.27) is a singular

X

perturbation whose symbol a (x,E,S) E Sv (v) has the following asymptotic X 1.0 expansion,

where (3.10.29) qa t X

and where

(x,c) = Da(,i<X(Y)-X(x)-X'(x) (y-x), 5 > Y

t x' is the transpose of

the Jacobian matrix

x i(x).

Proof. It suffices to consider properly supported singular perturbations. Let m

u E Co(V) and denote y

= x(x), v(x) = ( u

X) (x). One finds

where we have denoted (3.10.30) b (x,E,<)= e-i<X(x)rS>a(x,E,Dx)(ei<x(x)rS>).

X

Notice that Vx<x(x),S> = tx'(x)5 # 0, V (x,S) E U Hence, formula (3.10.1) with p = 151, g(x)

=

x

(lRn\{O)).

<x(x),w>, w

=

5 / 1 5 ] yields

(3.10.28), (3.10.29), while Theorem 3.10.4 implies that m

a (x,E,5) E sy,o(V). Therefore, any C -diffeomorphism

X

bijection

x,

:

x

:

U

+

V induces a

Sy,o(U) ++ sy,o(V) according to the formula (3.10.27), the

corresponding symbols being transformed according to the formulae (3.10.281, (3.10.29).

I

3. Singular Perturbations on Smooth Manifolds without Bounda y

318

C o r o l l a r y 3.10.10.

E S"(U)

L e t AE = O p a , a ( x , E , < )

of a ( x , € , D x ) . L e t

and l e t a o ( x , E , E ) b e t h e p r i n c i p a l symbol m

x

: U

V b e a C -diffeomorphism and l e t a X , O ( x , ~ , Eb)e

-f

t h e p r i n c i p a l symbol o f t h e t r a n s f o r m e d s i n g u l a r p e r t u r b a t i o n d e f i n e d by ( 3 . 1 0 . 2 7 ) . Then ( 3 . 1 0 . 2 8 ) ,

( 3 . 1 0 . 2 9 ) imply:

C o r o l l a r y 3.10.11. Let

m

x

: Rn

Rn b e a C -diffeomorphism which r e d u c e s t o a l i n e a r map

-t

o u t s i d e of some b a l l i n Rn Then f o r V s

E

R3

.

t h e diffeomorphism

x

i n d u c e s a b i j e c t i o n of each s p a c e m

H (S)

( R n ) o n t o i t s e l f . I n d e e d , u n d e r t h e C -diffeomorphism t h e i d e n t i t y

o p e r a t o r i s t r a n s f o r m e d i n t o a n o p e r a t o r J = op j , whose symbol j ( x , < ) E So ( E n ) and i s c o n s t a n t f o r 1x1 s u f f i c i e n t l y l a r g e . T h e r e f o r e , 1,o w i t h v ( x ) = (u o x ) ( x ) Theorem 3 . 4 . 1 i m p l i e s t h a t v E H ( I R n ) whenever

.

u E Hs(Rn)

Hence, t h i s p r o v e s Lemma 2 . 6 . 1 ,

w h i l e Lemma 2.6.2

is a direct

consequence o f Theorem 3 . 4 . 1 . Remark 3.10.12. m

x

A C -diffeomorphism

i s o m o r p h i c mappings: (3.10.32)

x

*

: U

x*$

V w i t h U and V open sets i n R n i n d u c e s * -1 m = $ 0 x , x*$ = ( x ) $, V $ E C (U), V IJJ E C m ( u ) , -f

m

m

: Co(V)

+

Co(U),

x*

: C m ( U ) + Co0(V)

m

a n d , by d u a l i t y between C ( V ) and D'(V) t h e isomorphism 0 (3.10.33)

x,

:

D'(U)

*

D'(V).

F u r t h e r m o r e , it g i v e s r i s e t o a map t o t h e formula ( 3 . 1 0 . 2 7 ) ,

i.e.

( 3 . 1 0 . 3 4 ) X,a

x -1 ,

= (a

o

X)

D

x,

:

op

V a(x,E,Dx)

S y , o ( u ) * Op S"

1,o

E

(V) according

Op . q y , , ( U ) ,

a s w e l l a s a symbol homomorphism (which i s a n isomorphism u p t o t h e smoothing symbols o f o r d e r (3.10.35)

-m):

(X,a) ( y , ~ , n )= a ( y , ~ , n ) , X

where a ( y , ~ , q )i s w e l l d e f i n e d a s y m p t o t i c a l l y by ( 3 . 1 0 . 2 8 )

X

smoothing symbol of o r d e r

-m.

up t o a

3.10. Diffeomorphisms and Singular Perturbations

319

F u r t h e r m o r e , t h e p r i n c i p a l symbols a r e t r a n s f o r m e d a c c o r d i n g t o t h e formula ( 3 . 1 0 . 3 1 ) , which c a n b e r e w r i t t e n a s f o l l o w s :

(x,ao) ( y , ~ , r ~ = )a o ( x , E , S ) l x = x - ' ( y ) . 5 = tX ' ( X ) l l .

(3.10.36)

m

W e c a n now d e f i n e s i n g u l a r p e r t u r b a t i o n s on a C

paracompact m a n i f o l d M.

D e f i n i t i o n 3.10.13. m

A f a m i l y of operators A €

i s s a i d t o be a s i n g u l a r p e r t u r -

: c ~ ( M ) -f c-(M)

b a t i o n i n t h e c l a s s o p Sv

1 to

and a cm diffeomorphism x

i f f o r any coordinate neighbourhood x c

(MI,

:

x

+

u c n n , t h e f a m i l y o f operators

M

A€

U

d e f i n e d by t h e commutative diagram

i s a singular p e r t u r b a t i o n i n op Sv

1to

I n o t h e r words A' -1 .

A E = (A'

o

x)

x

0

: C;(M)

+ Cm(M)

: X + U,

i s i n t h e c l a s s Op

1 s a s i n g u l a r p e r t u r b a t i o n i n Op

U consequence of Theorem 3 . 1 0 . 9 , :A

x

(u).

i s i n Op

sv

sv 1,o

sv (M), 1,o

if

(U). A s a

( U ) f o r any C -diffeomorphism

1 PO so t h a t D e f i n i t i o n 3.10.13 i s c o r r e c t , i . e . t h e c o n c e p t of

s i n g u l a r p e r t u r b a t i o n on m a n i f o l d d o e s n o t depend of s p e c i f i c c h o i c e of neighbourhoods and d i f f e o m o r p h i s m s . Furthermore, given t h e t r a n s f o r m a t i o n r u l e (3.10.36) f o r t h e p r i n c i p a l

*

symbols, t h e y a r e w e l l d e f i n e d f u n c t i o n s on t h e c o t a n g e n t b u n d l e T ( M ) . I n t h e same way one d e f i n e s s i n g u l a r p e r t u r b a t i o n s classes O p S " ( M ) and Op K v ( M ) ,

u s i n g t h e d i a g r a m above and t h e i n v a r i a n c e of the classes

S"(U), K V ( U ) (see D e f i n i t i o n s 3.3.2 and 3 . 6 . 4 ) w i t h r e s p e c t t o t h e

Cm-

diffeomorphisms. However,

one c a n g i v e an i n d e p e n d e n t i n t r i n s i c d e f i n i t i o n o f s i n g u l a r

p e r t u r b a t i o n s on a smooth m a n i f o l d . D e f i n i t i o n 3.10.14.

Let

M

be a

Cm

paracompact manifold. A f a m i l y of operators m

E

-f

: C;(M)

-f

c

(M)

i s s a i d t o be a s i n g u l a r p e r t u r b a t i o n on

i f t h e f o l l o u i n g c o n d i t i o n s are s a t i s f i e d : 1

'.

There e x i s t s a sequence k

,

7

J.

--, f o r

j

+

m,

such t h a t

M

3. Singular Perturbations on Smooth Manifolds without Bounda y

320

2O. E

There e x i s t s a sequence s . 3

+

-m

for j

+ m,

such t h a t f o r each given

E ( 0 . ~ ~one 1 has:

( 3 . 1 0 . 3 9 ) e- ip g (x)AE ( f ( x )e i p g ( x ) )

f o r v f E c;(M), v g E

-

S .

’

C b E ( f , g ) p I, p + m , j>o c ~ ( M ) , dg f 0 on supp f , g being r e a l

3O. There e x i s t s a sequence m . C 3

for

-m

j +

a,

valued.

such t h a t

E c ~ ( M ) , V g E c - ( M ) , dg # 0 on supp f , g being r e a l vaZued. Furthermore, t h e order v = ( v l , v 2 , v 3 ) of t h e singuZar p e r t u r b a t i o n is t h e l e a s t v E m3 such t h a t

for V

f

> v (’I dEf

(3.10.41) v

(k.,m.-k.,s.-m.+k.), V j 1 3 3 3 3 1

=

0,1,

...

Furthermore, t h e f u n c t i o n ( 3 . 1 0 . 4 2 ) uA ( f , g ) = e

-ig(x)A (feig(x)

E

) #

E

which is asymptoticalZy represented by (3.10.39) when I V g ( x ) [ each g i v e n

+

m

for

i s s a i d t o b e t h e symbol o f t h e operator A ~ . E R n and i n t r o d u c i n g t h e f u n c t i o n

E,

Taking g ( x ) = < x , < > , 5

one c a n r e w r i t e t h e o p e r a t o r A (3.10.44)

m

: CO(M) + C

m

(M)

i n t h e following fashion:

4

( ~ ~ u ) ( =x )( 2 7 ~ ) -1~ e IRn

where i t i s u n d e r s t o o d t h a t supp u b e l o n g s t o some p a t h V

x

: V

-f

C

U i s a diffeomorphism o f V c M o n t o some open s e t U

Furthermore,

homogeneous f u n c t i o n s o f

f u n c t i o n s on

M

X

Taking g ( x ) a

<

of d e g r e e s .

a j o b e i n g smooth

{Rn\{O}} which depend c o n t i n u o u s l y of E E ( 0 . ~ ~ 1 . = E-~<x,<>,

(f,E,S) AE

Rn.

( 3 . 1 0 . 3 9 ) becomes

with aE(f,:) 3

M and C

w i t h 5 E Rn\{O} g i v e n , and i n t r o d u c i n g

= 0A ( f , E - l S ) ,

3.10. Diffeomorphisms aird Singular Perturbations

32 1

( 3 . 1 0 . 4 0 ) becomes: (3.10.46)

uA (f,E,C)

-

Z a.(f,E,S),

0

E +

j>o

E

-1 , < ) f u n c t i o n s o f d e g r e e m . which where a . ( x , E , S ) are homogeneous i n ( E 3 3' a r e smooth f o r x E M , E Rn\{O}, f E R + .

I f v ( O ) = ( k , m -k ,s -m +k ) s a t i s f i e s t h e c o n d i t i o n 0 0 0 0 0 0 (3.10.47)

v(O) t "(1)

=

(k.,m.-k. s.-m.+k.), 3 3 3 ' 3 3 3

j = 1,2

,...,

t h e n t h e zero c o e f f i c i e n t on t h e r i g h t hand s i d e o f ( 3 . 1 0 . 4 6 ) i s s a i d t o be t h e p r i n c i p a l symbol o f A (3.10.48)

If

and i s denoted by o

AE.O'

U , ~ , ~ ( ~ , E , C )= a O ( f , E , 5 ) .

(3.10.47)

i s well defined,

i s s a t i s f i e d i . e . t h e p r i n c i p a l symbol o f

t h e n A~ E Op Sv

(0) (M)

.

I f , moreover, one has (3.10.50)

v ( O ) t v ( l ) t...> v ( 1 ) t

.. . ,

One h a s t o u s e Theorem 3 . 1 0 . 1 i n o r d e r t o show t h a t e a c h s i n g u l a r i n t h e sense of D e f i n i t i o n 3.10.13 i s a s i n g u l a r

perturbation A

p e r t u r b a t i o n i n t h e sense of D e f i n i t i o n 3.10.14 a l s o . W e s h a l l n o t e l a b o r a t e on t h i s p o i n t and l e a v e t h e d e t a i l s t o t h e r e a d e r . Example 3 . 1 0 . 1 5 . L e t R 1 denote t h e u n i t c i r c l e ,

nl

=

{eie E C

1

1

101 5 n } .

Define t h e

operator

n+

:

c;(nl)

by t h e formula (3.10.51)

( I I + ~ )( 8 )

1 lim 211

= -

6++0

and l e t

n-

= Id

71

Jr1-e

i (e-y+iS)

-1

1

u(y)dy

-Tr

-.'II

Consider t h e family o f o p e r a t o r s E

-f

A

:

Cm(nl)

+

Cm(Ql),

3. Singular Perturbations on Smooth Manifolds without Bounda y

322 (3.10.52)

A

=

Id

+

+

-

E D ~ (- ~Il ) ,

where I d i s t h e i d e n t i t y o p e r a t o r and D

e

=

-ia/ae.

The F o u r i e r series e x p a n s i o n

e s t a b l i s h e s a n isomorphism of C m ( R ) o n t o t h e s p a c e s of r a p i d l y d e c a y i n g 1 s e q u e n c e s , {u 1 -m 0 s u c h t h a t lun\< CNRs2 N of t h e s p a c e D(Tm), m b 1 i n t e g e r ) . U s i n g t h e F o u r i e r s e r i e s e x p a n s i o n o p e r a t o r F o n e c a n r e w r i t e (3.10.52) as f o l l o w s : (3.10.54)

AEu = F - l ( l + E / n l ) F u .

We s h a l l see t h a t AE i s a s i n g u l a r p e r t u r b a t i o n i n t h e c l a s s Op S ( o r o ' l ) ( R 1 ) and even i n O p K

( O r O r l ) (R1).

L e t U C R1 b e any p a t h

U

22

( i t s u f f i c e s t o t a k e f o r i n s t a n c e (-n/2,n/2)

E

m

m

x E

x

(R 1 , : 1 on some p a t h 0 1 U , one c a n r e w r i t e t h e o p e r a t o r AE a s f o l l o w s ( w e d e n o t e 8 and y by

and i t s r o t a t e s ) and l e t u

Co

(U). With

C

1 x and y , r e s p e c t i v e l y ) :

Introducing b(x,y)

=

-1 i(x-yl (i(x-y)) [l-e IX(Y)

I

one h a s : (3.10.55)

A

u

=

u(x)

+

+ ( ~ I T ) - ~ E Dl i m

I

i b ( x , y ) [ ( x - y + i S ) - l - ( x - y - i G ) -1 ] u ( y ) d y .

6+0 R Since

m

(x-y+i6)-1

= -i/ ei(x-y+iG)c d c ,

0 one c a n r e w r i t e ( 3 . 1 0 . 5 5 ) a s f o l l o w s :

(x-y-id)

-1

=

O

i J e -m

i(x-y-iS)c dE

3.10. Difleomorphisms arid Singular Perturbations

323

where

N o t i c e t h a t t h e p r i n c i p a l symbol o f t h e s i n g u l a r p e r t u r b a t i o n A

is

a o ( x , E , C ) = 1+E151. Using ( 3 . 1 0 . 5 4 ) , one c a n d e f i n e t h e i n v e r s e o p e r a t o r A - l :

O f course, A

R1

: A-lu(f?) =

-1

i s a f a m i l y of c o n v o l u t i o n o p e r a t o r s on

(rE * u ) ( f ? ) ,

where t h e c o n v o l u t i o n k e r n e l r (8) i s g i v e n

by t h e f o r m u l a :

rE(e)

=

z

(~n)-l

(l+Elnl)-leinf?.

nE Z Introducing t h e d i s t r i b u t i o n

m

one checks t h a t r

=

(e)-E-'r(f?/E)

EqE(f?), where q ( 0 ) E C

(R 1) u n i f o r m l y

w i t h r e s p e c t t o E. One c h e c k s i n t h e s a m e way as p r e v i o u s l y f o r A p e r t u r b a t i o n i n Op

s ( o r o ' - l )(R1)

Notice t h a t A i l

E'

-1 . t h a t AE i s a singular

whose p r i n c i p a l symbol i s ( l + ~ I f l ) - ' .

allows t o solve t h e following s i n g u l a r l y perturbed

boundary v a l u e problem Av(x) = 0 , (3.10.58)

x E B1 = { X E R

a ( 1 - E--)V(X) aNe

lQl

2

, 1x1

< l},

= u(f?)8

where Nf? i s t h e inward normal a t e

if?

E

Ql,

and E > 0 .

I n d e e d , u s i n g t h e F o u r i e r series e x p a n s i o n , one f i n d s t h a t

(3.10.59)

v(x)

=

?

(211)-'

--71

where x E B ' ,

x

=

1-1x1 2

I I

1-2 x cos ( e-y)

+

I

2 ( . % i 1 u )( y ) d y ,

if? (xie

.

N o t i c e t h a t t h e problem ( 3 . 1 0 . 5 8 ) where t h e inward normal i s r e p l a c e d by t h e outward o n e , i s n o t w e l l posed f o r a l l E > 0 , s i n c e i n t h a t case 1 t h e o p e r a t o r A = F- ( l - E l n l ) F i s n o t always i n v e r t i b l e when E + 0 . I t i s -E

3. Singular Perturbations on Smooth Manifolds without Boundary

324

s t i l l a s i n g u l a r p e r t u r b a t i o n i n Op l-ElC1

i s no l o n g e r i n v e r t i b l e ,

s ( O ' O t l ) (n,)

N o w we check t h a t t h e s i n g u l a r p e r t u r b a t i o n

i n t h e c l a s s Op K ( o n o f 1 ) ( S 2

E

A

(f)

=

E S"(n,)

i . e . t h e r e i s no symbol r

such

1 w i t h ro t h e p r i n c i p a l symbol o f r .

t h a t ( l - ~ l c l )0r( x , E , ~ )

one h a s f o r e a c h f

whose p r i n c i p a l symbol

1

)

is, i n f a c t ,

(3.10.52)

i n t h e s e n s e o f D e f i n i t i o n 3.10.14.

Indeed,

Cm(al)

f(8)+Efl(e),f,(e)

so t h a t ( 3 . 1 0 . 3 8 ) h o l d s w i t h ko Now, w e check ( 3 . 1 0 . 3 9 ) .

=

= D

e (n+-n-)f(e) E cm(nl),

0 , k l = -1, k . I

First, we notice t h a t

=

--, j

> 1.

(3.10.52) f o r u

E

m

C

(R1)

can be r e w r i t t e n as f o l l o w s

f o r each g Moreover,

E

,

Cm(lR)

g' ( 8 )

# 0 on supp f

( 8 ) , g being real valued.

(3.10.61) can b e d i f f e r e n t i a t e d with r e s p e c t t o

I n d e e d , assuming t h a t g ' ( 8 ) > 0 on supp f

e

and p .

( t h e case g ' ( 8 ) < 0 c a n b e

t r e a t e d i n a c o m p l e t e l y analoguous w a y ) , i n t r o d u c e t h e new v a r i a b l e t = g ( y ) - g ( 8 ) and l e t y

=

x e ( t ) .Then

where

al(e,t) al(B,-)

E

C:(R)

=

,V 0

F u r t h e r , with $ ( t )

E

t(e-y)-'a(e,y)

E

Q1,

m

CO!R)

(g*(y))-lf(y), y =

and a l ( B , O )

,

=

xe(t),

f(8).

$(t) F 1 f o r It1 S a , $ ( t )

0 f o r It1 2 2 a ,

a b e i n g s u f f i c i e n t l y s m a l l , one c a n r e w r i t e I ( 8 , p ) as f o l l o w s

3.10. Diffeomorphisms and Singular Perturbations

325

The function

-03

Therefore, one has I1(e,p) = O ( p part (or Proposition 3.8.1).

)

for

p +

m,

as shows the integration by

Furthermore, the function

is analytic for complex t # 0 , It1 < a. Using the Cauchy theorem, one can rewrite I ( 8 , p )

2

in the following

fashion Il(e,p) = e ipg(8)f(8) lim [

6++0

J t-'$(t)eiPtdt + lt/>6

dt + nil,

+ 6+6 t-leiPt where

C i =

{t E

C

1

I

It1 = 6, Im t > 0).

Therefore, with

ri

the contour consisting of intervals [-2a,-6],

[6,2a] and the half circle C i , one has (3.10.63) I 1 ( B , p )

=

eipg(e)f(8) [ J + t-'$(t)eiptdt +nil.

r6 The integration by part yields: (3.10.64) /+ t-l (t)eit dt

-m

= O(p

),

forp

+

m,

r6 so that one gets combining (3.10.62)-(3.10.64) the asymptotic formula

(3.10.61). Differentiation with respect to p or 8 leads to the same kind of integrals. Combining (3.10.611, (3.10.62), one find -m

(3.10.65) ewipgA (feipg) = (l+Epjgl(e)j)f(e)+EDef(e)+O(p

),

so that (3.10.39) holds, as well, with s = 1, s1 = 0, s . 0 3 -1 in . Setting p = E (3.10.611, one finds

=

p + m

-m,

j > 1.

3. Singular Perturbations on Smooth Manifolds without Boundary

326 when

E +

0 , so t h a t (3.10.40)

h o l d s w i t h mo

=

0, ml = -1, m .

3

= -a,

3 > 1.

One c o u l d have u s e d t h e f a c t t h a t t h e c o n v o l u t i o n o p e r a t o r ( n i )

-1

v . p . x-'*

can be considered a s t h e s i n g u l a r p e r t u r b a t i o n ( p s e u d o d i f f e r e n t i a l o p e r s t o r of o r d e r z e r o ) w i t h symbol sgn

c,

and have a p p l i e d Theorem 3.10.1

However,wehavepreferedtogivehere

i n o r d e r t o g e t t h e expansion (3.10.65).

a n i n d e p e n d e n t argument u s i n g s o m e d i f f e r e n t The o r d e r of A "(1) =

(-m,--,-m),

technique.

i s " ( O ) = ( 0 , 0 , 1 ) , f u r t h e r m o r e , one h a s " ( l )

7"

>

1. Hence, A E

E op

P ( O ) (fill,

=

(-l,O,O),

i t s p r i n c i p a l symbol

being the function

and i t s symbol b e i n g

u

(f,c)

E/clf(B)+(f(e)+EDef(e)).

=

AE

The r e a s o n f o r which t h e e x p a n s i o n s ( 3 . 1 0 . 3 8 ) - ( 3 . 1 0 . 4 0 ) perturbation A

d e f i n e d by (3.10.52)

f o r the singular

( o r e q u i v a l e n t l y , by ( 3 . 1 0 . 5 4 ) )

c o n t a i n o n l y two t e r m s i s t h a t it d i f f e r s o n l y by a smoothing o p e r a t o r

i s a l i n e a r f u n c t i o n of

1)

( 1 + I~D

from t h e s i n g u l a r p e r t u r b a t i o n

E Op

s (o'ofl)(pi) ,

whose symbol

and a p i e c e - l i n e a r f u n c t i o n of 5 , so t h a t

6

0 f o r l a [ > 1 and 5 # 0. 1 m 1 = {z(s) E C , s E [ 0 , 1 ] } b e any c l o s e d Jordan C c u r v e i n C . One

Da(l+EISl)

5

Let

r

can c o n s i d e r t h e corresponding o p e r a t o r A

: Cm(r) + Cm(r)

g i v e n by t h e

formula (3.10.66)

where D

(A u ) ( z )

=

-1 -1 (ri) I(z-5) u ( < ) d < , z E

u (z)+ E a(z)D v.p.

r

is the tangential derivative a t z E

r.

The same c o n c l u s i o n s are v a l i d f o r t h e o p e r a t o r (3.10.661,

K(ofo'l)

a s i n g u l a r p e r t u r b a t i o n i n Op

(r)

r

A

being

w i t h t h e p r i n c i p a l symbEl

/el.

a o ( z , ~ c )= 1 + E a ( z )

I-' 5 Cis>-1 , V ( z , n ) E r x R , t h e n a Op s ( o f o ' -(lr)) w i t h t h e p r i n c i p a l symbol

F u r t h e r m o r e , i f jao(z,c) singular perturbation R

E

r ( Z , E ~ )= ( l + ~ a ( zlt1)41 ) and s u c h t h a t i t s symbol r ( z , ~ < s) a t i s f i e s t h e 0 c o n d i t i o n s o f Theorem 3.10.7 h a s t h e f o l l o w i n g p r o p e r t y : (3.10.67)

A

0

E

R

E

=

I d + EQA'),

R

0

€

A

6

=

Id

+ EQ ( 2 ) E

'

3.10. Diffeornovphisrns and Singular Perturbations

E

where Q ( 1 )

Indeed,

r

if

s ( o ' o r o () r )

and I d i s t h e i d e n t i t y . 1,o ( 3 . 1 0 . 6 7 ) i s a consequence of C o r o l l a r y 3.10.8.

Op

= R , a(z) f a

(3.10.67)

i f RE

321

=

# 0 , l+alnl # 0 , V

Op ( 1 + E l C l ) - ' .

convolution o p e r a t ors R

E

I n t h e l a t t e r c a s e R:

= E-lr(x/E)

*

Notice t h a t

R , t h e n Q ( 1 ) : 0 , j = 1.2 i n

u , V u E C;(R)

i s a f a m i l y of

,

where t h e

d i s t r i b u t i o n a l k e r n e l r ( x ) h a s t h e form (3.10.68)

r(x)

=

(2n)

-1

J (l+lg])

-1 i x g e dg.

R One c a n u s e ( 3 . 1 0 . 6 8 ) f o r c o n s t r a c t i n g an o p e r a t o r R (3.10.67)

on a smooth c l o s e d c u r v e

r

of a p i e c e of z E

r

r

with p r o p e r t y

above. I n d e e d , t a k i n g t h e l e n g t h t

between a g i v e n f i x e d p o i n t zo

a s a g l o b a l c o o r d i n a t e , so t h a t z

E r

and a g e n e r i c p o i n t

z ( t ) , one r e w r i t e s A

=

i n the

following fashion ( A v ) ( t ) = v ( t ) + E a ( z ( t ) ) D tv . p .

J (t-s)-lK(t,s)v(s)ds,

(Ti)-'

R

where v ( t )

(u

=

m

o

K(t,s)

z) ( t ) E C O ( R ) , and =

(t-s)( z ( t ) - Z ( S ) ) - l Z ' ( S ) .

I n t r o d u c inq

one d e f i n e s t h e o p e r a t o r R

(REu

o

2)

a s follows:

(t)= J R

The p r i n c i p a l symbol of A p r i n c i p a l symbol of R

K E ( t , s ) (U

o

Z)

(s)ds.

b e i n g t h e f u n c t i o n a.

b e i n g ro

=

(1+Ea( 2 )

=

( l + ~ a ( z151 )

and t h e

15 I ) - l , C o r o l l a r y 3.10.8 i m p l i e s

the property (3.10.67). I t i s l e f t t o t h e r e a d e r t o check t h a t t h e s i n g u l a r p e r t u r b a t i o n

i s t i g h t l y connected with t h e o p e r a t o r B

(3.10.66)

U

C

R2

t h e i n t e r i o r of

problem: Aw z

E

r,

=

where N

mappings.

)

0 i n U, w

d e f i n e d a s f o l l o w s . With

r

and w i t h w t h e s o l u t i o n Ef t h e D i r i c h l e t

=

u on

r,

l e t (B u)

i s t h e inward normal a t z E

(2)

r.

=

(l-Ea(z)a/aNZ)u(z), ( H i n t : u s e conformal

3. Singular Perturbations on Smooth Manifolds without Boundary

328

3 . 1 1 . An a l g e b r a o f D i f f e r e n c e O p e r a t o r s I n t h i s s e c t i o n s e v e r a l c l a s s e s o f one p a r a m e t e r f a m i l i e s o f d i f f e r e n c e o p e r a t o r s are c o n s i d e r e d .

I n some r e s p e c t t h e y are s i m i l a r t o t h e c o r r e s -

ponding c l a s s e s o f s i n g u l a r p e r t u r b a t i o n s introduced i n Sections 3.1

-

( p s e u d o d i f f e r e n t i a l operators1

3 . 3 , 3 . 9 and a r e i n t e n d e d f o r n u m e r i c a l

t r e a t m e n t o f boundary and i n i t i a l v a l u e problems f o r t h e s e o p e r a t o r s . Throughout h i s s e c t i o n h

E (O,ho]

w i l l b e a s m a l l p a r a m e t e r and U

h a one p a r a m e t e r f a m i l y of u n i f o r m meshes ( g r i d s ) w i t h meshsize h i n an open s e t

u c - mn.

W e s t a r t w i t h t h e c a s e when U = R n

and Uh

=

<<

{ek}lsk=
,

= hZn,

.ih".

1 5 k S n be the s h i f t operator i n the positive x -direction k

a l o n g t h e c o o r d i n a t e v e c t o r e k , so t h a t f o r any mesh-function u : I€?n+ C one h a s Q u = u ( x + h e 1 . h k k Using t h e d i s c r e t e F o u r i e r t r a n s f o r m F

x+S , h

and i t s i n v e r s e F

-1 E-fx,h '

one c a n w r i t e

(0 u ) (x) k

where 8

k

(n)

= F

<+XI

v

FJk(h<) F

-1

h

X+E

k

so t h a t

I f a ( x , h , Q ) i s a polynomial i n O k ,

(3.11.1)

,

= exp(i<ek,rp) = exp(irik).

Denote by 0-l t h e i n v e r s e o f Elk,

which a r e Cm

u E C;(IRn)

,bur

-1

0 1 2 k 5 n , with c o e f f i c i e n t s , k ' f u n c t i o n s of x E R n and depend c o n t i n u o u s l y on (x,h)E R n X ( O , h o ] ,

a(x.h,@) =

Z

aa(x,h)Oa,

ClEZ"

t h e sum on t h e r i g h t hand s i d e o f

( 3 . 1 1 . 1 ) b e i n g f i n i t e , t h e n one can

a s s o c i a t e w i t h t h i s d i f f e r e n c e o p e r a t o r t h e 2n/h p e r i o d i c i n e a c h 1 6 j 5 n, function a(x,h,FJ(hS I ) ,

e(n)

< I'

= ( ' J l ( n ),..., e n ( r l ) ) , so t h a t one

has

-1

,

is i r r e l e v a n t , and one can u s e Bk ( 3 . 1 1 . 2 ) f o r d e f i n i n g a c o r r e s p o n d i n g d i f f e r e n c e o p e r a t o r f o r any " d e c e n t "

The f a c t t h a t a i s polynomial i n 9k ,

21r/h p e r i o d i c i n e a c h E j ,

1 C j 2 n , f u n c t i o n a ( x , h , h < ) . Now w e s h a l l g i v e

a p r e c i s e meaning to t h e word " d e c e n t " by i n t r o d u c i n g t h e c o r r e s p o n d i n g

i . t

3.11. A n Algebra of Difference Operators

329

classes of functions a(x,h,hC) which will be also called symbols Definition 3 . 1 1 . 1 . A f u n c t i o n a(x,h,n)

FY,o(u),

:

U

v = ( v l , v 2 ) E IR

zf f o r each compact

c

u,a there e x i s t s a constant

idhere 5

a,B,K

-vl

I

,...,cn), sk

= (il

<5>

+I

,

u and each m u l t i - i n d i c e s

V (x,h,5) E K

X

(O,hol

Bn,

x

(exp(ihCk)-l).

,...,w

1, w

w(c)

h

= (wl

<

introduced ;reviously

We shall denote by The partial ordering <,

-1

(ih)

=

K c

such t h a t one has:

IDBDaa(x,h,h<) 5 C h x s a,B,K

(3.11.3)

s a i d t o b e a symbol i n t h e c l a s s

is

(O,hol x : T 2 .

x

=

the vector 5 with h = 1 .

for vectors in W3

induces,

in a natural way, an order relation in R2 , if one identifies the vectors v

=

E

( v l , v 2 ) E B2 and ( v l , v 2 , 0 )

If a E

FY,,(U)

but a f F'

order of the symbol a: ord a

=

1 to v.

W

3

.

(U), V 1-1 < v, then v is said to be the

Let I$ (t) E C " ( K + ) be a cut-off function such that I$ (t) :0 for

6

6

4 (t) : 1 for t

t E [0,61,

6

2 26,

4 (t) t 0, v t E K + . 6

Definition 3 . 1 1 . 2 . A symbol a C Fv

1,o

(u) i s s a i d t o belong t o t h e c l a s s

a f u n c t i o n ao(x,n) E set

K c

u

x

r'7

one has:

( T ~\{o}) 1,rl

and moreover, i f f o r some 6

>

0 one has

< v. The j u n c t i o n ao(x,q) : I R ~x (T~\IO:)

f o r some Symbol

Fv (u), i f t h e r e e x i s t s

such t h a t uniformly on each compact

(TY \ { O j ) )

Cm(U x

1-1

Of

+

c is c a l l e d t h e p r i n c i p a l

F V ( 2 ) , h r l - ( v 1 + v 2 )a0 (x,hE) being

t h e symbol a E

homogeneous i n

(h-l,F) of degree v l + v 2 . Proposition 3 . 1 1 . 3 .

If a(x,h,hS) C F V ( u ) ther, t h e p r i n c i p a l symbol a (x,n) 0

:

U

x

(Tn \{01) 1,n

+

c is a cm-function which on every compact

and for each m u l t i - i n d i c e s a,B s a t i s f i e s t h e i n e q u a l i t i e s : (3.11.6)

ID:D:ao(x,n)

1

5 Ca , O , K I w ( q )

I

v2-lal

, v (X,ri) E

K

(T>{Oj),

K c

u

3. Singular Perturbations on Smooth Manifolds without Boundary

3 30

where t h e c o n s t a n t s c

depend onZy on t h e i r s u b s c r i p t s . v +v -In1

a,B,K

Proof. Multiplying ( 3 . 1 1 . 3 ) by h -

and letting h

-f

0, ( 3 . 1 1 . 4 )

I

yields ( 3 . 1 1 . 6 ) .

m

The class of functions a (x,n) E C ( U

0

.

(TY,n\{O})) satisfying (3.11.6)

X

is denoted by HV2 (U)

Definition 3.11.4.

A symbol a E F V ( u ) i s s a i d t o beZong t o t h e cZass G' ( u ) , i f t h e r e exist a (j)= (j) (j)) =2 , v = v (0)> v ( l ) > . . _ ,v ( j ) + v ( J )4. -m, sequence v ( v l ,v2 1 2 .(I)

for j

and f u n c t i o n s a .(x,n) E H

+ m,

(u) such t h a t f o r any given

7

0 holds:

integer N

>

(3.11.7)

(a(x,h,h5)-$6(1
-

(v; j

h

) +V ( j) )

2

(N) aj(x,hS)) E

Fv

1,o

O<j
(U).

The forma2 sum

i s ca2Zed t h e graded def a .(x,h,hS) = 1

order

=

9'1)

(x,h,F,)E

u

x

symbo2 a s s o c i a t e d w i t h a E G ' ( u ) , whiZe -(,,;J)+,,(I) 2 h a.(x,hC) i s s a i d t o be homogeneous symboZ of 3

, a ,(x,h,h5) b e i n g a we22 d e f i n e d cm function of 1 , ht # 0 (mod 2 n ) , homogewous of degree v 1( 1 ) + v2( J ) Rn , hc # 0 (mod 2 7 ~ ) . The class of a l l homogeneous

( V ~ J ), v : y

R+

i n ( I I - ~ , ~E) IR+

x x

)

R

symboi's a(x,h,hS) of order v

=

(v1,v2) and such t h a t a(x,l,n) E H

v2

(U),

i s denoted by H'(U).

Theorem 3 . 1 1 . 5

L e t ar be agraded symbol, -(V1

(3.11.9) a (x,h,q) =

h

C

( j)

+ v (1)

(j)

a .(x,rl), 7

j 20

w i t h v = v") > v ( ' ) t ..., and w i t h v ~ J ) + v ~ J4.) Then t h e r e e x i s t s a symbol a(x,h,h<) E G ' ( u ) ,

a. E 7

-- f o r

j

H~ -f

(u)

-.

for which ( 3 . 1 1 . 8 ) i s

t h e graded symbol. ~

Proof. With 4 (t) the cut-off function introduced above and {tj}j20 6 t . C 0 for j + let b.(x,h,hc)=$6(tj1<1)a.(x,h,hS)and let 3 3 1 de f ( 3 . 1 1 . 1 0 ) a(x,h,hE) = C b,(x,h,hC). (a,

jro

J

,

33 I

3.21. An Algebra of Difference Operators For e a c h g i v e n h E (O,ho!

and f o r each g i v e n ( x , S ) E lRn

Let

%,

k = 0,1,2,...,

lR;

X

,

t h e r e are

(3.11.10).

o n l y f i n i t e 1 . y many t e r m s on t h e r i g h t hand s i d e o f

b e a Sequence of compact s u b s e t s , e x h a u s t i n o U .

V (x,h,S)

E

%

(O,hol

X

X

mn,

f o r lal+lBi+k S j . L e t t o = 1. The L e i b n i t z formula y i e l d s :

O b v i o u s l y , one h a s f o r x E LY.:

S i n c e $ , ( t , l <El )0 f o r t . 1 5 1 5 6 , one f i n d s f o r x E I 7

If la-yl

=

5

1 > 0 i n t h e sum on t h e r i g h t hand s i d e o f

using the l a s t

(3.11.12) then a f t e r

q time d i f f e r e n t i a t i n g @ S ( t . I < q l 5) ,1, one g e t s tqhl-q a s a f a c t o r . Taking I 1 i n t o a c c o u n t t h a t t h e s u p p o r t of any d e r i v a t i v e of b e l o n g s to t h e s e t -1 , one g e t s f o r each t e r m i n t h e sum ( 3 . 1 1 . 1 2 ) {6 6 tkl
k

t h e same e s t i m a t e

Therefore,one has f o r x (j-l)-"(j) -"(I) "(j-i)-lUl 1 2 v2 t. 2 h <5>

(3.11.13) since h 5 C l i 1 - l .

/ D R D C l b .( x , h , h S )

x 5 1

5 C

a,B,k

E

3

when Denoting by m , t h e maximum of a l l C 3 a,B.k t . t o be

I a1 +I R 1 +k

5 j , and c h o o s i n g

I

one g e t s t h e i n e q u a l i t i e s

%:

(3.11.11).

One c h e c k s e a s i l y t h a t t h e f u n c t i o n a ( x , h , h C ) d e f i n e d by

(3.11.10) with

332

3 . Singular Perturbations on Smooth Manifolds without Boundary (j)

b ( x , h , h 1 E F" (U) s a t i s f y i n g ( 3 . 1 1 . 1 1 1 , 3 1# O ( 3 . 1 1 . 3 1 , so t h a t a ( x , h , h c ) E F" (U). 1 to

s a t i s f i e s the inequalities

C o r o l l a r y 3.11.6.

k

E

L e t a (x,h,hC)

k

Gv(U),

1 , 2 , have t h e same g r a d u a t e symbol a

=

r'

(j)

ar(x,h,n)

Z

=

a.(x,h,n),

v

=

v(O)

>

"(l)

...,

>

Then k ( x , h , h < )

=

E ffv

a.

I

j20

I

(U),

" ( J ) + " ( j ) + -m f o r j + m. l1 2 a ( x , h , h c ) - a ( x , h , h c ) i s a symbol i n

*

s 1(-a,-) ,o

(U), i . e .

f o r e a c h i n t e g e r N t 0 , e a c h compact K c U and e a c h p a i r of m u l t i - i n d i c e s

a,R t h e r e e x i s t s a constant C (3.11.14)

a , B ,N,K

such t h a t -N

,

5 C n,0,N,K<5>

/D:D;k(x,h,hc)]

I n d e e d , t h i s i s a n immediate consequence o f

symbol k

r

V (x,h,C) E K

x

(O,hol

x

R;.

( 3 . 1 1 . 7 ) , s i n c e t h e graded

of k v a n i s h e s i d e n t i c a l l y .

P r o p o s i t i o n 3.11.7.

L e t a ( x , h , h ( ) E F"

1t o

(U), b ( x , h , h E ) E F'

1 ,O

(U). Then

and (3.11.16) a b

E

F"+'(U). 1,o

Furthermore, i f w i t h some c o n s t a n t c (3.11.17) la(x,h,hC)-'l

5 C

h

0 one has:

-v 2 , V ( x , h , C ) E U

vl

x

(O,h,,]

X

Rn,

then ( 3 . 1 1 . 1 8 ) a ( x , h , h C ) - l E F;Yo(U). P r o o f . All t h e s t a t e m e n t s above a r e a n immediate consequence o f t h e c h a i n ___ rule. D e f i n i t i o n 3.11.8.

L e t a ( x , h , h S ) E Fv a(x,h,hD)

m

:

Co(u)

1 ,o +

(U). The corresponding f a m i l y o f d i f f e r e n c e operators

D ' ( u ) i s d e f i n e d by t h e formula

(3.11.19) a ( x , h , h D ) u = F

where

5

-1 C+X

a(x,h,h<)F

X-S

u+\u,

i s a one parameter f a m i l y o f operators t h a t can b e extended a s a

(continuous in h E [O,hO])f a m i l y of continuous l i n e a r mappings from E ' ( u )

333

3.1 1. An Algebra of Difference Operators

into

Crn(U1.

The n o t a t i o n Op a w i l l a l s o b e u s e d f o r d e n o t i n g t h e d i f f e r e n c e o p e r a t o r a s s o c i a t e d w i t h t h e symbol a E

Fv

1 to

(U).

Remark 3 . 1 1 . 9 . Using t h e P o i s s o n formula r e l a t i n g i n t e g r a l and d i s c r e t e F o u r i e r t r a n s f o r m s and t h e 2 1 r - p e r i o d i c i t y of q (3.11.20) a ( x , h , h D ) u

=

F

-f

a ( x , h , q ) , one c a n r e w r i t e ( 3 . 1 1 . 1 9 ) a s f o l l o w s :

-1 a(x,h,hS)F u+K u E+x,h x-fS.h h

-1 t h e d i s c r e t e F o u r i e r t r a n s f o r m and FS-fx,h i t s i n v e r s e . with F x+S , h makes s e n s e f o r any meshThe d e f i n i t i o n ( 3 . 1 1 . 2 0 ) (modulo t h e t e r m \ ) function u ( h ; x )

:

RE+ C , whose F o u r i e r t r a n s f o r m i s a 2n/h p e r i o d i c

d i s t r i b u t i o n on t h e t o r u s Tn S ,h. Examples 3.11.10. 1. L e t a = ( al , . . . , a n )

<

E

Z n and l e t e"(h5)

=

e x p ( i < h a , S > ) , Ba(hS) E FY,oAIRn)

The c o r r e s p o n d i n g d i f f e r e n c e o p e r a t o r i s t h e f a m i l y o f s h i f t o p e r a t o r s 0

h hZn, Oau(x) = u ( x + h a ) . Of c o u r s e , t o t h e complex c o n j u g a t e symbol h Ba(hS) * = e x p ( - i < h a , S > ) c o r r e s p o n d s t h e i n v e r s e s h i f t o p e r a t o r h ' O-au(x) = u ( x - h a ) . h : E Op Fo (U), V a E Z n . Obviously, 0

on

=

a. 2 . L e t < . ( h , h S ) = ( i h ) - l ( e J ( h < ) - l ) and l e t S * ( h , h S ) b e i t s complex 7 c o n j u g a t e . Obviously < , , c * E F ( O ' ' ) ( R n ) and t h e c o r r e s p o n d i n g f a m i l i e s 3 7 1;o of d i f f e r e n c e o p e r a t o r s D ,., h , D j , h a:-e ( m u l t i p l i e d by - i ) f o r w a r d and back-

ward f i n i t e d i f f e r e n c e d e r i v a t l v e s , D

3th

u = ( i h ) - l ( u ( x + h e . ) - u ( x ) ) ,D* u = ( i h ) J J rh

The symbols 5

a,h

=

-1 (ih) (e"(hS)-l),

-1

(u(x)-u(x-he.)). 7


define

t h e ( m u l t i p l i e d by -i) f o r w a r d and backward f i n i t e d i f f e r e n c e d e r i v a t i v e s i n t h e d i r e c t i o n a E Zn. 3.

To

]
t h e symbol

One h a s : =

Obviously, (-Ah) 4 . With

E

=

h

-2

a E Zn.

F(Or2)

of - A ,

A b e i n g t h e Laplace o p e r a t o r ,

(2u( x )-u ( x + h e , -u ( x - h e , 1 ) 1

16j5n

E op

E F ( O r l ) (&,V

I<.<*E F'Or2' (R*')i s a s s o c i a t e d t h e u s u a l

1 7 f i n i t e d i f f e r e n c e a p p r o x i m a t i o n -Ah -A u ( X I h


(mn).

1

3. Singular Perturbations on Smooth Manifolds without Bounda y

334

<

d e f i n e s a n o t h e r f i n i t e d i f f e r e n c e a p p r o x i m a t i o n of t h e L a p l a c e e q u a t i o n 4 o n t h e mesh f o r which t h e a p p r o x i m a t i o n e r r o r i s O ( h ) when h + 0 . With

t h e symbol (h,hC)

a:')

=

(-151

2

+a(h,hS)-b(h,hS))

E

F(o'2)(iK*1j

1, o

d e f i n e s a f i n i t e d i f f e r e n c e a p p r o x i m a t i o n of t h e Laplace e q u a t i o n on 6 w i t h t h e a p p r o x i m a t i o n e r r o r O(h ) when h -f 0 . One c h e c k s e a s i l y t h a t a ( ] ) (h,hC) E F ( O r 2 '

a

5. L e t a ( h , h E )

=

(mn).

(l+I
'-*) E IR. O i v i o u s l y , a E F (1O,o

.

(BR)

The c o r r e s p o n d i n g d i f f e r e n c e o p e r a t o r a ( h , h D ) s o l v e s t h e e q u a t i o n on (3.11.21)

(1+D D * ) u ( x ) h h

=

5:

f(x),

I?)

i.e. u

= a ( h , h D ) f = F-l - l Fx+5,h f . (1+16 S-fx, h E v i d e n t l y , one c a n r e w r i t e a ( h , h D ) a s a f a m i l y of c o n v o l u t i o n

operators,

We a r e g o i n g t o f i n d an e x p l i c i t formula f o r t h e k e r n e l G ( x ) which i s h nothing e l s e b u t t h e Green's function f o r t h e d i f f e r e n c e equation (3.11.21) One h a s

'h,C The l a s t i n t e g r a l c a n a l s o b e r e w r i t t e n a s f o l l o w s :

With e + ( h )

=

2

l + ( h / 2 ) - h ( 1 + h 2 / 4 ) ' t h e z e r o of t h e e q u a t i o n

e2 -

2 (2+h ) e + 1 = O

i n s i d e t h e u n i t c i r c l e 181 = 1 , one f i n d s u s i n g t h e residuum c a l c u l u s :

3.12. An Algebra of Difference Operators Notice, t h a t t h e r e i s a point-wise s o l u t i o n G(x)

=

335

convergence of G ( x ) t o t h e fundamental

9

( 1 / 2 ) e x p ( - [ x [ ) f o r t h e o p e r a t o r 1-d / d x 2 , when h

I t i s e a s i l y seen t h a t ( 1 + / < l 2 ) - ' E

-f

0.

F ( 0 r - 2 ) (U). 4

6. L e t a ( h , h E ) = a l ( h , h S ) + a 2 ( h , h 5 ) w i t h a l ( h , h S ) = hi
Obviously, a

( u ) , a2 E

~ ( O t ~ ) ( u S) i.n c e n e i t h e r ( - 1 , 4 )

>

=

151

(0,2)

2

.

1 ( - 1 , 4 ) , t h e symbol a = a + a 2 does n o t have a w e l l d e f i n e d 1 p r i n c i p a l symbol, s o t h a t a E Fv (U) w i t h v E iR2 t h e l e a s t v e c t o r such 1 ,0 t h a t v > ( - 1 , 4 ) , v > ( 0 , 2 ) , b u t a B F v ( U ) . Obviously, i n t h e c a s e c o n s i d e r e d , nor (0,Z)

v

=

>

(0,3).

Theorem 3.11.11.

L e t a ( x , h , h S ) E Fv

1.0

(3.11.23) h

v1

Then uniformly w i t h r e s p e c t t o h E [O,h 0] one has:

(U).

a(x,h,hD)

:

C;(U)

-f

Cm(U).

m

P r o o f . One h a s f o r e a c h u E C (U):

0

~

Since

G ( S ) i s rapidly decreasing for I S ]

-f

= and a ( x , n , h t ; ) s a t i s f i e s

( 3 . 1 1 . 3 ) one f i n d s immediately t h a t h V 1 a ( x , h , h D ) u b e l o n g s t o a bounded s e t

I

i n C"(U), V h E [O,h,,I.

Theorem 3 . 1 1 . 1 2 . de f L e t Ah = a ( x , h , h D ) E Op Gv and

be i t s graduate symbol, a , ( x , ~ E) v = v(O) > v ( l ) t

1

_ _.. Then

-(v;j)+v(J)) k t ay(x,h,n) (j) v2

H

f o r each

i h - <x

1 h jt0

(u), ( v ~ J ) + v ~ J )4)

rl

a.(x,n) 7 -m

E Tn\{O) and each f

tQ>f ( x ) )

P r o o f . The d e f i n i t i o n ( 3 . 1 1 . 2 0 ) y i e l d s : __

The T a v l o r f o r m u l a v i e l d s :

=

-

-, E c i (U) hoZds

for j

+

3 . Singular Perturbations on Smooth Manifolds without Bounda y

336

where R

M

i s t h e remainder,

(3.11.25) R (x,h,n,hS) = M

C

hM ( a ) a (x,h,rl+yhS)Sa, y E [ 0 , 1 1 .

IaI=M '!

Furthermore, s i n c e a ( x , h , h S ) E G v ,

(3.11.7) y i e l d s :

v a

V x E K C C U , V h E (O,ho],

E

Zy,

v

N 2 0

Now w e e s t i m a t e t h e r e m a i n d e r . On t h e s e t

one h a s f o r

1011

= M >

v

x E

2'

K C C U, h

f (O,hol:

Further, ( 3 . 1 1 . 2 8 ) / h M a ( a )( x , h , n + y h S )

-V

/

5 CM,K h

V

'<<>

-M

,

V x E K, V h, V

5 E

En

( 3 . 1 1 . 2 8 ) , one c a n e s t i m a t e t h e r e m a i n d e r t e r m

Combining ( 3 . 1 1 . 2 7 ) ,

i n t h e following fashion: (3.11.29)

IRM(f)I 5

m

Since f E C o ( U ) , O ( h m ) when h

+

t h e l a s t i n t e g r a l on t h e r i g h t hand s i d e o f

0.

Hence, M-(V

(3.11.30)

RM(f) = O(h

+V

1

2

) ),

h + O

(3.11.29) i s

I

3.11. A n Algebra of Difference Operators

u

u n i f o r m l y on any compact s e t ( x , q ) E

337

101.

x Tn\

11

I n o r d e r t o f i n i s h t h e p r o o f , one h a s o n l y t o show t h a t (3.11.31) e-ih

-1

<x,n>

ih-l<xtl^l>f( x ) )

'i,( e

u n i f o r m l y on any compact s e t i n

u

x

T~

O,,,a,,

=

~

\{oI.

1,ri One p r o c e e d s as f o l l o w s . By S c h w a r t z ' s k e r n e l t h e o r e m , one c a n w r i t e

%

:

E ~ ( u -+) crn(u)a s f o i i o w s

(3.11.32) K u ( x ) h

=

,

V u E E'(U), m

where t h e k e r n e l f u n c t i o n k

h

( x , y ) E C (U

U) uniformly with r e s p e c t t o

X

h E [O,hol. Let

x

m

E Co(U),

x

: 1 on some open subset U

kh(XrE)

1

s u p p f . Then i n t r o d u c i n g

2

Fy+s k h ( x r Y ) X ( Y )

=

one can w r i t e , a s p r e v i o u s l y f o r a ( x , h , h < ) : -ih-l<x,n> K

ih-l<x,ri> f ( x ) )

=

h(e

S i n c e k h ( x , s ) i s r a p i d l y d e c r e a s i n g (uniformlywithrespecttohandx)w h e n 5 - t m , o n e g e t s conclusion ( 3 . 1 1 . 3 1 ) , u s i n g t h e sameargumentas i n t h e e s t i m a t e o f t h e remainder R M

.

Now a n a n a l o g u e o f t h e a s y m p t o t i c f o r m u l a ( 3 . 1 0 . 1 ) w i l l b e s t a t e d and proved f o r d i f f e r e n c e o p e r a t o r s . Theorem 3.11.13.

Let f E

E

Cm(U), g

0

Cm(E),

LeR a ( x , h , h D ) E Op F"

1.0

integer

N 2 0

: C:(U)

holds: -1

( 3 . 1 1 . 3 3 ) e-ih

and d g ( x ) # 0 , V x E s u p p f . + C m ( U ) . Then for each

g being real-valued

(U), a(x,h,hD)

9(X)a(x,h,hD

(f(x) e

ih-lg (x)

)

=

where (3.11.34) $ (x,Y) 9

= g(X)-g(y)-

and where t h e remainder s a t i s f i e s on any compact K c u ( w i t h a c o n s t a n t

3. Singular Perturbations on Smooth Manifolds without Boundary

338

c

depending only on i t s s u b s c r i p t s ) t h e f o l l o w i n g e s t i m a t e

NrK

(3.11.35)

( R (x,h) N

1

5 C

N,K

,

hN-"/21-vl-"2

P r o o f . A f t e r t h e change o f v a r i a b l e s ~

5

-f

V (x,h) E K

x

(O,hol.

h c , one g e t s t h e f o r m u l a

-1 ( 3 . 1 1 . 3 6 ) e-ih

g ( X ) a ( x , h , h D x )( f ( x ) e i h - l g ( x ) ) =

where

and t h e i n t e g r a t i o n on t h e r i g h t hand s i d e o f

5.

r e s p e c t t o y and a f t e r w a r d w i t h r e s p e c t t o Then f o r

151 2 R ,

(3.11.35) i s f i r s t with Let

IV g(y)[ 5

C,

Y

V y E

u.

R sufficiently large, the operator

i s w e l l d e f i n e d a n d , moreover, one h a s :

S p l i t t i n g t h e i n t e g r a l on t h e r i g h t hand s i d e of where t h e i n t e g r a t i o n w i t h r e s p e c t t o {

151

< R} and

> R},

(3.11.36)

i n t o two p a r t s

5 i s r e s p e c t i v e l y over t h e sets

u s i n g t h e o p e r a t o r ( 3 . 1 1 . 3 8 ) and t h e f o r m u l a

( 3 . 1 1 . 3 9 ) , one g e t s , a f t e r [ n / 2 ] + l p a r t i a l i n t e g r a t i o n s i n t h e second p a r t , a r e p r e s e n t a t i o n o f t h e l e f t hand s i d e o f

( 3 . 8 . 3 6 ) by a b s o l u t e l y and

uniformly convergent i n t e g r a l s . Using t h e same argument a s i n t h e p r o o f o f Theorem 3 . 1 0 . 1 ,

one g e t s t h e

c o n c l u s i o n t h a t t h e s t a t i o n a r y p h a s e method i s a p p l i c a b l e t o t h e i n t e g r a l ( 3 . 1 1 . 3 6 ) , t h e o n l y s t a t i o n a r y p o i n t of t h e p h a s e

on t h e r i g h t hand s i d e of function

+

( d e f i n e d by ( 3 . 1 1 . 3 7 ) ) b e i n g t h e p o i n t M(x) = ( x , V x g ( x ) ) ; b e s i d e s ,

t h e same argument a s p r e v i o u s l y i n t h e p r o o f of Theorem 3 . 1 0 . 1 M(x) i s r e g u l a r . Now u s i n g ( 3 . 1 0 . 2 3 ) w i t h q proof of

(3.10.11,

C o r o l l a r y 3.11.14. graded

t h e formula (3.11.33). If

4,

=

=

V x g ( x ) , one g e t s , a s i n t h e

I

a ( x , h , h D ) , with a(x,h,hS)

symbol a r i s w e l l - d e f i n e d .

shows t h a t

E G"(U),

then i t s

Indeed, t h e c o e f f i c i e n t s of t h e

asymptotic expansion (3.11.24) a r e uniquely determined.

3.11. An Algebra of Difference Operators

339

The n e x t r e s u l t i s concerned w i t h t h e c o n t i n u i t y p r o p e r t i e s of d i f f e r e n c e o p e r a t o r s as mappings between s p a c e s ff Theorem 3.11.15.

v

-v

L e t a E Fv

1t o

(Rn),

v

(s),h("h) * = ( v , , v 2 ) E R2 , and

h l < < > 2 a ( - , h , h S ) E S ( R z ) uniformly w i t h r e s p e c t t o ( h . 5 ) E (O,hol x Rn

Then t h e f a m i l y (3.11.40)

is

5 '

%,

,,(.lf:)

a(x,h,hD) : H(s+V) 2 equicontinuous, V s E R :=

-f

H

.

( s ), h

h E (O,hol,

(<),

P r o o f . W e u s e more o r l e s s t h e same argument a s i n t h e p r o o f of Theorem __

F

After t h e d i s c r e t e Fourier transform

3.4.1.

x+S , h

applied t o z

= A

h

u,

one f i n d s :

(2n)-nJ

-n

where

-h a

h ( 5 ) = Fx,5,hz(x),

ah (5-rl , h ,hn

)

,h Fx+5,ha(x,-,.),

(n)d o ,

u-h

'rl

(5,.,-)

=

Gh(t)

= Fx+5,hu

a r e t h e corresponding d i s c r e t e Fourier transforms. Denote C e = < ( h , h < ) . I n t r o d u c i n g Gh(rl)

one can r e w r i t e

-(sl+vl) = h

2

+v 2 -h

-h u (ri), w



--s

(5)

'
= h

5

>

s2 -h z

(5),

(3.11.41) i n t h e following fashion def

(3.11.42) Gh(h)

s

=

(ylGh)

(5)

=

J

(2n)-"

K(h,c,n)Gh(n)dn

Tn

n th

where

Kh

so t h a t ( 3 . 1 1 . 4 0 ) i s e q u i v a l e n t w i t h

: L2(Ti,h)

+

5 ,h ) uniformly

L2(Tn

with r e s p e c t t o h E [O,hol. S i n c e x + a ( x , . , . ) E S(R:)

,

a E

Fv1 ,o

( E n ), t h e f o l l o w i n g i n e q u a l i t i e s

h o l d w i t h any i n t e g e r N 1 L 0 :

Indeed,

( 3 . 1 1 . 4 4 ) i s a consequence of

(1.3.10) w i t h k

Using ( 3 . 1 1 . 4 4 ) and P e e t r e ' s i n e q u a l i t y << >'<<

S

r

l

>-'

=

0

5 2"'<<

j = N1'

E-n

>, V y E R ,

one f i n d s f o r any i n t e g e r N 2 0 : (3.11.45) where C

lK(h,c,q)l 5 C

N rho

Ntho

<<5-n>

-N

,

V ( h , t , n ) E [O,hol

may depend o n l y on N and h o .

T h e r e f o r e , w i t h any N 2 0 h o l d s :

l R5n x

xn n'

3 . Singular Perturbations on Smooth Manifolds without Boundary

340

Applying t h e i n e q u a l i t y

and t a k i n g N = n + l , one g e t s t h e c o n c l u s i o n t h a t t h e norms o f t h e o p e r a t o r s

K

: L

~ ( T ) ~+ L 2 ( T n

5 rh h by t h e c o n s t a n t

Remark 3.11.16.

5 .h

a r e u n i f o r m l y bounded w i t h r e s p e c t t o h E [O,ho]

)

E F" (Bn) , a(x,h,hE) = am(h,hC)+ n1 ,o E S(IRx) u n i f o r m l y w i t h r e s p e c t t o h

I t i s obvious t h a t i f a

a ' ( x , h , h c ) where h V 1 < p - Y 2 a ' ( * , h , h S ) and 5 , t h e n 9 , = Op a , h

E

(0,h

0

1 i s stillequicontinuousbetweenthesarnespaces:

A s i m i l a r argument shows t h a t i f a

E F"

1.0

(U) t h e n u n i f o r m l y i n h

E (O,ho]

holds : (3.11.46)

a(x,h,hD)

:

H

~

(s+u),h('h)

H(l0C)

( s ) ,h

(Uh),

v

s E R

2

.

Now we c o n s i d e r t h e c o m p o s i t i o n of d i f f e r e n c e o p e r a t o r s . Theorem 3.11.17.

Let

9, =

Gp a ( x , h , h C ) , a E

Fu

1 ,o

(U) and Let Bh

Then f o r each $I E cm(u) t h e composition ch 0

=

= A,

F' (U). 1t o i s a difference

Gp b ( x , h , h C ) , b E 0

I)

0

B~

operator i n Gp F"+'(u)

and for each x such t h a t $I s 1 i n some neighbourhood 1 ,o symbol c ( x , h , h S ) has t h e asymptotic expansion:

of x i t s (3.11.47)

c(x,h,hC)

-

C

l

a

z ( a C a ( x , h , h 5 )( D > ( x , h , h < ) ) ,

/a(?O

where t h e asymptotic r e 2 a t i o n (3.11.47) i s i n t e r p r e t e d in t h e fol2owing sense: f o r each i n t e g e r N > 0 one has: (3.11.48)

with hN

=

rN(x,h,h5)

def = (c(x,h,hS)-

Z / 4 < N

1

=aa

5

a(x,h,hS)D" b ( x , h , h E ) ) E X

V+U-(~,N).

We g i v e h e r e a n h e u r i s t i c argument, which c a n b e e a s i l y made r i g o u r o u s by a p p l y i n g t h e s t a t i o n a r y p h a s e method w i t h p = h - l I 5 I

as a l a r g e parameter

I n d e e d , one h a s f o r e a c h u E C m ( U ) and h s u f f i c i e n t l y s m a l l

0

3.1 1. A n Algebra of Difference Operators

34 1

The complete proof of (3.11.47), that is the justification of i3.11.48), can be done in the same way as the proof of Theorem 3.10.5. It is left to the reader (see also [Fr, 9 , l o ] ) , where the proof is given without use of the stationary phase method). A different commut'ition formula for the symbol c(x,h,h<) of the operator Ah

D

$J

0

Bh may be useful, especially, when the symbol a(x,h,ht)

of Ah is a polynomial in

. Namely,

(<,<*)

holds for each x E U such that

)I

5

the following asymptotic formula

1 in some neighbourhood of x:

*

where iD and iD are difference forward and backward derivatives, x,h x,h respectively. Of course the sum on the right hand side of (3.11.49) is finite, when a is polynomial in ( < , < * I . We shall not give here a complete proof of (3.11.49), leaving it to the reader, and shall only present a formal argument which leads to (3.11.49). -h One has for the difference Fourier transform v ( 5 ) of the function v

= C

u the formula

h

where 5

<(h,hq) and the upper -h stands for the difference Fourier

=

transform of corresponding functions. Obviously, one has: 5,

with

eT

=

=

( e T ,...,eT

Notice, that

1

<;

c,+GT
n

eT

=

<:+e,

-1

*

cq-T,

= exp(ihT ) ,

k

k

so that the Taylor formula yields:

3. Singular Perturbations on Smooth Manifolds without Bounda y

342

where F i s t h e d i f f e r e n c e Fourier transform. x+5 , h Therefore(after t h e inverse Fourier transform F

-1 1, ( 3 . 1 1 . 5 0 ) E+x,h

can

be formally r e w r i t t e n i n t h e following f a s h i o n

t h e sum on t h e r i g h t hand s i d e of t h e l a s t a s y m p t o t i c f o r m u l a b e i n g a s y m p t o t i c a l l y c o n v e r g e n t i n t h e same s e n s e a s i n ( 3 . 1 1 . 4 8 ) . W e d o n ' t p r o v e t h i s l a s t s t a t e m e n t , l e a v i n g i t s proof t o t h e r e a d e r . Example 3.11.18. With a ( x , h , h < )

=

< ,

A

h

= D

x,h'

b ( x , h , h S ) :b ( x ) , t h e f o r m u l a ( 3 . 1 1 . 4 7 )

becomes

g i v e s i n t h i s case

w h i l e (3.11.49)

c(x,h,h5) = b(x)i(h,hS)+D b(x)O(hS). x.h Now t h e c o r r e s p o n d i n g r e s u l t w i l l b e s t a t e d f o r t h e a d j o i n t o f a d i f f e r e n c e o p e r a t o r , t h e p r o o f of t h i s r e s u l t b e i n g i n any r e s p e c t , v e r y s i m i l a r t o t h e one i n t h e c a s e of s i n g u l a r p e r t u r b a t i o n s . With

9, =

Op a ( x , h , h E ) , a E

F"1 ,o (U),

l e t tA

h

be i t s a d j o i n t defined

by t h e r e l a t i o n : (3.11.51)

(

4,u , v I h

= ( u , ~ A ~ v ) ~V, u , v E C i ( U )

( w i t h h s u f f i c i e n t l y s m a l l , where ( family of l a t t i c e s

( u , v ) ~=

,

jh

i s t h e i n n e r p r o d u c t on t h e

g h r

C u(x)v(x)*hn. xERn x,h t The f o l l o w i n g a s y m p t o t i c formula h o l d s f o r t h e symbol a ( x , h , h E ) o f

3.11. An Algebra of Difference Operators

where as usual, a

*

is the complex conjugate of a.

a(x,h,S, S * ) t asymptotic formula for a Again, if a

343

5

then one can also write a different

the latter being more convenient in applications, since, usually, difference approximations of differential operators have symbols which are polynomial in

(<,<*).

Example 3.11.19. With a(x,h,hS)

=

a(x)i(h,hS), 9,

t =

~ ( X ) D ~ one , ~ has ,

9,

*

*

= Dx,h a (x).

Formula (3.11.52) qives in this case

while using the formula (3.11.521, one finds:

Remark 3.11.20. As in the case of singular perturbations, one can use amplitudes ( U X U ) for introducing difference operators. Under the a(x,y,h,hSi F F" 1 ,o assumption that the corresponding operators are properly supported (see

Definition 3.7.3, which can be adequately modified in this case), the (UxU) coincide with the classes Op F" ( U ) , the relation classes Op F" 1 .o 1,o between an amplitude a(x,y,h,hS) E F" (UXU) and the corresponding symbol 180

a(x,h,hE) E F"

(U) being the one given by formula (3.7.12) and the formal 1 ,o symbol of a(x,h,hD) being given by the asymptotic expansion (3.7.13).

Example 3.11.21. Let q

=

q(x,y) E C m ( R

x

a(x.y,h,hE)

=

0

Obviously, a E F 1 , O ( ~x One finds easily

9,

IR) and let

-2 . 2 hc -1 (1+4q(x,y) sin -) 2

.

IR). = Op a(x,y,h,hS). Indeed,

3 . Singular Perturbations on Smooth Manifolds without Boundary

344 where

%(x,y)

=

-1 Fc+(x-y) ,h a(xrYchrh5).

Introducing f3 = exp(ihc), one finds using the residuum calculus:

Applying (3.7.13), one finds an asymptotic expansion of the symbol a(x,h,hE;)of Ah: a(x,h,hS)

1

-

C

N

+o

a aDaa(x,y,h,hS)

I

a! 5 y

Y=x

,

so that

where r(x,y,h,hS) E F0

1 to

(R x

a).

Introducing the difference operator 2

p(x,h,hD) = l+q(x) (2-0-0-') with 0 the shift operator, Ou(x1 = u(x+h), putting q(x,y) = q(y) one finds, applying the commutation formula (3.11.47) that =

Id

+ hR(l)

=

Id

+

h

'

hR(2) h '

where Id is the identity operator and R(1) are difference operators with h their amplitudes in Fo ( R x IR) (thus, RLJ) have their symbols in 1 .o

FY,O(IR)

and under the assumption that q(x)

=

q+q'(x) with q' E S ( R ) , Rh( 3 )

are uniformly with respect to h E (0,h ] bounded mappings from H (IR' ) 0 (s),h x,h into itself, as a consequence of Theorem 3.11.15). The formula (3.11.53) means that p(x,h,hD) and a(x,y,h,hD) with q(x,y) = q(y) are quasi-inverse of each other (inverse up to a small operator for h

+

O), so that using the Neuman's series one can construct -1 -1

the inverse operators p(x,h,hD) h

and a(x,y,h,hD)

when h E (O,hol with

sufficiently small. Of course, the fact that p(x,h,hE;) + Const (1 in

0 this case) when h

+

0, h5

+

0 is the reason that the quasi-inverse of

p(x,h,hD) can be constructed using the symbol

calculus. The same is true

3.11. An Algebra of Difference Operators

345

for the multidimensional difference operator

with q(x) symmetric positive definite matrix, V x E R n , q(x) E Cm(wn) ,

*

(x), q' (x) E S(Rn) , and iDx,h, -iD the difference forward x,h andbackward gradients, respectively.

q(x) = q,+q'

In this case one can take as an amplitude of the quasi-inverse difference operator the same symbol-function a, a(y,h,hS) where 5

=

2

(l+h )-1,

=

* -1 ( 5 l,...,
* k of 5 ) are symbols of D and Dx,h, respectively. x,h Now we are going to consider the group of the linear automorphisms of

<

and the corresponding transformations of symbols of difference operators. Denote by ISO

(q< I ;

the group of linear transformations mapping

onto itself, and by SL(n;Z) the group of nxn matrices, whose entries are integral and the determinant 21. Proposition 3.11.22. SL(n;Z) and I S O ( R ~ ;<) are isomorph.

Proof.

If a E SL(n;Z) then ax

as well, the mapping x

-f

<

E

,

V x E

<.

Given that a-1 E SL(n;Z),

ax is onto Rn. h and let {he,}lcj5n be the standard

Reciprocally, let a E ISO(Ri; <)

<

1

basis on RE, e . being the j-th coordinate vector. 3

One has a(he.) E 3

, Vj,

1 6 j S n and, thus, the matrix ( a ) of a with

respect to the basis {he.} < , < nhas integer entries. So it is also for a-l, J 1=]= -1 E ISO(R~; RE) , too. Therefore, det a and det a-' are both since a

I

integral and necessarily equal to +1. Let

4,

=

F"1 r o (Rn) . is periodic and smooth in n , it can be

Op a(x,h,hC), a(x,h,hE) €

Since a(x,h,n)

:

U

x

(O,ho] x Tn

n

+ C

expanded into a Fourier series

so that using the expansion (3.11.541, one can write

(3.11.55) (%u) (x) =

C

vEZn

a (x,h)u(x+hy), V u E C;(Rn).

3. Singular Perturbations on Smooth Manifolds without Boundary

346

P r o p o s i t i o n 3.11.23.

L e t A,, x

a

=

=

Fv ( R ~.)Then a f t e r t h e change of v a r i a b l e s 1 ,o t -1 E S L ( n ; z ) t h e symbol of A,, becomes a ( a y , h , h ( a ) 5 ) , where

Op a ( x , h , h o , a E

ay w i t h

c1

t .

z-s t h e transpose o f a .

P r o o f . Denote v ( y ) -

(3.11.56)

(\u)

=

(x) =

u ( a y ) and u s e (3.11.54),

Z

(3.11.55). I t g i v e s :

-1 a ( a y , h ) u ( a ( y + h a y))

-1 a ( a y , h ) v ( y + h a y).

C

=

yEz"

yEZn

The symbol of t h e d i f f e r e n c e o p e r a t o r on t h e r i g h t hand s i d e o f

(3.11.56)

is

We a r e g o i n g t o i n t r o d u c e some d i f f e r e n c e a n a l o g u e s of t h e F o u r i e r I n t e g r a l Singular Perturbations. Let

u.

t

3 -

,

RnJ

j = 1,2.

D e f i n i t i o n 3.11.24.

The f u n c t i o n g ( x , y , h , n ) , ( x , y , h , q ) E U 1

X

U2 x R +

x

Tn

n

i s s a i d t o be a

phase f u n c t i o n if t h e f o l l o w i n g c o n d i t i o n s are s a t i s f i e d 1". $J

is r e a l valued, $ ( x , y , h , h E ) E F::;') ( U 1 x U2) u1 x u2 t h e r e e x i s t s a c o n s t a n t cK such t h a t

2O. f o r each c o q u e t s e t K c

A s a consequence o f

h

+

0,

lhEl 2

no

(3.11.57) t h e f u n c t i o n ei'

> 0 f o r some g i v e n

i s r a p i d l y o s c i l l a t i n g when

no.

D e f i n i t i o n 3.11.25.

L e t $ ( x , y , h , h c ) be a phase f u n c t i o n and l e t a ( x , y , h , h E ) be an amplitude in FY,o ( U 1 x u2). The operator

is c a l l e d t h e d i f f e r e n c e Fourier operator. I f a ( x , h , h c ) E F"(U1 ao(x,y,h,hS!

x

U2) ( D e f i n i t i o n 3.11.2) t h e n , a s p r e v i o u s l y ,

s t a n d s f o r t h e c o r r e s p o n d i n g p r i n c i p a l symbol.

Using t h e same argument as i n t h e p r o o f o f Theorem 3.9.3, t h e following

one p r o v e s

3.11. An Algebra of Difference Operators

341

Theorem 3.11.26. h

Let A

@

be a difference Fourier operator. Then uniformly with respect t o

h E [O,hol hoZds

Indeed for establishing (3.11.59) , one has to use, instead of the operator the one defined by the formula:

(3.9.3),

L @ (x,y,h,hS,Dy,h,ag)= =

-i( ID

$ 1 2+1

Y ,h

where Dy,h

5 (h,h5)I

2

/ v 5 $ / ')-'

(+l5 (h,hS)/'
5 $,Vg>),

(DYl,h,...,Dyn,h), iDyj,h being the forward difference

=

derivative with respect to y . .

I

Again, one has in this case

and an integration by part with so that a summation by part in y E Rn Y th respect to 5 lead to a representation of Ah in the form similar to (3.9.5) @ h (A (u))(x) $

=

(Ah

+to

(u))(x) +

Z Ah u) (x), 15j5n $.j'"xjrh

where the amplitudes of the difference Fourier operators A @ , j , 0 5 j S n, FV-(O,l) (U1 x U2). 1 ,o

belong to

Example 3.11.27. Consider the following difference Fourier operators

The functions

solve the Cauchy problem:

3. Singular Perturbations on Smooth Manifolds without Boundary

348 where

Ah

-

=

* -2 * C D D = h 1 (0).-2+0.) 1s jsn xj f h xj ph lsjsn J

'

is the difference Laplace operator. h The distributional kernels of E+(t) are -

(3.11.60) E+(h;x,y,t) -

=

( 2 1 ~ h ) - ~eih-l(<x-y'n>'t~u(n)l)dn, /

t 2 0,

Tn 17

where, as previously, w ( n )

=

L(1,n).

One can apply the stationary phase method for computing the asymptotic expansion of E+ when h

0.

-f

-

An

easy computation leads to the following equation for the stationary points

on T ~ :

n

lw(q)/-lRe w ( n )

=

T t-'(x-y)

so that for Ix-yl > t there are no stationary points and the singular 2n+1 support of E+ is contained in the cone = { (x,y,t) E R + , Ix-yl 5 t}. -

One can show that in fact,sing supp E+ -

=

V, while for the corresponding

distributional kernels of the wave operation (h aV = {(x,y,t) E R 2n+l j jx-yj = tI.

=

0 ) the singular support is

precisely

Let A be a symmetric positive definite nxn matrix and let * l

A (3.11.61) E+(h;x,y,t)

-

=

(21~h)-"/ e Tn

ih-I (<x-y,n>+t 2 ) dn,

t 2 0.

One checks easily that (3.11.62) s i n g sup

Ed

=

i(x,y,t) E IRINC1

I

5 t}.

A Of course, E+ are solutions of the Cauchy problems: -

2

2

<,

*

+
>)Eh(h;x,y,t) = 0, x E t > 0 x ,hIDx,h at aE; L Ei(h;x,y,O) = 6 (x-y), -(h;x,y,O) = f OP < A < , < > 2 6h(X-Y). h at (-

2

Example 3.11.28. Let w ( n ) ,

n E

T" be as previously defined and let

n

2 2 (3.11.63) B ( p ) = 1-(p /2)+ip(l-p /4)',

2 p 2 = p2(n) = r <w(n),w*(n)>,

where the parameter r satisfies the (hyperbolicity) condition:

-t (3.11.64) 0 5 r 5 n

.

3.11. A n Algebra of Difference Operators A s a consequence o f

v

ri

349

( 3 . 1 1 . 6 4 ) , one h a s p 2 5 4 , so t h a t I f 3 ( p ) f

1

=

1,

E T ~ v, r E [ o , n t I . rl

Consider t h e f o l l o w i n g d i f f e r e n c e F o u r i e r o p e r a t o r s :

where (3.11.66)

++- ( x , y , t ; h , h S )

=

-1

<x-y,C>

I n f3 ( p (hg)1 .

f t (irh)

h

h

-

-

I t i s e a s i l y s e e n t h a t t h e f u n c t i o n s v + ( x , t ) = ( E + ( t ) u )( x ) s o l v e t h e

f o l l o w i n g Cauchy problems

where, a s p r e v i o u s l y , IR+

=

+

TZ ,

Z

+

=

{k E Z

t , T

I

k > 01.

One c h e c k s e a s i l y u s i n g t h e s t a t i o n a r y p h a s e method, t h a t a g a i n i n t h i s

case t h e s i n g u l a r s u p p o r t of t h e c o r r e s p o n d i n g k e r n e l s of t h e o p e r a t o r s

E:(t), -

i s contained i n

(3.11.68)

7

=

( a c t u a l l y , c o i n c i d e s with) t h e cone:

2n+l { ( x , y , t ) E iR+

1

Ix-yl

5 t}.

N o t i c e t h a t when r + 0 one g e t s a g a i n t h e d i f f e r e n c e F o u r i e r o p e r a t o r s h E + ( t ) from Example 3.11.27.

Eh ( t ) t h e d i f f e r e n c e F o u r i e r o p e r a t o r s d e f i n e d by ( 3 . 1 1 . 6 3 ) , f ,A (3.11.66) with

Denote by (3.11.65), (3.11.69)

p2 = p

2 A

(ri)

=

2

r

where A i s a symmetric p o s i t i v e d e f i n i t e nxn m a t r i x and r s a t i s f i e s t h e (hyperbolicity) condition: (3.11.70)

o

5 r 5

(nj (A1

I)-+.

The c o r r e s p o n d i n g f u n c t i o n s vh

f ,A

of t h e d i f f e r e n c e equation r e p l a c e d by
*

(x,t)

=

(Eh f

,+A

( t ) u )( x ) a r e s o l u t i o n s

where ( 3 . 1 1 . 6 7 ) on IRn x R x,h t , ~

x ,h ‘Dx,h>. Our c o n j e c t u r e i s t h a t t h e s i n g u l a r s u p p o r t of t h e k e r n e l s

E+ A ( h ; x , y , t ) of - I

the operators

Eh+,A ( t ) c o i n c i d e s w i t h t h e cone ( 3 . 1 1 . 6 2 ) .

is

3 . Singular Perturbations on Smooth Manifolds without Boundary

350

If n = 1 , r = 1 , one f i n d s an e x p l i c i t formula f o r t h e k e r n e l s

E+(h;x, y, t) of the d i fferen ce Fourier opeators

-

(3.11.65), (3.11.66).

h

E -+ ( t ) d e f i n e d by ( 3 . 1 1 . 6 3 1 ,

Indeed, i n t h i s c a s e O + ( p ( q ) ) = c o s q f i l s i n q l , -

SO

t h a t a n e a s y c o m p u t a t i o n shows t h a t (3.11.71) E + ( h , x , y , t )

=

(ZTih)-'

;(e ,.

-ih

-1

(x-y-t)n+eih

-1

(x-y+t)q

)

dn

and t h e same formula ( u p t o t h e s i g n ) h o l d s f o r E - ( h , x , y , t ) . Of c o u r s e , i n t h i s case t h e c o r r e s p o n d i n g d i f f e r e n c e F o u r i e r o p e r a t o r c a n b e r e w r i t t e n a s a d i s c r e t e c o n v o l u t i o n o p e r a t o r i n t h e form:

w i t h E + ( h , x , y , t ) g i v e n by ( 3 . 1 1 . 7 1 ) . AS a consequence o f

h

-t

( 3 . 1 1 . 7 1 ) , E + ( h , x , y , t ) converges i n D ' ( I R )

when

0 t o the distribution

t h e l a t t e r b e i n g t h e s o l u t i o n of t h e Cauchy problem

I

= Op(lSI) i s t h e s i n g u l a r p e r t u r b a t i o n ( i n f a c t , t h e where o f c o u r s e ID p s e u d o - d i f f e r e n t i a l operator) w i t h t h e symbol 151 E S ( 0 , l r O ) ( R ) .

W e a r e g o i n g t o r e s t r i c t t h e c l a s s of p h a s e f u n c t i o n s and s h a l l

c o n s i d e r , from now on t h e p h a s e f u n c t i o n s # ( x , y , h , h c ) which s a t i s f y t h e following C o n d i t i o n 3.11.29.

There e x i s t s a c o n s t a n t 6 ( h , n ) E IR+

x Tn

> 0

w i t h I < ( h , q )I

such t h a t for any ( x , y ) E u1 2 6

x

u2 and any

holds:

-1 (3.11.72) $ ( x , y , h , q ) = h $ ( x , y , l , n ) .

For such phase f u n c t i o n s t h e c o n d i t i o n (3.11.57) w i l l be s t a t e d a s foZlows:

3.11. An Algebra of Difference Operators

35 1

On t h e o t h e r hand, we s h a l l e x t e n d t h e c l a s s o f t h e o p e r a t o r s c o n s i d e r e d i n t h e f o l l o w i n g way. We s h a l l d e n o t e by

Kh t h e c l a s s o f o p e r a t o r s o f t h e form

where t h e f a m i l y o f k e r n e l f u n c t i o n s h s e t in Cm(U1

x

U2)

E

for h

: Cm(U

4

0

2

K ( h , x , y ) b e l o n g s t o a bounded

[O,hol w i t h a g i v e n h

We s h a l l u s e t h e n o t a t i o n l i n e a r mappings Ah

+

F"

m l,o

+ C

0' f o r t h e f a m i l i e s of continuous

(4)

(U1),

v

h C ( 0 , h 1, which c a n b e 0 4 satisfying

r e p r e s e n t e d i n t h e form ( 3 . 1 1 . 5 8 ) w i t h t h e p h a s e f u n c t i o n C o n d i t i o n 3.11.29,

i.e.

a ( x , y , h , h S ) E Fy,,CU,

x

(3.11.72),

(3.11.731, and with an amplitude

U2) (mod K h ) .

h One can a s s o c i a t e w i t h a d i f f e r e n c e F o u r i e r o p e r a t o r A + E -V

its distribution kernel h

FY,O(d)

'A4 ( h , x , y ) , where

V ( x , y , h ) E U1

x

U2

(O,hol.

X

W e are g o i n g t o l o c a l i z e t h e s i n g u l a r i t i e s o f A ( h , x , y ) .

d

F i r s t , introduce t h e set

where t h e u p p e r b a r , a s u s u a l d e n o t e s t h e c l o s u r e o f t h e c o r r e s p o n d i n g s e t . I n t h e same way, a s p r e v i o u s l y f o r Theorem 3 . 9 . 7 ,

one p r o v e s t h e

following Theorem 3.11.30.

The s i n g u l a r support o f t h e f a m i l y h

+

A (h,x,y)

0

of t h e d i s t r i b u t i o n a l

k e r n e l s d e f i n e d by ( 3 . 1 1 . 7 5 ) i s contained i n t h e s e t (3.11.76), (3.11.78)

(3.11.77): s i n g supp A ( h , x , y )

d

5

Q4.

Q

4

d e f i n e d by

3. Singular Perturbations on Smooth Manifolds without Boundary

352

Now t h e c l a s s of H y p e r b o l i c D i f f e r e n c e O p e r a t o r s w i l l b e i n t r o d u c e d and t h e c o r r e s p o n d i n g F o u r i e r D i f f e r e n c e O p e r a t o r s w i l l b e c o n s i d e r e d f o r Hyperbolic D i f f e r e n c e Operators with c o n s t a n t c o e f f i c i e n t s . One d i s t i n g u i s h e d ( t i m e - l i k e ) v a r i a b l e i s g o i n g t o p l a y a s p e c i a l n+l - R n X R and c o n s i d e r g r i d s x,t x + = (hZn) x ( T Z ) w i t h two mesh-sizes h and T = r h , where r > 0 i s a

r o l e . Hence, w e s h a l l d e n o t e W IRE::

g i v e n c o n s t a n t . The d u a l v a r i a b l e s w i l l b e d e n o t e d by ( S , E ; ) 0

A s usual, 0

E

Rn x R

5

50

and 0-1 a r e , r e s p e c t i v e l y , t h e f o r w a r d and backward s h i f t

o p e r a t o r s on t h e g r i d R

while 8

= TZ,

t , T

symbols. F u r t h e r , D

=

(iT)-'

t t T

-1

( T S ~ ) ,B0 ( T S0 )

*o ( e T - l ) , Dt,-,

-1

= (iT)

stand f o r t h e i r are

(I-@-')

( m u l t i p l i e d by -i) forward and backward d i f f e r e n c e d e r i v a t i v e s and

A(T,TS

*

-

(T,?< ) d e n o t e t h e i r r e s p e c t i v e symbols. A s u s u a l , 0 ORn x n+l = with R = { t E R t > 0 ) and x t,+ t ,+ n+ 1 n+l w i t h R+ t h e c l o s u r e o f R ),

*;+I=

I

%,-,,+

~

.

D e f i n i t i o n 3.11.31. n+ 1

A symbol a ( x , t , h , h E , ~ < ~E ) Fv(P(+

)

,

is s a i d t o be s t r i c t l y

v = (vl,v2)

hyperbolic i f t h e following conditions are s a t i s f i e d :

i s an i n t e g e r ;

1". v2 > 0 2'.

a can be r e p r e s e n t e d i n t h e form:

(3.11.79) a ( x , t , h , h S , T S O ) = h

'1

*

-k

( ~ E ; ~ ) p ( x , t , h , X , C) ,, <

O0

(0,v2)

where k 2 0 i s some i n t e g e r and t h e symbol p E F

(R:+l)

i s polynomial

i n A o f degree v 2 , 3 " . w i t h p o ( x , t , h , X , C , < * ) t h e p r i n c i p a l symbol o f p , t h e z e r o s 1 5 j 5 v2, (3.11.80)

of t h e equation

*

po(x,t,l,~,w,w)

a r e aZl d i s t i n c t f o r (3.11.81)

u 1. ( x , t , ~ ) ,

=

0, w

IE T>{O},

11+irU.(x,t,q)l = I , 7

=

(ul,

(x,t) E

v

...,w n ) ,

= -i(e

w

iqk

-l),

7 -

and, moreover,

( x , t , q ) E x;+l

x T ~ .

II

D i f f e r e n c e o p e r a t o r s w i t h s t r i c t l y h y p e r b o l i c symbols a r e s a i d t o be s t r i c t l y hyperbolic. It i s e a s i l y seen t h a t t h e o p e r a t o r

s t r i c t l y hyperbolic with v

=

(O,l), n

=

(3.8.117)

i n Example 3 . 8 . 2 3 i s

1.

F u r t h e r , t h e d i f f e r e n c e a p p r o x i m a t i o n ( 3 . 1 1 . 6 7 ) of t h e wave o p e r a t o r A -2:

i s s t r i c t l y h y p e r b o l i c w i t h v = ( 0 , 2 ) , p r o v i d e d t h a t (3.11.64)

is

3.11. An Algebra of 3ifference Operators

*

s a t i s f i e d a s w e l l a s Dt , T

353

*

D t,T - w i t h A nxn symmetric p o s i t i v e

d e f i n i t e matrix i s a s t r i c t l y h y p e r b o l i c d i f f e r e n c e approximation of t h e wave o p e r a t o r D 2

t

=

, p r o v i d e d t h a t ( 3 . 1 1 . 7 0 ) i s s a t i s f i e d .

x

x

We s h a l l c o n s i d e r t h e Cauchy problem f o r s t r i c t l y h y p e r b o l i c d i f f e r e n c e o p e r a t o r s w i t h symbols which do n o t depend on x , t and s u c h t h a t i n t h e r e p r e s e n t a t i o n ( 3 . 1 1 . 7 9 ) one h a s : v 1 p(x,t,h,X,<,<*)

po(h,h,c,<*) = Am

supposed t o b e p o l y n o m i a l i n

*

=

0, k

=

0 , v2

m > 0,

=

+ terms o f lower o r d e r i n

<,< , a s

A,

po b e i n g

well.

Hence, c o n s i d e r t h e Cauchy problem:

m

where v E C (R) i s g i v e n .

0

Denote a g a i n by

u.(n)

(3.11.83) po(l,u,w,w*)

3

=

the zeros of the equation

0 , w = (wl

,...,w n ) ,

I t i s e a s i l y s e e n t h a t t h e s o l u t i o n of

in mk = - i ( e

k-l).

( 3 . 1 1 . 8 2 ) i s g i v e n by t h e f o r m u l a

C L: a ,(h,hg)ei~l'x~yft'h'hS)v(y)hndg, (3.11.84) u ( x , t ) = J h T~ y ~ m f : l<j<m 1 ht5 where

(3.11.85) $ . ( x , y , t , h , h S )

l

=

-1 <x-y,S>+t(Ti) l n ( l + i u .( h S ) ) , 3

and where

I n d e e d , w i t h $ . d e f i n e d by ( 3 . 1 1 . 8 3 ) , one f i n d s u s i n g t h e f a c t t h a t -1 po i s homogeneous i n ( h ,g) o f o r d e r m:

since

u , (n) s a t i s f y ( 3 . 1 1 . 8 3 ) . 1 Hence, t h e f u n c t i o n ( 3 . 1 1 . 8 4 ) i s a s o l u t i o n o f t h e d i f f e r e n c e e q u a t i o n

3. Singular Perturbations on Smooth Manifolds without Bounda y

354 (3.11.82).

F u r t h e r , one h a s f o r any polynomial P ( A ) o f d e g r e e m w i t h s i m p l e z e r o s A L and w i t h l e a d i n g c o e f f i c i e n t 1 t h e f o l l o w i n g f o r m u l a e :

where y i s a c o n t o u r i n C p'(A)

=

which e n c l o s e s t h e z e r o s h l ,

...,A rn

of p(X) and

dp(A)/dX.

Using a g a i n t h e f a c t t h a t w i t h $ , d e f i n e d by ( 3 . 1 1 . 8 3 ) one h a s :

I

and a p p l y i n g (3.11.871, (3.11.84)

one g e t s t h e c o n c l u s i o n t h a t u (x,t) d e f i n e d by h s a t i s f i e s t h e i n i t i a l c o n d i t i o n s i n ( 3 . 1 1 . 8 2 ) , a s w e l l , so t h a t

it i s t h e s o l u t i o n o f t h e Cauchy problem ( 3 . 1 1 . 8 2 ) .

Using t h e homogeneity of a . ( o f o r d e r m-1) and 4 , ( o f o r d e r 1 ) i n 3 I -1 ( h , < ) , one c a n r e w r i t e ( 3 . 8 . 8 4 ) i n t h e f o l l o w i n g f a s h i o n : (3.11.88) u ( x , t ) = h

C lsjsm

(A

h

v)(x),

'j

where t h e d i f f e r e n c e F o u r i e r o p e r a t o r Ah function j '

has as i t s k e r n e l t h e following

One h a s f o r e a c h t

where

AS a consequence of t h e h y p e r b o l i c i t y c o n d i t i o n t h e v e c t o r

is real-valued. and l e t K =

L e t K . b e t h e c l o s u r e of t h e range of F . 1 I

U K.. 16j<m J

(n)

: Tn\{O}

n

+

Rn

3.1 1. An Algebra of Difference Operators

355

Introduce K* = {(x,y,t)

E IRn

En

x

X

1

R+

t-'(x-y)

E K},

so that K*

Q

U

=

(t).

tzo,j j' As a consequence of Theorem 3.11.30, the singular support of the kernel of

the operator, which solves the Cauchy problem (3.11.821, belongs to K*. Obviously, K* is a conic set in Rn

x

Rn

x

R+

.

Assume that p (h,D 0 t,T ,Dx,h,Di,h) is a difference approximation of a hyperbolic differential operator pO(Dt,Dx) which is necessarily homogeneous in (D D ) of order m. Moreover, assume that the set po(So,<) = 0, 5 E l R n , t' x 151 = 1, consists of a finite number of ovals containing each other. In that case the smallest oval V is always convex (see [John, 11 and also for the sufficient conditions for such a situation, which are that m and n to be even and po(O,E) # 0 , V 5 E R n , 161 = 1 ) . Let K be the cone in

Rn+' having the smallest oval V as its base. Let V* be the dual cone of V (the cone of rays). We conjecture, that the singular support of the kernel of the operator which solves the Cauchy problem (3.11.81) for po (h,Dt,T,Dx,h,D:,h)

in this case is always contained in the closure of

the interior of the dual cone V*. This is precisely the case for the hyperbolic approximation (3.11.67) of the wave operator, as discussed in Example 3.11.28. Further,we consider one class of hyperbolic difference operators with variable symbols and investigate the asymptotic behaviour of their oscillatory solutions when the mesh-size goes to zero. Let lR;+l

= {x =

(xo,x')

I

xo E R+ , x' E Rn-'1 and let

R,,+ n+l

be the

with the mesh-size hk = rkh, h E W+ with rk > 0 , 0 5 k 5 n,

greed in

given positive constants. Further, let Vn = { ( h , C ' ) E Let X(x,h,<') 1".

X E

E + X

n+1

: R+

x

Vn

Cn +

1

Il+ih 5 1 = 1, 1 2 k 2 n}. k k

C have the following properties:

m

C

in variables (x,h,C') for

<' #

0;

Z 0 . X(x,h,<') s X(-,h,<') for 1x1 sufficiently large;

3'.

X(x,t-'h,K')

5

--n+l

th(x,h,<'), V t > 0, V (x,h,<') E R +

4 " . h(x,l,o') with w '

= (wl

satisfies the conditions:

,...,w n ) ,

l+irkwk = exp(ir 5 1 , k k

x

Vn;

1 5 k 5 n,

3 . Singulav Perturbations on Smooth Manifolds without Boundary

356

and

These conditions mean that def A(x,h,<) = G o + $6(15'1)h(x.h,5') with C 0 = ( i h o )

-1 (exp(ihOcO)-l)and $,(t)

above ( $ 6 ( tZ) 0 for t

E

[0,6],

the cut-off function introduced

$&(t) E 1 for t t 26) is a hyperbolic

symbol in F(OS1) (my+1),whose principal symbol is A 0(x,h,<)=
0 2 k 5 n, and using 3 O ,

in the form:

-1 AO(x,h,<) E h o(x,S), 5 = ( c o , 5 ' ) , rkSk = nk, Oskin, rl E T;+l. Consider the Cauchy problem on the greed

("(xA

I

Dxo,h,DX , , ) uh ( X) = 0 ,

-1

uh (0,~') = f(x') exp(ih

where f E

m

c0

g:: x E Iqt

$o(x')),

(Rn)and $o E Cm(Rn), $q : Rn

+

R , are given.

We seek uh as an asymptotic series:

-1 k uh(xo,x') = exp(ih $(xo,x')) C h pk(xo,x') k>O with Q(xq,x') and pk(xo,x) t o be determined as smooth functions of (xo,x') n+ 1 in R . Substitution of uh(xo,x') given as a formal asymptotic series above and the use of Theorem 3.11.13 yields the relations:

4

and pk being subject to the initial conditions: $(O,X') = $,(x),

po(O,x') = f(x'), pk(O,x') = 0, k = 1,2,...i

here o(x,<) = hA 0 as defined above, the coefficients Ck (x) are expressed

3.11. A n Algebra of Difference Operators recursively in terms of u and $, Lk variables x', L-l

=

357

are differential operators in

0.

A s a consequence of the hyperbolicity condition, the Cauchy problem for

with $o (x') real-valued, has a well-defined smooth solution for

$ (xo, X I )

xo E ( 0 , 6 ) , x' E Rn with 6 > 0 sufficiently small. The same hyperbolicity condition guarantees that -1 (x,Vx$)) , 1 2 k S n, are real-valued, V x E 10,s)

x Rn , as 50 ' k it will be shown below, so that the Cauchy problem for pk, k 2 0, is well0

(x,Vx$)( U

posed. It can be shown (but will be not done here) that the formal asymptotic m

expansion thus obtained,is asymptotically convergent in C (K) with -n+ 1 K C W+ any compact. Hence, one can determine successively pk(xo,x') for (xo,x')F [0,6)X Rn. We shall discuss a little more in detail the non-linear Cauchy problem for $(xo,x') stated as follows: U(X,V

$1 = 0, $(O,X') = $o(x').

One finds differentiating the equation for $ with respect to xk, 0 5 k 6 n: 0 =

v

u(x,V$) = [ $

v

xx ll

I

u(x,ll) + v x u ( x , l l ~ l

ll=Vy$

is the matrix of second derivatives of 9. x I'O
where $

xx

=

I l ' x

k j

0 0

A s a consequence of the hyperbolicity, the equation u(x,So,S')=O has the

real zero <,(x,S')

(well-defined up to an additive term 2rLri1, L E Z), so

that differentiating this equation with respect to that

I?5

-'

k ~ c o=-

-aco/ac,

5,.

1 4 k 5 n, one finds

are real-valued. Hence, the system of ordinary

differential equations

with the initial condition x(0) = ( O , s ) , s = (sl,. .. ,sn) E Rn defines the n+ 1 , passing through the point unique curve without singularities in R (0,s)

With S(t) = Vx$(x(t)), x(t) being the curve defined above, one finds

3. Singular Perturbations on Smooth Manifolds ioithout Boundary

358

'Th-refore,x(t),E,(t)

is the solution of the following Hamiltonian type

system:

and along the trajectory x(t),<(t) one has:

Taking the initial condition for x(t) and $(t) in the form x(0) = ( 0 , s ) E Rn+l, $'

0

= $'o(s),

i t is readily seen that the initial conditions for C ( t ) are

0 0

where 5 ( s ) is the zero of the equation

The Hamiltonian type system for x(t),S(t) along with the equation for $ and the initial conditions above, define the curve x = x(t,s),

in iR

2n+2

5

=

S(t,s)

and the function $(t,s) :

The equations x = (xo,x') = x(t,s), $I = @(t,s) give a parametric representation for $. One can eliminate (t,s) by solving the equation x(t,s) = x and find 0 as a function of x, provided that the Jacobian J = D(x0,x')/D(t,s) is non-singular.

It is readily seen that = 1, det J (t=O = det D(x')/D(s)~~=~

so that for t sufficiently small det J # 0 and one can determine $ as a

3.11. An Algebra of Difference Operators function of x by eliminating (t,s). Therefore, for t E [ 0 , 6 ) with 6 sufficiently small the asymptotic series for the solution uh of the Cauchy problem above is well-defined. For the first coefficient po(xo,x) in the asymptotic expansion for

uh one has the following equation (cf Theorem 3.11.13):

Introducing the vector field v tangential to the solutions of the system of ordinary differential equations (the bicharacteristics of the equation for p ) 0

:

dx /ds = 1, dx'/ds = ( u 0

50

(x,V$))-~V ,u(x,Vd),

5

one can rewrite the equation for po in the form:

Using the Liouville's theorem for first order autonomous systems of n+ 1 ordinary differential equations in R

which says that the determinant of the Jacobian

satisfies the differential equation: (d/ds) In det J = div h(x), one finds for Fo:

Further, given that a-1 50

=

exp(-iroEO), one gets

359

3. Singular Perturbations on Smooth Manifolds without Boundary

360

=

given

-ir

exp(-ir @ ) u 5 (x,V$) = ir u exp(-ir$ ) , OXO k O xo xO

C @ 06k6n xO%

that the differentiation of the equation u(x,V@) = 0 yields +

0

xO

z u (x,vQ)@x = 0 OSkSn k ' k O

Therefore, one finds :

Hence, one gets for Po with some constant C:

If SP(UxS) + ox

= 0,

0

then po = C(det J)-'

with some constant C z 0 (see [Mas,

1

1,

where this

kind of computation is done for pseudodifferential equations). This argument can be easily extended to higher order strict hyperbolic difference operators which are polynomial in D

xo

or to first Ih0

Id + A(x,h,D' with A(x,h,S') a pxp order systems of the form D xo rho x',h) matrix-symbol in F(O'l) (IRy+l), whose principal symbol AO(x,l,w') has real eigenvalues A , (x,1 ,w 7

')

such that

[w'~-l/A.(x,l,w')-~ 9. (x,l,w'P 2 6

n+ 1 with some 6 > 0 , V x E IR+

3

,...,w

V w ' = (wl

),

l+ir w =exp(ir 5 k k k k

,

# 1.

Consider several specific examples. With u

=

exp(i<0)-exp(i51), which corresponds to the approximation of the

-u = 0 by the finite difference equation uh (xO+h,x1 x1 0, one finds for the solution uh of this equation with the -1 initial condition u (0,x ) = exp(ih Q (x ) ) , the following equation for h 1 0 1 Q(xo,x): equation u

xo uh(xO,xl+h) =

exp(i$

)-exp(i$x xO

)

= 0,

2

x E R+

1

with the initial condition $(O,x1 ) = @ 0 (x11 . For the first coefficient Po(xo,xl) in the asymptotic expansion of u h'

3.1 1 . A n Algebra of Difference Operators

36 1

one has the Cauchy problem:

One finds easily: $(x) = $ (x +x ) + 2nkxO, 0 (x) = f(xo+xl), 0 0 1 0 2

so that on the greed IF$,

one has: r+

uh(x) = exp(ih

-1

$ (x +x ) ) (f(x +x )+O(h)),

0 0 1

0 1

h

+

0.

In fact, in the case considered O(h) : 0 . Now consider the following Cauchy problem on

( ( f ) (l+r)+(l-r)Oxl,h)Dx ,h -D h)\=O, 0 0 xl' where ho = rh, hl

=

2

%,+ :

uh (0,x1 ) =exp(ih-l$O(xl))f(xl),

h.

One has: u

=

-1 (2r) ((l+r)+(l-r)exp(iS1))(exp(irSO)-l)-(exp(i~l)-l),

and the following equation for $:

tJX

=

0

-1 r arg((p+exp(itJx ) ) (l+p exp(i4x ))-l), 1 1

p

=

-1 (1-r)(l+r)

where arg z is the argument of z E C-{01. Let $(O,x ) = wxl with some w E IR 1

. Then

+(x) =y(w)xO+uxl, y ( w ) =r-l arg((p+exp(io)) (l+p exp(iw))-'). Further, one finds easily that

(u

-1

u

)

(V$)

=

I-'.

-41 (l+r)exp(i$x )+(l-r)

50 51

1

Hence, with $ = y(w)xo+wxl as above, one gets for po the equation (polx - a(@) (polx = 0 , 0 1

a(w)

=

41 (l+r)exp(iw)+(l+r)

so that Po(x) = f(a(w)xo+xl) and u h (x) = exp(ih-'(y(w)x0+ox1)) (f(a(w)xo+xl)+O(h)),

h -+ 0 .

The argument above can be extended to higher order hyperbolic operators.

,

362

3. Singular Perturbations on Smooth Manifolds without Boundary

Consider as an illustration, the wave equation u

-u = 0 and its x x 1 1 finite difference approximation (D D* -D = 0, on the xO,ho xo,h0 x1 ,hDxl,h)% x x

oo*

greed

2

I$,+ in Rf

with the meshsize ho = rh, hl = h, the latter being

hyperbolic provided that r E (0,l) One finds easily:

u

=

2 2 4(r-2sin (rc0/2)-sin (c1/2)) = o+o-,

where

u'(~)

= 2(r-lsin(r~~/2)3sin(51/2)).

Consider for instance, the asymptotic behaviour of the wave associated with

. ' 0

One finds for the phase function @ the following equation: -1 r sin(r9 /2)-~in($~/2) = 0. xO 1

Again assume that $(O,x

1

) =

axl, I w / < a / 2 . Then

~(x) = y(w)xo+wxl, y ( w ) = 2r-larc sin(r sin(w/2)). For po one gets in the same way as hereabove: po(x) = f(y'(a)xO+xl),

y'(w)

= dy/dw,

so that for the wave moving to the right one has:

uh (x)

=

-1

exp(ih

(y(w)xo+wxl)f(y'(w)x0+x1) .

As another illustrative example, consider a hyperbolic difference approximation of the system atu+pglaxp = 0 ,

atp+p o o2 a xu

= 0,

(x,t) E

E x

R+ =

2

n+ ,

which describes the propagation of plane acoustic waves of small amplitudes in a medium

at rest, u(x,t) being the speed of the propagation, p(x,t) the

presure, the constants

p o > 0,

co > 0 being the density and the

compressibility characteristic of the medium, respectively; here, as previously a = a/at, ax = a/ax. 2 t Let %,+ be the greed in =

z:

(h,T),

T =

(x,t) E R

X

z+1

with the meshsize

rh, r > 0 being a given constant.

Consider the following finite difference approximation of the system 2 above on the greed B,,+ :

3.11. An Algebra of Difference Operators

363

where, as previously, 0 is the shift operator in the greed h I$, = {x E IR I x =kh, k E Z } , (0 v) (x) = v(x+h), and iDt,,, iDx,h are h forward finite difference derivatives in t and x, respectively, on the 2

greed ph,+ * Multiplying the second equation by ( p c

-1

0 0

,

adding it to and after-

wards subtracting it from the first one, one gets the following finite 2 + -1 difference equations on If$,+for the Riemann'sinvariants u- = u+(poco) p of the differential system above:

where we have denoted: L2(rc0,Oh,Dt,, ,Dx,h) = if ( 4 ) (l?rco)+(lfrco)@h)Dt,TfcoDx,h}, the latter being hyperbolic finite difference approximations of the operators at+coax. If the initial conditions for u and p are highly oscillatory functions 0)

of the form f (x)exp(ih-l$o(x)) with $o E C ( I R ) , fo(x) = p 0(x) for

o

m

f (x) = u (x) in CO(R 1 , then the initial conditions for the Riemann's 0 O + (x)). invariants u- will be of the same form u:(x)exp(ih-'$ 0 Seeking the asymptotic expansion of u'(xlt)

one gets for $'(x,t)

as h

+

0 in the form

the following Hamilton-Jacobi equations:

+

with the initial conditions $-(x,O) = $ (x). 0

The transport equations for u'(x,t) k as above.

are derived in the same way

Finally, for the multidimensional wave equation U

x0x0

-

16k,j(n

akjUx x k j

with positive definite matrix A approximation

=

= o

I [ akj 1 I

and for its finite difference

3. Singular Perturbations on Smooth Manifolds without Boundary

364

1 akjDxk,hDx,,h)uh=O,h0 =rh,h1=...=h =h, h D* x ,h (DxO' 0 0 0 l
Again consider the wave associated with o+ = 2(r-' sin(r~~/2)-p(~')).

If f (x') = (planewave), then p (x) = f (
uh = exp(ih-l(<w,x'>+y(w)x

)

0

(f(<X,x'+xoVwy(w))+O(h)).

We finish this section by introducing some classes of CiEference singular perturbations affected by the pr-eseme OF two small uarameters

E

and h.

Definition 3.11.32 A f u n c t i o n a(x,E,h,n)

u

x

( 0 . ~ ~x 1 (0,h 1 0

i s said t o be a symbol in (vl,v2,v3) E n3 , i f for each compact x

Tn

rl

( u ) , with vo E m , v = 1,O u and each multi-indices ci,B there e x i s t s a constant c

the class F K c

:

V0,V

a,B ,K

such t h a t

holds:

V (x,E,h,S) E K

If v0 = 0 we shall write

Fv1 r 0 (U) instead of

X

( 0 . ~ ~ X 1 (O,hol X Rn

Forv(U).

1r o Analogously to Definitions3.11.2,3.11.4oneintroduces

symbolclasses

FvO'"(U) and Gvorv(U),as well as the graded symbols in the latter class.

.

3.1 1. A n Algebra of Difference Operators

365

Examples 3.11.33.

1. Let

U C

W

and let q E C m ( u ) , q(x) 2 qo > 0. With -2

-p

a1(x,€,h,n) = (1-e and

)

-2

a2(x,E,h,n) = 1+4p

11-e

p =

h/E the functions

in-(p/q(x)) 2

I

2 q (x)sin2 n/2

vo,v

VO,"

are symbols in F

(U)

(U)) with v,

(and even in G

=

0, v = (0,0,2);

besides, they coincide with their graduate symbols,which contain in this case just one term. Both symbols al and a2 are approximations (when h + 0) of the symbol 2 2 1 + ~q ( x ) C 2 in the classes s ( o r o ' 2 ) (U) and K ( o p o r 2 ) (U) defined in Section 3.6. 2

Vo,V

I < I 4+1 < /

2 . The symbol E

belongs to the class G

v = (0.0.2). It approximates (when h

+

(U)

,

U

C

Rn , with v

0

=0,

0 ) the symbol E ~ ~ S ! ~ + in ! S the ! ~

class K(or2r2)(U) in Section 3.6. The symbols Q ( x , E , ~ , ~ ) 0 ,= ( 0 l,.. . ,en), Bk = exp(ihSk) which are -1 polynomial in 8 k, Bk , k = 1,2 ,...,n, Q E F v o r v ( U ) with v 2 , v 3 nonnegative integer, will be of special interest in Chapter 4 where boundary value problems for elliptic difference singular perturbations will be studied. The difference singular perturbations with symbols in corresponding F v o r v ( U ) and GVo"(U)

classes F V o " ( U ) ,

are defined as in Definition 3.11.8.

The proof of the following statement is just a copy of the one in Theorem 3.11.15.

-

v ,v

vorv

( W n ) the subclass of symbols in F

Denote by F 0

f o r each symbol a(x,~,h,hS)E Fvo

1v

(Wn) such that

(Rn) there exists a symbol a_(e,h,hE)

such that a-a-, as a function of x E lRn , belongs to S(lRn) Proposition 3.11.34.

Let

Ah,,

=

vo,v

op a(x,h,~,hE)w h e r e a E F

1 to

.

-

(w"). T h e n u n i f o r m l y w i t h r e s p e c t

to h E (O,hol, E E (O,ho] h o l d s : (3.11.91)

V O% , E :

Here the spaces H

(Rt)

H(s+v) , ~ , h

+ H(s) ,E,h( < I .

are the ones introduced in Definition 2.8.1.

(s),E.h (<)

The following statement can be proved using the same argument as in the proof of Theorem 3.11.5. Theorem 3.11.35.

.(j)

L e t a.(x,E,h,hS) E F 3

where v i 1 ) + v i J )

J.

180

-- f o r

(u), j

+

j = 0,1, m.

..., w h e r e

v = v")

t v(')

>

... and

3. Singular Perturbations on Smooth Manifolds without Boundary

366

Then t h e r e e x i s t s a symbol a ( x , E , h , h < ) E Fv

1 #O

(u) such t h a t for each

i n t e g e r N > 0 holds:

The proof of t h i s statement i s l e f t t o t h e r e a d e r . One can a l s o formulate (and prove i n e x a c t l y t h e same way) a statement analogous t o Theorem 3.11.12

f o r difference singular perturbations i n

Op G v ( U ) , v = (vl,v2,v3). Theorem 3.11.36.

LEA

a ( x , ~ , h , h D )E Op G v ( U ) , v = (vl,v2,v3) and l e t

:=

- (v:j)+v a r =

1

"(1)

(1)

2

h

)

a , ( x , p , n ) , p = h/E, b e i t s graduate 3

(l+p-')

j>O

symbol, where a . s a t i s f i e s on each compact K c c u t h e i n e q u a l i t y 3 (j) .(I) la4(x,p,n) J

and where v

= v")

I

cK l w ( r i )

1'"



3

,

>v

g i v e n p E R+ , P = h/E ho Zds :

3.12. E l l i p t i c Singular P e r t u r b a t i o n s In t h i s s e c t i o n a p r i o r i e s t i m a t e s a r e e s t a b l i s h e d and parametrix c o n s t r u c t i o n s a r e c a r r i e d o u t f o r E l l i p t i c p s e u d o d i f f e r e n t i a l and d i f f e r e n c e s i n g u l a r perturbations. We s t a r t with a g l o b a l v e r s i o n of t h e d e f i n i t i o n of symbols S v ( U ) introduced i n Section 3 . 3 . D e f i n i t i o n 3.12.1. a ( x , E , < ) i s s a i d t o be i n t h e symbol class L'(R") following conditions are s a t i s f i e d

A function

i f the

( i )a E s'(R") ( i i )t h e r e

x

-f

e x i s t s a symbo2 a,(E,c) E s'(R") such t h a t t h e @ n e t i o n : = a ( x , ~ , c ) - a - ( ~ , cbeZongs ) t o s(R:), i . e . t h e folZowing

a'(x,c,c)

i n e q u a l i t i e s hold:

3.12. Elliptic Singular Perturbations

367

where C > 0 are some constantswhichmay depend on t h e i r s u b s c r i p t s . With a (x,E,S) the corresponding symbol in the representation (3.3.2) (with 0 a . and r satisfying (3.3.3), (3.3.4)),we shall denote (with some abuse of notation) also by a (x.E.~)the homogeneous extension of a . (as a function 0 of ( E -, C~) ) to (x,E,S) E Rn X IR+ x Rn , which is also called the principal symbol of a. Definition 3.12.2.

A symbol a E L'(IR~)

i s s a i d t o be e l l i p t i c of order v E

m3

i f i t s principal

symbol a0(x,E,<) s a t i s f i e s t h e c o n d i t i o n :

where c

0

i s some c o n s t a n t .

The corresponding s i n g u l a r p e r t u r b a t i o n a(x,E,D)

= op a

E op LV(I??") i s

s a i d t o be e l l i p t i c of order v. The following definition is a natural extension of Definition 3.1.7. to v n symbol classes L (IR ) . Definition 3.12.3. 0

The symbol a (x,E) E 1 a(x,E,c) E L'(IR") (3.12.3)

-v2

<<>

(O,V2,O) ( 7 ~i ~s )s a i d t o be a reduced symbol o f

i f v1

(E

<E<>

2 0 a(x,~,S)-a(x,S))E L-e,

e

=

(l,-l,l),

0 a (x,C) being a reduced symbol o f a(x,~,C),t h e operator Op ao

=

0 a (x,D)

i s s a i d t o be a reduced operator of t h e s i n g u l a r p e r t u r b a t i o n op a v

= ~(X,E,D)

n

Lemma 3.1.8. extends automatically to the symbol classes L (I?? ) . (0,V2.O) If a E LV(Rn) has the reduced symbol ao E L (Rn) , then we shall denote by aOO(x,S) the principal homogeneous in 5 (of order v-+v L 3 0 part of its principal symbol aO(x,E,<) and by a (x,<) the principal homo0 0 geneous in 5 (of order v2) part of a (x,S). Remark 3.12.4. v n As a consequence of Proposition 3.3.3 a symbol a E L (I?? ) . order v iff its principal symbol a

is elliptic of

satisfies the condition

3. Singular Perturbations on Smooth Manifolds without Bounda y

368

x ( S ) aO(x,E,S) that

x

5

-1

E L-"(Rn) , where

0 for 151 5 6,

If a E Lv(IRn)

x

m n E C (IR )

x /Si

E 1 for

is a cut-off function such 0 2 2 6 with some 6 > 0 .

L(orv2po) (Rn) , then

h a s the reduced symbol ao E

the

ellipticity condition for a can be stated in the following equivalent fashion: Proposition 3.12.5.

L e t a E Lv(Rn) and l e t aO,aO0 and ao be its p r i n c i p a 2 , p r i n c i p a l homo0

geneous and p r i n c i p a l reduced symbols, r e s p e c t i v e l y . Then a i s e l l i p t i c of order v i f f t h e f o l l o w i n g c o n d i t i o n s are s a t i s f i e d :

z

v (x,w) E mn x an (ii) agO(x,u) P 0 , v (x,w) E nn x an E xn x (R~\{oI); (iii) ao(x,l,S) z 0 , v (x,~;) here an = { C E mn 1 151 = 11. (i) aOO(x,w)

0,

Proof. ___ It is quite obvious that ( 3 . 1 2 . 2 ) (3.12.2)

by

and letting

E~~

E +

implies (i)-(iii). Indeed multiplying

0, one gets

(i)i afterwards, multi~lying

v -v (3.12.2)

by

and letting

E

E +

immediate consequence of ( 3 . 1 2 . 2 ) of order vl+v2.

one gets (ii); finally (iii) is an

m,

and homogeneity of a (x,E,C) in 0

Reciprocally, using again the homogeneity of . a aO(x,E,S)

z

v (x,E,c)E mn

0,

x

R+

x

( E - 1, 5 )

and (iii), one finds:

(IR~\{oI).

Further, using (ii) one gets for E~ sufficiently small: -v 1 jao(X,E,W)j 2 CE , d' (x,6,w) E Rn X ( O , E ~ ] X an, and,as a consequence of (i), one finds: ja0(x,E,w)1 t C provided that

~

, ~v

(-X , E~, W )~

E IRn x [El,m)

x

nn,

is sufficiently large. Thus, one gets that

E

1-V

I

E

la (X,E,W) b c e 0

1<E>v3,

v

(X,E,<) E

mn

x

w+

x

an, where

C > 0 is some -1

(x,E,S) in 0

constant. Using one more time the homogeneity of a

(E

,S)

of order v +v2, one gets finally ( 3 . 1 2 . 2 ) . 1

Proposition 3.12.6.

Let

x

for

1XI

:

mn X

+

Rn be a diffeomorphism such t h a t X(X) Y

s AX w i t h A

E ISO(R~,IR~)

s u f f i c i e n t l y l a r g e . Then t h e e l l i p t i c i t y of s i n g u l a r p e r t u r b a t i o n s

i n op LV(mn) i s an i n v a r i a n t p r o p e r t y w i t h r e s p e c t t o such diffeomorphisms. Proof. This statement is an immediate consequence of ( 3 . 1 0 . 3 1 ) . Our purpose now is to establish a two-sided

a

priori estimate for

3.12. Elliptic Singular Perturbations

369

elliptic singular perturbations in Op L V (Rn) which is their characteristic property. It has been already proved (see Corollary 3.4.2) that if a E then uniformly with respect to op a

: H

(123

( s ),E

-t

E

E

(O,E

1

holds:

E

R 3 , i.e. one has with some

0

H ( ~ - ~ )(XIn) , ~ ,V s

constant C, which does not depend on

Lv (Rn ) ,

and u,the following inequality:

E

It turns out, that a kind of reciprocal inequality holds for elliptic singular perturbations of order v and that the ellipticity condition is also necessary for such inequality to hold. To establish this property some auxiliary results are needed. Lemma 3.12.7.

For each

(s2,s3)

E

R and each

R X

E

E (0,11 t h e following i n e q u a l i t y

0

holds: 1-s (3.12.5)

where c and N

0

depend only on s2 and

s

N +1

-S

5

2<Eq>

/<5>s2<E5>s3-s2<Ell>s3~<~>

C

, v

E

E

(o,Eo]

3‘

Proof. It is immediate that I V

5

<S>’I

6 lp1<5>p-1 ,so that the Lagrange formula for

the difference on the left hand side of (3.12.5) and this last inequality yield:

1

1 <5>s2<E5>s3-s2<Er)>s3 5

s 2 -1

<<-n>(ls21

1-s

-S

s3 <Ece>

5

2< E I ) >

+€IS

3

I<<

s -1 3

>’ e

<ECe>

1-s

)

-S 2<Eri>

3

,

= <-e(n-5) with some 0 E ( 0 , l ) . and Now using Peetre’s inequality to estimate e >P1-’l P -P <€Ce> 2 < ~ r p 2 with p 1 = s2, p 1 = s2-1 and with p 2 = s 3 , p2 = s3 -1, one

where C e

gets (3.12.5) with

c = 2maxI 1 s 2 12

I

I S 2 - i / + j s3 I

N~ = max{ s2-l

Is2/+ls3-1

I + I s3 I 1 s 2 I +Is3-l I }

noticing also that ~ < n > 5 <EW, V

E

E

(0.11.

I },

,/s3/2 I

I

In several applications the following auxiliary statement (known as Schur‘s

3. Singular Perturbations on Smooth Manifolds without Bounda y

370

lemma) turns out to be very useful Lemma 3.12.8. 0

If s(x,y) E C ( R ~x wn) and supn I Is(x,y)Idx 4 C, yElR lRn

then t h e i n t e g r a l operator s norm

:

sup x d

L'(R~)

+

I

Is(x,y)Idy 5 C,

wn

L'(IR~)

w i t h k e r n e l s(x,y) has i t s

SC:

I Is1

2
Proof. Cauchy-Schwarz inequality yields:

I (SU Estimating the last integral by C and integrating with respect to x, one gets

I Isu I Lemma 3.12.9.

-s

L e t a E l"(Rn) and l e t pS(~,S)= with respect t o (3.12.6)

E

E

(O,E

0

s

I<<> 2<~<>s3,s E m 3 . Then uniformly

E

I one has:

[ps(~,D),a(x,~,D)l: H

(s+v-e2)

,E

(wn)

+

H

(0),E

(Rn) ,

where [p (E,D),~(X,E,D)]stands f o r t h e c o m t a n t of t h e singuLar p e r t u r b a t i o n s p (E,D) = op p Proof. _ _ Since [p (€,D),am (E,D)I that x

+

and a(x,e,~)= op a, and e2

=

(0,1,0).

0, one can assume without loss of generality

a(x,E,S) belongs to S(R:)

,

as a function of x, V (E,E;)

Denote K(x,€,D) = [p (E,D),a(x,~,D)1. It is easily seen that

is an integral operator with the kernel

E ( O , E ~X] an

3.12. Elliptic Singular Perturbations where a(C,E,n)

=

37 1

Fx+C a(x,S,rl).

With V(c) = Ps+u-e ( E R E ) 2

U(<),

(3.12.6) is equivalent to the following inequality:

and where the constant C does not depend on Given that a E L'(Rn)

E

E

P-u(E,rl)/a(<,E,rl)I

,

and a(.,E,S) E S(iR:)

one has the following inequalities for ; ( C , E , ~ ) (3.12.9)

(O,E

5 CN<<>-N,

v

0

V (E,€,)

v

5

E

(O,E~]

x

Rn,

a(.,~,rl):

= Fx+<

N 2 0,

2 E L (Rn).

1 and

(E,n),

C may depend o n l y on N. N Now, (3.12.8), Lemma 3.12.7 and (3.12.9) with N

where the constant

(3.12.10)

J j;(S,~,n) / d S 5

J Ig(S,~,rl)/dn5

C,

N +n+2 yield:

=

0

C,

Rn

Rn

where the constant C does not depend on E , S , ~ I . Hence, as a consequence of Lemma 3.12.8, one has (3.12.7), so that

I

(3.12.6) holds, as well.

The following statement is the local version of a priori estimates for the solutions of elliptic singular perturbations with sufficiently small support. We recall that for each s E Rn

,

V

and a . V are defined by (2.1.13), 1 s

(2.1.15). Lemma 3.12.10.

L e t a E 1' (mn)be e l l i p t i c of order u and l e t u, ( 0 <

E

5 E ~ belong ) t o a bounded

s e t i n c t ( u 6 ) , where t h e open s e t u 6 does not depend on

E

and has diameter

6. 3

Then f o r each s E R and each such t h a t f o r 6 and holds:

E~

s'

E

a,rs, t h e r e e z i s t s a c o n s t a n t c s , s '

s u f f i c i e n t l y s m a l l , t h e fo'olowing a p r i o r i e s t i m a t e

3. Singular Perturbations on Smooth Manifoolds without Bounda y

372

Proof. L e t xo be any g i v e n p o i n t i n U6. I n t r o d u c i n g t h e n o t a t i o n

a (x) = a (x,E,D), 0

0

R

0

a ( x ) = a(x,E,D),

(x) = a (x )-a ( x ) , R(x) = a ( x ) - a ( x ) ,

0

0

0

0

one c a n w r i t e (3.12.12) a ( x ) = a o ( x o ) + R o ( x ) + R ( x ) . By ( 3 . 3 . 4 ) w i t h ( a = @ = 0 ) and as a consequence o f C o r o l l a r y 3 . 4 . 2 , has

(X-x0)

-1 6

Ro(X)

: H

uniformly w i t h r e s p e c t t o

E

(v)

,E

E (0, 1~. 0

(Rn)

,

one

3.12. Elliptic Singular Perturbations

373

By Proposition 3.1.11 one has:

I

1 lul I ( s ) , E

(3.12.17)

+I

5 C ( / la0(Xo)UI (s-v)

for each s ' E a,Vs, where

C

IuI I ( s ' )

1

may depend only on s and s ' but not on

E

and u.

Finally, combining (3.12.13). (3.12.15)-(3.12.17) and using interpolation inequality 2.1.12 (with 61 instead of 6) in order to estimate the

I

I

term l u l (s-e2) , E by A l l IUJ I ( s ) ,E and CI l u l on 6,, s and s ' , one gets the estimate :

I

with

( s t )

C

which may depend

I

(3.12.18)

I (s-v)

1 Iui 1 ( s ) , E

where the constant Now choosing

5 c ( j ja(x,E.~)ul

does not depend on

C

E

~

6 , and 6

E

,E

and u.

sufficiently small, (3.12.18) gives (3

1

Remark 3.12.11. If a E Lv (Rn) is elliptic of order v and u Ur

=

m

E L

m

( (o,Eo] ; Co(U,)

{x E Rn , 1x1 > r}, then for r large enough and

E~

)

,where

sufficiently small,

(3.12.11) still holds. Indeed, the fact that the diameter of supp u is < 6 was used in the proof of Lemma 3.12.10 only in order to estimate the term with R (x) by 0

Noticing that R (x) 0

=

(a (x)-a 0

0

(-))

C6.

E S(Rn) as a function of x, the same

argument as previously, yields the inequality (3.12.16). Now we are in a position to prove the global version of the a priori estimate (3.12.11) for elliptic singular perturbations of order v . Theorem 3.12.12.

L e t a E L'(IR") 3 each s E IR ,

be e l l i p t i c of order v. Then f o r s'

E~

s u f f i c i e n t l y small and

E alVs one has:

where t h e c o n s t a n t c does n o t depend on

E

and u.

Proof. Let B6

=

{x

E

Rn

I

1x1

C

6-l) and let {UjllSjsJ be a finite covering of B 6 n -

by the open sets U . whose diameters are < 6, so that, with U j + l

=nRxyB6,

the collection of open sets IUj}15jsJ+l is a finite covering of

JK

1

.

Let {$j}lsjsJ+l be the partition of unity subordinate to the covering

3 . Singular Perturbations on Smooth Manifolds without Boundary

314 "j

m n 15j<J+1' so that for each u E L ~ ( ( O , E ~ ]Co(Rx)) ; ,one has:

(3.12.20) u

=

I: Jlju. 15j<J+l

If 6 is chosen to be sufficiently small, then, as a consequence of

Lemma 3.12.10 and Remark 3.12.11, one finds, using (3.12.20):

Applying again Lemma 3.12.9 and interpolation inequality (2.1 gets the estimate:

with some constant s,s',v,6 and

C

which does not depend on

E

and u , but may depend on

E

0' Now, (3.12.21), (3.12.22) yield (3.12.19).

I

We show the necessity of the ellipticity condition for the validity of the a priori estimate (3.12.19). Theorem 3.12.13.

If f o r a E L'(R") and

E alvs,

s'

the a priori estimate (3.12.19) holds with some s'

<

s,

s

E R

then a is elliptic of order v .

Proof. Let

rl

E R3 and let $ E Ci(Rn)

.

We shall substitute into (3.12.19) the

functions u = exp(ic-'<x,Q>)$(x). using (3.6.31), one finds for each 3 q s E R :

and for each s E R',

s'

E alVs,

depends only on @ and

where C m,rl

s

rl.

<

s':

3

3.12. Elliptic Singular Perturbations

375

One can show also directly that (3.12.23), (3.12.24) hold. Indeed,

the brackets on the right hand side of

Denoting the difference within

(3.12.25), by g(E,n), using the triangle inequality for the norms and Lemma 3.12.7, one finds: s -1

(3.12.26) (g(EITl)I5 C E ( I E

S

-1

<En>

s N +1 3<5> $(5)11 . L (Rt)

Noticing that

one gets, using (3.12.25)-(3.12.27),the formula (3.12.23) with

A similar argument can be used for proving directly (3.12.24).

Now, using (3.12.23), (3.12.24). the a priori estimate (3.12.19) with u

rl

as chosen above, yields,after letting

where C does not depend on

n E

Wn\{O>

E

+

0:

2 and $ E L (Rn).

The inequality (3.12.28) is equivalent with (3.12.2), since a E -1 and a . is homogeneous in ( E , 5 ) E lR+ X Rn of order v +v I 1 2'

L v (Rn )

Our next step is to present the elliptic theory and, particularly, a parametrix construction for elliptic singular perturbations with their (U), S " ( u ) or K " ( U ) , which will be subsequently carried out to symbols in S" 1 PO m elliptic singular perturbations on C -manifolds without boundary.

We start with the concept of local ellipticity.

3 . Singular Perturbations on Smooth Manifolds without Bounda y

316

Definition 3.12.14.

A symbol a E s ~ , ~ ( ui s) said t o be e Z l i p t i c of order s t r i c t l y p o s i t i v e continuous on

u functions

v i f

there e x i s t

6(x), r(x) and q(x) such t h a t

for each x E u one has: -v (3.12.29)

la(x,E,S)\ 2 q(x)E

v

2<~5>v3, V

'151

E

E (0,6(x)1, V 5 E n n , 151 >r(x)

The corresponding singular perturbation op a = ~(x,E,D)i s called e l l i p t i c o f order v . For symbols a E S v ( U ) the ellipticity condition (3.12.29)

can be

stated in the following equivalent way by using the homogeneous extension of a principal symbol a0 (x,E,<) of a to R~

X

R+

X

(IRg\{Ol)

(by an abuse

of notation this homogeneous extension, which is well defined, will also be denoted by . a (x,E ,C)

.

Proposition 3.12.15.

A symbol a E

sV(u)

aO(x,E,<) onto

i s e l l i p t i c of order v i f f the homogeneous extension

xn

x R+ x (R~\{o}) of i t s principal symboz s a t i s f i e s the 5 condition: there e x i s t s a s t r i c t l y p o s i t i v e continuous on u f u n c t i o n q(x)

such t h a t f o r each x E

u holds: -v

(3.12.30)

/aO(x,E,S)\t q(x)E

v

'151 2 < ~ S > v 3 , V

(E,<)

E R+

x

(Rn\{Ol)

Proof. It is quite obvious that (3.12.29)

implies the same inequality for

/
2 r(x). Now the homogeneity of a (x,E,<) a0(X,E,~) when 0 < E 6 6(x), 0 in ( E - ~ , < ) of order vl+v2 implies ( 3 . 1 2 . 3 0 ) .

conversely, since la-a

0

I

5 Cla

0

I

when 0 <

E 6

6(x),

151

b r(x) with

a constant C which can be chosen as small as one wishes when 6 ( x ) is sufficiently small and r(x) sufficiently large, it is quite obvious that (3.12.30)

implies ( 3 . 1 2 . 2 9 ) .

Before the next statement will be formulated, we introduce a useful notation. For symbols a and b in Sv

180

(U) denote by a-b a symbol in Sv 1,o(")

whose asymptotic expansion is

In fact, as a consequence of Theorem 3.6.2,

formula (3.12.31)

the (non-commutative) multiplicative group structure on

defines

11 Sv (U), the VER3 l r 0

3.12. Elliptic Singular Perturbations product of two symbols or order

u

377

and v being well defined modulo a symbol

of order ( v +p1,--,v3+u3). Thus, with the natural additive group structure 1

and topology on

u

(U) defined by the semi-norms (3.3.1)' U Sv (U) becomes 1.0 v 1,o an algebra with addition and multiplication as continuous algebraic operations Sv

L,

with respect to its topology. Proposition 3.12.16.

Let a E

The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

S' l,o(u).

(i) a i s e l l i p t i c of order v. (ii) There e x i s t s a r i g h t i n v e r s e a-l E S-'(U)

(iii) There e x i s t s a l e f t i n v e r s e ' ; a

E

(3.12.33) (a;'*a-l)

S 1( ,o o f - m r o()u )

E s-'

1to

( u ) of a, i.e.

.

-1

Moreover, a l can be chosen i n such a way,that -1 a E

of a, i . e .

-1 -1 -1 al =ar = a , any i n v e r s e symbol

s;~,(u) being w e l l d e f i n e d i n t h e f a c t o r aZgebra

(-V1,-",-V3) (U) . s-v 1 .o (u)/sl,o Proof. -

Let a E

Sv

1.0 as follows:

(U) be elliptic of order v . Define the functions a-l recursively, k

and for k = 1,2,..

Since a E when 0 <

(U) is elliptic of order v, the symbols a;'(x,E,<) are welldefined 1 to -1 6 6 ( x ) , 151 Z r(x). We shall denote by ak ( x , E , S ) any smooth

Sv E

extension of these symbols onto

E ( 0 . ~ ~X 1lRn.

An elementary -1 v(k) computation shows (using induction in k) that a E S1,o ( u ) with - (-V1,-v2-k,-v ) . Thus, Theorem 3.6.2 guarantees the existence of a V(k) 3 -1 symbol a ( X , E , ~ )E S-' (U) , having the asymptotic expansion: 1to (3.12.36) a-'

-

X k>O

(E,<)

-1 a k'

Now, (3.12.34), (3.12.35) are derived to satisfy (3.12.32) for a-l defined by (3.12.36), and that ends the proof of (ii).

3. Singular Perturbations on Smooth Manifolds without Boundary

378

Now, suppose that (ii) holds. This implies that -1 (U), so that for each compact K (a(x,E.E)a (x,E,E)-I) E s 1 ,o there exists a constant CK such that la(x,E,S)a-'(x,E,S)-l~ 5

cK< ~ > - l ,v

(x,E,<) E K

x

C

U,

(o,Eo] x

mn

2 rK, one gets for Choosing rK > 0 ,ko large to have C i0-l < f , V K such 5 and V E E (O,E~],V x E K with (perhaps a different) constant Ck

the inequality:

+

-1

5 la(x,E,<)a

( ~ , E , Ej )5 c;ja(x,E,E)

u1 / E

<s>

-v

-V

2<E~>

3

and that implies the ellipticity of a of order v. We have to show the equivalency of (i) and (ii). Using the same argument as previously and seeking again the asymptotic expansion of -1 E S1),o(U) in the form a 1 a-l

N

c k>O

a-l k

-1 and the one gets, using (3.12.31), the same formula (3.12.34) for . a -1

following recurrence relations for ak , k > 0:

to satisfy (3.12.33). Now, the previous argument applies. To end the proof, we have to show that one can choose the same right and

left inverse symbol in S-v (U) for an elliptic symbol in Sv

1 ,o

180

(U). This is

an immediate consequence of the associativity of product (3.12.31). Indeed,

Corollary 3.12.17.

Let a ( x , ~ , ~be) in op sv

( u ) . Then the following conditions are equivalent:

1 ,o (i) a(x,E,D) is e l l i p t i c of order v.

(ii) There e x i s t s a-'(x,c,~) i n op s-'

1 to

(u) such t h a t

a(x,c,D) 'a -1 (x,€.D)-Id : 0 (mod Op S (0,-",O) (U)) 1 ,O

3.12. Elliptic Singular Perturbations

where Id is the identity operator and

0

379

stands for the composition

(multiplication) of singular perturbations, as defined in Section 3.7.

I

Definition 3.12.18.

A s i n g u l a r p e r t u r b a t i o n a-'(x,~,D) s a t i s f y i n g (3.12.37) i s s a i d t o be a

p a r m e t r i x f o r a(x,e,D) E op

S~,~(UI.

Corollary 3.12.19.

( u ) . Then a(x,~,D)has a pararnetrix a-l -1 (x,E,D)E op s-' 1 .o I,O(U) i f f a(x,E,c) i s e l l i p t i c of order v. Furthermore, a (x,E,D)i s e l l i p t i c of L e t a E sv

order -v and i s w e l l d e f i n e d i n S;~o(~)/~:~~l'-m'-v 3) (U).

I

Corollary 3.12.20.

Let a E E

Sv

1 to

(U).

Then a(x,E,D) i s e l l i p t i c i f f uniformly w i t h r e s p e c t t o

E ( 0 . ~ ~one 1 has:

(3.12.38)

-1 where a (x,E,D)i s any pararnetrix of a(x,~,D),HComP ( u ) i s t h e subspace ( s ) ,E (U) o f a l l U(X,E) w i t h t h e i r support i n some compact K c U and i f l H H

V

loc(s) , E (U) i s t h e space of a l l U(X,E) such t h a t x(x)u E H (S),E

x E

u do n o t depend on

Co(U); here K and

( s ) ,E

(U)

,

E.

Indeed, the top line in (3.12.38) i s a consequence of Theorem 3.4.1, since one can extend x

X

1 on an open set

a(x,~,S)to x E Rn by introducing x(x)a(x,~,S),

+

U',

U 3 3

U'

3

K,

continuous (uniformly with respect to

for any given compact K

C

x E)

m

E C0( U ) ;

further, there is a natural

embedding:

U.

The bottom line in (3.12.38) follows by the same argument, given that the -1 existence of a parametrix a (~,E,D) E op s;y0(u) of a(x,E,~)E op s Y,O(U) is equivalent with the ellipticity of a E S y , o ( U )

of order v.

Corollary 3.12.21.

If a E s ~ , ~ ( iu s) e l l i p t i c of order

v, t h e n any s o l u t i o n uE E m

E ' ( u ) of t h e

c (u) t o o , and moreover, i f EvlfE,O < E 2 E ~ ,belongs t o a bounded s e t i n cm(u) and uE, 0 < E 2 E 0 belongs t o a bounded s e t i n E ' ( u ) , t h e n f o r any g i v e n compact K c u one has: equation a(x,E,D)uE

=

fE w i t h f E E cm(u) i s , i n f a c t , i n

3. Singular Perturbations on Smooth Manifolds without Boundary

380 uE, 0 <

E

5

E

beZongs to a bounded s e t in Cm(K).

0'

--Y

Indeed, writing u b(x,E,D) f Op

S

1 -1

a

(x,E,D) r"la(x,E,D)u+b(x,&,D)u with

( o f - m f o )(U) (as 1#O

a consequence of (3.12.33)), one gets

= 6

a3

immediately the conclusion that {u}o<EcE belongs to a bounded set in C ( K ) , 0

since {b(x,E,D)u } O < E L E and -V

Cm(U) and

1

6

{ E V 1 a(x,s,D)u)O<E,E

0

-1 a (x,E,D)E Op

are in a bounded set in 0

(0,-V2,-V3) S

(U) has the pseudolocality

1#O

I

property (see Theorem 3.5.4). Remark 3.12.22.

The elliptic singular perturbation theory extends automatically to systems of singularly perturbed pseudodifferential operators, whose

symbols are pxp matrices depending on (x,E,<)E

u

(O,E

x

0

1

x

En.

In all

the definitions and results above the absolute value of symbols should be interpreted as the matrix-norm in Hom(CP;CP). Remark 3.12.23. Let M be a smooth paracompact manifold without boundary and let a(x,E,D) E Op

Sv

1

(M) be a singular perturbation of order v on M (see

Definition 3.10.13). If a(x,E,D) f Op S"(M), then its principal symbol a ( x , E , ~ is ) well defined 0

and, as a consequence of (3.10.31). aO(x,E,5) is a function on R (E

E 1R+

,

*

X

T (M)

(x,S) E T*(M), T*(M) being the cotangent bundle on M) with values

in Hom(CP;CP). Thus, the ellipticity concept can be extended to singular perturbations on M, the ellipticity condition for a E S v ( M ) being the

v requirement: a (x,E,<) E ISO(C~;C~), (ao(x,E,<)I 2

CE

E

R+

-" 16 I "2<E5>"3, v x E M, where

,v

(x,~) E T*(M), 5

C > 0

+ 0,

is some constant. The

same argument as previously in the case M = R n , shows that the ellipticity condition is equivalent with the a priori estimate (3.12.19) where the norms are in corresponding spaces H

( s ), E

(M).

One can also show (but it will not be done here), that if a(x,&,D) E Op S"(M)

is elliptic and M is compact, then the kernel of m

a(x,E,D)

:

H

V 6 E (O,Eol;

( s ) , )E'(

+ H ( s - v ) ,E

(M) belongs to a bounded set in C (M),

furthermore, the range of a(x,~,D) is the orthogonalcomplement

of thekernel ofthe adjoint a(x,~,D)*. Therefore, theindex dim coker a of an elliptic singular perturbation a(x,&,D) H

(s-v)

fact, 0

( M ) does not depend on s

X(E) :

H

=dim ker a ( s ),E

(M)

*

E 1R3. It will be shown later, that, in

,E

X(E)

5

X ( 0 ) if a(x,s,D) has the reduced pseudodifferential operator 0

) the index of a (x,D). a (x,D), where ~ ( 0 is

3.12. Elliptic Singular Perturbations

38 1

Example 3.12.24. The singular perturbation (3.10.52) on the unit circle

a1

is elliptic of

order v = (0,0,1), since its principal symbol a (x,E,S) = l+EIE;I 0

-

<ES>.

As a consequence of (3.10.54) kernel and cokernel of this singular

perturbation are both trivial and its index

X(E)

=

0 , t/

E

2 0.

The next example is concerned with Stokes' singular perturbation (3.2.43). Example 3.12.25. Consider the singular perturbation S (E,D)whose symbol is the following

e

nxn matrix s

E

(E,S):

S ~ ( E , S )= where w = 5 / 1 5 / , w

E

2ie+E21
w X w T) ,

T .

is the transpose of w and wxwT is the matrix with the

entries wjok, 1 5 j , k 5 n. Any vector u E Cn such that <w,u> = 0 is an eigenvector of s (E,S) e 2 2iE 2 = E (e + 15 ) , while w is an eigenvector with zero

I

with the eigenvalue u eigenvalue. Thus,

is not elliptic. However, considered on the

se(E,S)

orthogonal complement of

w

in Cn it becomes elliptic of order (-2,0,-2).

We are going to introduce a more general ellipticity concept for systems of singular perturbations in order to include in the elliptic theory Stokes' singular perturbation (3.2.44). Introduce again the notation

and introduce

Definition 3.12.26.

A matrix symbol a(x,E,S) = 1 lakj(x,E,S) 1 1 with a E LvkJ(mn) is said to kj be elliptic in the sense of Douglis-Nirenberg, if there exist vectors vk E IR 3 , LI; E n 3 , 1 5 k 5 p, such that the symbol ;(X,E,~), N

(3.12.40) a(x,E,S) = q,,(E,~)a(x,E,S)q,(E,~)

is elliptic of some order v in the sense of (3.12.2) where the norm in Hom(CP;CP) .

I. 1

stands f o r

3. Singular Perturbations on Smooth Manifolds without Boundary

382

A matrix singular perturbation a(x,e,~)is said to be elliptic in the sense of Douglis-Nirenberg, if its symbol a(x,~,S)is elliptic i n the same sense. The following result is proved by repeating word for word the proof of Theorem 3.12.12.

Theorem 3.12.27.

Let a(x,s,D) be e2Ziptic in the sense of Douglis-Nirenberg and let 1-1k' 1 5 k 5 p and v be the corresponding vectors in Definition 3.12.26. Then uniformly with respect to (3.12.41) 119

-P

where s' E R

3

(E,D)u/

1 (s)

E

,E

E (O,E I 0

-j

one has:

I

Iqu,(E,D)a(x,E,D)ul ( s - v )

is any order s.t. s' < s-uk, 1

,E

+

I lul I

(s'), €

2 k 5 p.

Example 3.12.28. Consider Stokes' singular perturbation the symbol a(A,S):

where, as previously,

ST

is the row-vector transpose of 5 f lRn

.

With any u = ( u t,O)T such that <S,u'> = 0, one finds that 2 T a(A,S)u = ( A + l < l ) u , so that on the orthogonal complement of Span{(S,O) ,en+l] 2 n+ 1 a(A,S) reduces to ( A + / < ] )I in C (n-l) . Furthermore, an easy (n-l) computation shows that the other two eigenvalues of a(A,S) are, respectively,

Hence, if u - = (S,p-) is an eigenvector associated with the eigenvalue restricted to the orthogonal complement of u- in CnTi has 2 (X+lSl 2), while X/El2 when its norm of order (X+lS ) , since x+

X-,

then a ( A , S )

I

(A+l<12)

-t a.

-

-

This suggests that a(X,<) might be elliptic in the sense of

Definition 3.12.26. It is easily seen that for larg X I < 77 ( 1 8 1 < IT/^), 5 E Rn\{O} -1 inverse matrix a(X,S) exists and is given by the formula

the

3.12. Elliptic Singular Perturbations where, a s previously, w

=

5 / 1 5 ] and

w

T

383

1s the transpose of o.

1 5 k 5 n+l in Definition

The latter suggests a choice of pk,L$, 3.12.26 as follows: (3.12.44)

uk

= p i = 0, 1 5 k 5

n, !Jn+l=

=

(Z,-1,2),

a reasonable candidate for v in the definition being v = (2,O.Z). Indeed, with p . chosen as in (3.12.44) and with 7

C2 =

A-',

one finds,

using (3.12.39),(3.12.40):

N

The matrix a(A,c) on the orthogonal zomplement of Span{ ( 5 , O )

,en+l

is s t i l 1 the same multiple of the (n-l)x(n-1) identity i.e. (X+/S/2)I(n-l)x(n-1), N

and an easy computation shows the two other eigenvalues of a(X,c) are, respectively,

( € , E l above by introducing the

Modifying slightly the definition of q

N

factor i on the right hand side of (3.12.39::

one gets a(h.6) which is the

following symmetric matrix

whose eigenvalues are ( A + 15 1 Y

a(A,C)

2

)

and ( 3 ) (1+45)(X+ 15

can be estimated by ( f ) (fikl)( A + ( S /

2

)

1 2)

so that the norm of

frombelowandabove, respectively.

N

Thus, a ( h , S ) is elliptic of order (2,0,2) in the sense of Definition 3.12.2. The estimate (3.12.41) becomes in this case: (3012.45)

c

lSjSn

+

I bjI I ( s ) , 6

+

I

I (s-pn+l)

lUn+ll

N

,€

I IUI I (s'),E

with any s' < s,

s'

< s-pnCl, where e . is the j-th coordinate vector in Cn 3

3 . Singular Perturbations on Smooth Manifolds without Bounda y

384

With u

=

(u',u~+~) a(A,D)u ~, = (f',fn+l)T, one can rewrite

(3.12.45)

in the following fashion:

where, as previously, w = (2,O.Z) and e2 The reduced symbol for :(A

while the inverse :(A,<)-'

=

(O,l,O).

, 5 ) is :

has the form:

Introducing

one can write (3.12.50)

a(X,S)

=

(A+l5l

2

)

0

S(h,S)a ( 5 ) S ( h , S ) ,

ao(C) being the reduced symbol for a(A,S). An easy computation shows that

Hence, (3.12.52)

a(A,S)-'=

( A + 1 5 / 2 ) -1 diag(Inxn,(A+15/ , 2))a 0 (5)-'diag(In,,,(X+~S~*)),

v

E

C,

/arg A / <

T,

v c

E Rn\tO},

and the multiplication from the left by (A+1S12)-1S(h,S)-1 right by

S(A,S)-l

and from the

reduces a(A.5) to a0 ( 5 ) .

In x-representation the left reducing operator is the convolution with diag(A -1 A. -2 (h-fIxl)Inxn,l)

3.12. Elliptic Sinplar Perturbations

385

2 and the right one is the convolution with diag(1 nxn' (X+ ID1 ) 6 (x)) , where A;(E~X~J is the family of distributions, defined by (3.2.21) with E E C, /arg € 1 < ~ / 2 ,E =

-1

x '.

Example 3.12.29. Consider the Oseen linearization of the system of partial differential equations in the Nonstationary Magnetohydrodynamics under the noncompessibility assumption upon the moving fluid (see, for instance [La-Li, 21). It has the following form:

at

- aAv + v

+ VP

=

0

T (3.12.53) VTv = 0 , V H = 0

aH at

BAH + H

=

0

where v(x,t) is the velocity field, P(x,t) is the pressure, H(x,t) is the magnetic field and where the linearization is taken at the stationary magnetic field H

0 and the stationary velocity field vm : Const,the

density of the fluid being p

?

1; here, as usual V is the gradient operator

and the upper index T stands for the transpose, so that VT is the divergence operator; further a > 0, 6 > 0 are given. If the initial value H (x) for H(x,t) satisfies the condition 0 T V H (x) = 0, then, as a consequence of the last equation in (3.12.53) one 0 T gets that V H(x,t) E 0, V t 10, so that under this assunotion upon the initial value for H in the overdetermined system (3.12.53) one can drop the equation V'H

=

0, thus getting a determined system of equations.

After the Laplace transform with respect to t one gets the Stokes-Oseen problem in Magnetohydrodynamics, whose symbol a(X,c) has the form:

(3.12.54) a(X,E) =

Further, introducing new unknown functions

one reduces (2.12.53) to the Stokes-Oseen problem with the principal symbol aO(X,5),

3 . Singular Perturbations on Smooth Manifolds without Boundary

386

With

1x1

=

E - ~ ,/arg XI <

T,

a (h,D) is an elliptic singular 0

perturbation in the sense of Definition 3.12.26, as it follows from Example 3.12.28. Notice that for each given A E constant

Co >

for B = C a with some 0 in (3.12.55) is also elliptic in

0%-and

0, a << 1, the symbol a .

the sense of Definition 3.12.26, the order v appearing in this definition being v = (0,0,2), while for a and 6 fixed and A

-t

m

this order is (2,0,2).

The next example is concerned with von Karman's equations for the buckled state of a thin elastic plate. Example 3.12.30. The following system of equations

describes the buckled steady state of a thin elastic plate; here h is the thickness of the p1ate.h << 1, <(x) is the deflection and x(x) is the stress -1 = Po(x)h , P (x) being a given exterior forces'density.

function, P(x,h)

0

The linearization of (3.12.56) at 5 following singular perturbation (h = (3.12.57) (E21DI4+)c XO

=

E)

=

0,

x

=

x0 (x) leads

to the

for the deflection:

P

where

If

( x ) 2 y(x)Id with some continuous y(x)

A

0 , then the singular

XO

perturbation (3.12.57) is elliptic in the sense of Definition 3.12.14 of order v = (0,2,2). The ellipticity of (3.12.57) is equivalent to the strict convexity of the given deflection

x 0 (x)

at each point x C

U.

3.12. Elliptic Singiilar Perturbations

387

Next the elliptic theory of difference singular perturbations will be considered. We start with one parameter families of difference operators introduced in Section 3.11. Definition 3.12.31. A symbol a E F v ( u ) i s s a i d t o be e l i p t i c of order v

=

(vl,v2) i f i t s

p r i n c i p a l symbol ao(x,rl) s a t i s f i e s t h e c o n d i t i o n (3.12.58) ]ao(x,n)I 2 m(x) /w(n)I

v2

where m(x) i s some continuous s t r i c t l y p o s i t i v e f u n c t i o n on U. A f a m i l y of d i f f e r e n c e operator,. a(x,h,D) E op Fv(u) i s s a i d t o be

e l l i p t i c of order v, i f i t s symbol a(x,h,hS) i s e l l i p t i c of order v. We shall specifically treat the case U = lRn, using the following global version of symbol classes

Fv ( U ), Gv ( U ).

Definition 3.12.32.

-

A symbol a E FV(mn) ( r e s p . a E G'(IR") ~

FV(wn) ( r e s p .

G'(IR~)

I is

s a i d t o b e in t h e c l a s s

I if t h e r e e x i s t s a symbol a (h,hC) E F'(E~)

am(h,hE) E GV(Rn)I such t h a t t h e symbol

(resp.

hu1-v2ma'(x,h,hE)=

h~'<~>-"2(a(,,h,h5)-a_(h,h~)), a s a f u n c t i o n of xERnbelongs t o S(Rz) UnifOYW?ly %n (h,S).Asymbol aEFV(mn) i s s a i d t o b e ( g l o b a l l y ) e l l z p t i c o f order v = ( v , , v ~ ) ,

=

if t h c r e e x i s t s a p o s i t i v e c o n s t a n t m such t h a t f o r t h e p r i n c i p a l symbol a (x,n) of a(x,h,h<) one has: 0

(3.12.59) la0 (x,rl)I 2 m/w(rl)I

v2

, V (x,rl) E IRn

X

(T>{O}).

The l a r g e s t m such t h a t (3.12.59) i s v a l i d , i s c a l l e d t h e e l l i p t i c i t y c o n s t a n t f o r a(x,h,hS). Remark 3.12.33. As a consequence of Proposition 3.11.23, ellipticity is an invariant

property of difference operators with respect to the group of transformations SL(n;Z). The following statement can be proved in exactly the same way as the analoguous statement in Theorem 3.12.12 (so that its proof is left to the reader) : Theorem 3.12.34.

L e t a E Fv(En) be g l o b a l l y e l l i p t i c of order v. Then f o r each couple of orders

(s,s')

E m2x IR'

w i t h s' < s, s;

= sl,

there e x i s t s a constant c

s,s'

3 . Singular Perturbations on Smooth Manifolds without Boundary

388 such t h a t

Now a parametrix construction (analogous to the one for elliptic singular perturbations) will be briefly sketched for families of elliptic difference operators. First, using (3.12.31), one provides U Fv (U) = Fl,o(U)with a multiv 1,o plicative group structure , the product a1*a2 of two symbols ( 1 ) +\) (2) k (U) with the (u), k = 1,2, being any symbol c E Fv a E FYj:) 1 #O asymptotic expansion (3.12.31) in the following sense : for each integer E J > 0 the difference

where

pN =

~(')+v(~)-Ne~, e2 = (0.1)

Furthermore, Proposition 3.12.16 carries over to the elliptic s y h o l s in classes Fv (U) with the only difference that instead of Theorem 3.6.2 1 PO one will have to use the corresponding statement, analoguous to the one in

Fv'(k) (U) with 1.0 Further, for a -1 and a-l in given elliptic symbol a E Fv (U) the corresponding symbols a 1 1,o F;Yo(U) are defined by the same recurrence procedure (3.12.34), (3.12.35). (-v1 , m ) -1 -1 (U) and Corollary modulo a symbol in F Again, one has a = al 1n o 3.12.17 carries over to the elliptic difference operators with their Theorem 3.11.5, which extends to any sequence of symbols in

> v(l) > v ( ~ )

... > v ( ~ )> ...,

V:~)+V;~) +

-m

for k

+ m.

symbols in the classes Fv (U), as well as all the other corollaries and 1 .o remarks which follow Proposition 3.12.16, with the only difference that the spaces H

(s)

(U) should be replaced ,P' H(s) ,E

by the corresponding

spaces of families of meshfunctions H (s),h(If) . Next, several examples of elliptic difference operators will be considered. Example 3.12.35. h Consider the family of difference operators aZ whose symbo-1 h o(a-) = (cl ,h+ic2,h) E F("') (Rn) approximates the Cauchy-Riemann operator z h a = wl+iw2. This approximation of a_, the principal symbol of o(a-1 being . z

2

a-

is not elliptic, since the circles Il+iXI = I and Il-XI in the complex

plane C

always have an intersection at a point X o # 0, so that there is

always a point

T.

o

E T2\{O} r

l

such that aO(rlO) = 0 .

3.12. Elliptic Singular Perturbations

389

Rewriting the Cauchy-Riemann operator 2- as a system for the real and imaginary parts of the unknown complex-valued function, one can indicate h a whole family At,depending upon the parameter t E R , o f approximations of

a-2' h At

=

it

all of them being elliptic, but the one corresponding to t

=

*

t ; here, as

previously, D are the (formal) adjoint operators for D x . ,h x .,h' 1 1 h Indeed, one finds immediately that /det(-iAt)I = 2 2 2 *2 21 z2 = I(t t(1-t) )1#+t(l-t)(5 +5 2 (2t-1) 1 5 : .

)I

Considering a more general system for pseudoanalytical functions (3.12.62) $x -T(X)$x -0(X)ix = 0 , $x +0(X)Qx -T(X)$x 1 1 2 2 1 2

=

0

> 0 (see [Bers, 1 I ) , One checks easily that 0 the approximation by finite differences of (3.12.62) with the symbol

which is elliptic if u(x) 2 u

is elliptic and one can indicate in this case, as well, a one parameter family of elliptic approximations of (3.12.62) similar to the case of the Cauchy-Riemann system considered above. Example 3.12.36. Consider the following approximations by finite differences of - A . Obviously, the approximation with the symbol u1 = / 5 1 2 is elliptic and it corresponds to the classical approximation of -A. Next, consider the approximation with the symbol u2 = [ 5 ' principal symbol lw'12+w;

I

2+
vanishes when

it is not elliptic, since the

ri

approximation of -A with the symbol u 3 =

= IT

(wn

=

2i) and I w ' /

nl 5 I 2- (h2/6)

=

2. The

2

I Ckl I Sj I

Z

is

l(k<j$n elliptic if n 5 3; for n > 3 it is not elliptic, since its principal

Iw(q)

[ sufficiently small and strictly negative when

lw(rl)

I

=

It is easily seen that the approximation of -A with the symbol

2 , n > 3.

3 . Singular Perturbations on Smooth Manifolds without Boundary

390

is elliptic in any dimension, the accuracy of this approximation being 6

O ( h ) when h

-f

0.

Example 3.12.37. The difference operator p(x,h,hD)

=

l+q(~)~(2-0-0-~) with 0 = exp(ihD) the

1

shift operator on F$, , is elliptic,of order ( O , O ) , provided that 2 2 /Re q(x) i qo > 0 , V x E IR . It approximates the identity operator with the 2 accuracy O(h when h + 0. Besides, as discussed in Example 3.11.21, the 2 2 -1 difference operator a(x,h,hD) with the symbol a(x,h,h<) = (l+q(x) /w(h5)) )

I

is a quasi inverse of p(x,h,hD), i.e. p where

41)

0

are some difference operators with the symbols o(R(1)) in Fl,O(IR). h More precisely, using Theorem 3.11.17, one finds the asymptotic expansions for the symbols of a(RL1))

0

-

c

ia

a>O

(2)) (Rh

ia a>O

where, as previously, for each symbol b(x,h,hE) we have denoted: b(a) (x,h,q) = Dab(x,h,rl), b (x,h,n) = D>(x,h,n), D = -id/dx. (a)

Using the recurrence procedure as in (3.12.341, (3.12.351, and the Corresponding analogue of Theorem 3.6.2 for symbols in Fv (U), one can 1 to find a symbol p-'(x,h,ht) such that for the corresponding difference -1 N (1) -1 operator p-l(x,h,hD) holds p o p -Id = h Rh,Nr P D p - Id = hN$yA, with any integer N > 0 and with some R(k) E Op FG ( W ) , k = 1,2. Of course, h,N 1 ,o -1 (x,h,hD) is also elliptic of order ( 0 , O ) .

p

Example 3.12.38. The following system appears in the linear elasticity theory: (3.12.63) P.(D

)

=

u~~ nx n+ ( p + ~ ) v x v ~ ,vxvT

I / a x , a x I I16k,j6n,

=

I

k

where I is the identity in Hom(Cn,Cn), V is the gradient and VT is the nxn divergence, !J > 0, X 2 0 being given coefficients. The system (3.12.63) is elliptic in the classical Petrovsky sense, since n 7n det A ( S ) t !J 151- , v 5 E IRn. T

Denote by Vh and Vh the forward difference approximation on the grid

.If: of the gradient V

and the divergence V

T

,

respectively, and let

3.12. Elliptic Singular Perturbations

*

391

T *

be the corresponding backward approximations of V and V Vh, (Vh following two approximations of (3.12.63) are elliptic

T

. The

where Ah is the classical approximation of the Laplacian with the symbol

12.

li(h,hS)

It is also easily seen that the whole family of approximations tAh(Dx,h)+(l-t)<(D x,h) , t E [O,ll, is elliptic, the accuracy of the approximation for t = f being higher than that for other values of t.

Ah(Dx,h) t

=

Example 3.12.39. The system (3.12.63) is a specific example of strongly elliptic operators in the sense of M.I. Vishik, the general form of these systems being (3.12.65) A(Dx)

z

=

la1

with A

a,B

=I 51-

Dz(Aa,6(x)D6) + terms of lower order,

(x) symmetric matrix satisfying the condition:

with some constant y o > 0. It is quite obvious that the family of difference operators

is an elliptic approximation of the principal part of A(D

)

in (3.12.65).

To end this paragraph, elliptic difference singular perturbations will be considered. W e start with the corresponding symbol classes with two small parameters

E

E (0, 1~ and h E (O,hol. 0

Definition 3.12.40.

A f u n c t i o n a(x,c,h,q) : u x ( 0 . ~ ~x1 (O,ho] x T i C is s a i d t o be a symbol i n t h e cZass F Y , o ( u ) , v = (vl,v2,v3)E ~ 3 i ,f for each compact K c u and each p a i r of m u l t i - i n d i c e s a,@ t h e r e e x i s t s a c o n s t a n t c +

a,B,K

t h a t holds:

such

3. Singular Perturbations on Smooth Manifolds without Bounda y

392

If a E

but a B F;,,(U),

FY,,(U)

V

u < v, then v i s said t o be the order

of a.

i s said to belong t o the c l a s s FV(u) i f there

A symbol a E F;,,(U)

e x i s t s a f i n e t i o n ao(x,p,n) a .

E

(Tl,n\{O})

Cm(u x

each compact s e t

w i t h some u $6(t) E $6(t)

=

K c

u

:

0

n C (u x

R+

u

x

x

R+

R+

;l,il + c r (T~,~\{O}) such t h a t uniformly on

x

one has:

(T:,~\{o})

x

(pl,p2,u3)s: v such t h a t

IA + p

1

v 1+v 2; here, as previously,

<

2

cm(E+)i s a cut-off f u n c t i o n such t h a t $6(t) : 0 f o r t E [ o , S l , 1 f o r t t 26, $ 6 ( t2) 0 , v t E E,, 6 > o being a given s u f f i c i e n t l y

small number. The f u n c t i o n aO(x,p,n) s a t i s f y i n g (3.12.67) i s called the principal symbol of a E FV(u). Proposition 3.12.41.

L e t a E F v ( U ) . Then i t s principal symbol ao(x,p,n) i s well defined and f o r

each compact cowstalzt c

u and each pair of m l t i - i n d i c e s a,8 there exists a such t h a t hold:

KCC

V (x,p,n)

E K

X

R+

X

(TY

rn

\{o}).

Proof. _ _ Rewriting (3.12.67) in the form -V

(3.12.69) ((s+h)v1

-V

2a(x,E,h,h5)-h

-V -V 2 $ 6 ( 1 5 / ) < 5 > 2ao(x,h/E,hS))<EC>

E

FV-V

1 ,O(U)' setting

I- =

hc,

p =

h/E and letting h

+

0,

E

+

0 in such a say that

=

h/E

be constant, one gets the conclusion that the limit on the left hand side of (3.12.69) is zero, since

u1

5 vl,

ul+u2

<

vl+v2, u < v, so that ao(x,p,n)

is well defined by the formula:

v +v (3.12.70) ao(x,p,q) = lim h h+O

2(1+p-1)v

at each given point (x,p,n) E

U

X

R+

X

1

a(x,hp

-1

,h,n),

(TY,n\{O}).

The same argument leads to the following formula

v

+V

(3.12.71) D BDaa ( x , p , n ) = h 1 x n o

-la/

2

(l+p-l)' l D ; D : a V (x,p,n)

(x,hp-l, h , 0 )

E

U x

R+

X

,

(TY,n\{O}).

3.12. Elliptic Singular Perturbations

393

Now,using (3.12.661, (3.12.71), one g e t s (3.12.68).

I

D e f i n i t i o n 3.12.42.

A symboZ a E F V ( u ) i s s a i d t o be e l l i p t i c of order v

p r i n c i p a l symbol a

0

(x,P,I)

=

(vl,v2,v3), if i t s

s a t i s f i e s the condition: there e x i s t s a

s t r i c t l y p o s i t i u e continuous on u f u n c t i o n q ( x ) such t h a t for each ho Zds : (3.12.72) / a o ( x , p , n )

I

2 q(x)]w(ri)

x

E u

1 v 2<~-1w(n)>v3

The g l o b a l v e r s i o n of D e f i n i t i o n s 3.12.40, 3.12.42 and o f P r o p o s i t i o n

3.12.41 f o r U

=

b a t i o n s i n Op

L v ( Rn )

Rn

i s t h e same, as p r e v i o u s l y for t h e s i n g u l a r p e r t u r -

-

and d i f f e r e n c e o p e r a t o r s i n Op Fv(Hln)

.

In particular,

t h e g l o b a l e l l i p t i c i t y means t h a t i n (3.12.72) one c a n t a k e a p o s i t i v e c o n s t a n t q i n s t e a d of a p o s i t i v e c o n t i n u o u s f u n c t i o n q ( x ) .

W e a r e g o i n g t o g i v e a n o t h e r e q u i v a l e n t d e f i n i t i o n of t h e e l l i p t i c difference singular perturbations

which have a reduced symbol.

D e f i n i t i o n 3.12.43.

A symboL a E Fv

1 ,o

0

a (x,h,hE,) E

FV’

1,o

-V

(3.12.73) <<>

where e

=

2

(u),

v = (v1,v2,v3), i

,

(U)

v ‘ = (v -V

((E+h)”<E<>

v ),

2’ 3 3

s s a i d t o have a reduced symbol

i f holds:

a(x,E,h,hS)-a

0

(x,h,hS))

E Fi:O(U),

(I,-I,~).

I t i s q u i t e o b v i o u s t h a t t h e reduced symbol i s u n i q u e l y d e f i n e d ,

i f it

e x i s t s , a n d , moreover, one h a s

0

(3.12.74) a ( x , h , h S )

= lirn

(E+h)’l

a(x,E,h,hS)

E+O

for e a c h g i v e n x E U , h E ( 0 , h

1,

0

ri = i 5 E Ty,n.

L e t a E Fv(U) be e l l i p t i c o f o r d e r u and assume t h a t a h a s t h e reduced symbol a’. a.

L e t a o ( x , p , n ) b e t h e p r i n c i p a l symbol of a . O b v i o u s l y , 0 h a s t h e reduced p r i n c i p a l symbol a o, a s w e l l . B e s i d e s , i t i s e a s i l y

seen t h a t

0 (3.12.75) a,,(x,n) U

=

1

l i m ( h V 2 l i m (ECh)” h+O E O ’

a ( x , ~ , h , n ) )=

3. Singular Perturbations on Smooth Manifolds without Boundary

394

Furthermore, introduce the principal quasi-homogeneous symbol aO0(X,n) by the formula: (3.12.76)

aOO(x,n) = lim h

v +v 2 3

ao(x,h,n).

h+O

It is quite obvious that if an elliptic symbol a E F " ( U ) a

of order

V

has

reduced symbol, then its reduced principal symbol and principal quasi-

homegeneous symbol are both elliptic of order v 2 and V2+v3, respectively, in the sense of Definition 3 . 1 2 . 3 1 . The global version of the previous definitions and Proposition 3 . 1 2 . 4 1 in the case of U = Bn is the same, as in the case of the symbol classes Lv(lRn)

,

-

FV(IRn)

,

yet considered in this section.

Remark 3 . 1 2 . 4 4 . It is easily seen that if a E F v ( U ) is elliptic of order v , then its principal symbol a ( x , o , n )

0

satisfies the inequalities:

'd (x,E,h,S) E U

(O,EO1

X

X

(O,hol

Bn,

where q(x) is a strictly positive continuous on U function. Our aim now is to establish two-sided a priori estimates for elliptic difference singular perturbations with their symbols in F " ( U )

v-

or F (Rn)

(global version) and to outline the corresponding parametrix construction. A

difference singular perturbation op a E op

F"(U)

is defined by its

symbol a E F u ( U ) as in ( 3 . 1 2 . 2 ) . using the discrete Fourier transform. It is said to be ezziptic of order v , if its symbol is elliptic of order v . Example 3 . 1 2 . 4 5 . The symbols a.(x,E,h,n), x E 1

3.11.33

u c

n a n d the symbol

E

2

4 151

+1<12

from Examples

are elliptic of order ( 0 , 0 , 2 ) and ( 0 , 2 , 2 ) , respectively. The

corresponding elliptic difference singular perturbations are finite difference approximations of the differential singular perturbations 2 l + s q(x)2D:

and E ~ A ~ - A ,respectively, where, as usual, Dx = -id/dx and A

is the Laplace operator. Proposition 3 . 1 2 . 4 6 .

The eZZipticity of a difference singular perturbation in op F V ( u ) is an

invariant property w i t h respect

to

the autornorphisms of

<.

3.12. Elliptic Singular Perturbations Proof. -

395

Indeed, this statement is an immediate consequence of Proposition

I

3.11.23.

Proposition 3 . 1 2 . 4 7 .

L e t a E F V ( u ) be e l l i p t i c of order v. Then f o r E ~ ,ho s u f f i c i e n t l y small and R s u f f i c i e n t l y large h o l d s : v

-v

ja(x,e,h,hS)I Zq(x) (E+h)

(3.12.78)

2<~<>v3, V x V h

E (O,ho],

V

E U,

E (O,E~]

V E

5 E mn with 151

b R,

where q(x) > 0 i s continuous f u n c t i o n of x E U. Proof. -

Let a (x,p,n) be the principal symbol of a, so that ( 3 . 1 2 . 7 7 ) holds, 0

on the one hand, and, on the other hand, one has:

with some li < v such that ~ ~ + < l v i1+v2 ~ and with some continuous q 1 ( x ) > 0, V x E U, as it follows from ( 3 . 1 2 . 6 7 ) .

Now, combining ( 3 . 1 2 . 7 7 ) that ho,

E~

and R

-1

with ( 3 . 1 2 . 7 9 ) , one gets ( 3 . 1 2 . 7 8 1 ,

provided

are sufficiently small.

Corollary 3 . 1 2 . 4 8 .

L e t a E F V ( u ) be e l l i p t i c o f order v. Then f o r (s;,s;)

w i t h any

= (vl,s;,s;)

s'

E m2 one has: -V

(3.12.80)

V x

E U,

-v

S'

(/a(x,E,h,hSj+(E+h) '"<E<> V E

3,

2 q(x) (E+h)

v 3 '<<> 2<E<>v ,

E (O,E~], V h E (O,hol, V 5 E Rn ,

provzdea t n a t cO arid ho are s u f f i c z e n t l y smal2; h e r e , a s p r e v i o u s l y q(x) > 0

is some continuous f u n c t i o n on u. 2 R , then ( 3 . 1 2 . 8 0 )

Indeed, if

]<]

follows from ( 3 . 1 2 . 7 8 ) , >

6 R one can always find a constant CR , s ' , v

s'

s'

< p 2<E<>

V

L

cR , s ' ,v<'>

V

2 < ~ < >3 , V

<

and for

0 such that

with 151 5 R .

Proposition 3 . 1 2 . 4 9 .

L e t a(E,h,hC) E F V ( m n ) be e l l i p t i c of order s and any s' E

v

=

(vl,v2,v3).

~ , that a 1v s t h e r e e x i s t s a c o n s t a n t c ~ , such

Then f o r any

3. Singular Perturbations on Smooth Manifolds without Boundary

396

prcvided t h a t Proof. -

As

E~

and h0 a r e s u f f i c i e n t l y small.

a consequence of Parceval's identity, (3.12.81) is equivalent with

the inequality

with some other constant

C'

s,s"

the latter being equivalent with (3.12.801,

since <(c+h)<> can be estimated from below and above by ,V h E (O,hol, V E E ( 0 . ~ ~ 1with , some constants which depend only on ho.

Remark 3.12.50.

If (3.12.81) holds for an elliptic difference singular perturbation Op(a) Of order v with some s ' < s such that s ; < sl, then, in fact, the term

I I.'

( s ' ) ,E,h

on the right hand side of (3.12.81) can be deleted.

Lemma 3.12.51.

For any (s2,s3) E R and (E,h) E (0,tl

x

(0,41

the following inequality

holds:

6

c<<

5-11

= <(h,h<) and where c and N~ depend only on s 2 and s3. E Proof. One checks easily that -

where 5

where 7

5

is the gradient with respect to 5 .

Further, denoting the left-hand side in (3.12.82) by g(E,h,E,n), the inequality (3.12.83) and the Lagrange formula for the difference on the left-hand side of (3.12.82) yield:

3.12. Elliptic Singular Perturbations p1 P to estimate << > -” and <(E+h)< > ’<(Eth)< >-”

X

X

P1 = s -1 and with 2

p2 =

s3 or

p2 =

397

with

p

= s

1 2 OK s -1, and using also the obvious 3 11

inequality

N~ = maxI Is2-l/+Is31,ls21+Is3-1l},

noticing also that (E+h) 5 <(E+h)< > if (E,h) E L O , + ]

n

Lemma 3.12.52.

-

X

[O,+]

.

L e t R(x,~,h,hC)E F’ (mn) Then f o r any s = (sl,s2,s3)E n3 and 1 .o u n i f o m Z y w i t h r e s p e c t t o E E lo,+], h E (o,?] h o l d s : -S

(3.12.85) [ E

s2 s3 <(E+h)Dx,h> ,Op(R)]

1

:

H (s+iJ-e2), ~ ,(<) h

-f

where [A,B] is t h e c o r n t a t o r of o p e r a t o r s A and B and where e2

H ( 0 ),E,h(<)’ = (O,l,O). -

Proof. We recall that a symbol R(x,~,h,hS)is in the class F’ (Rn) if it 1r o is in the class F’ (Rn)and there exists a symbol Rm(~,h,hF)E F’ (Rn) such 1,o 1 no that ( E + ~ ) ’ ~ < C > - ~ ~ < E ~ > - ’ ~ ( R -as R ~a)function , of x , belongs to S(IR:) uniformly in ~ , andc. h F u r t h e r m o r e , w i t h o u t r e s t r i c t i o n o f generality, one canassume that h R_(~,h,hc)E 0. Denote by KE the commutator on the left-hand side of

kt

its difference Fourier transform, i.e. the operator, (3.12.85) and by -h u = F ( Khu ) . It is easily which acts according to the formula: K,.? x+C,h x+S,h E . a farr.ily of integral operators with the kernels seen, that K-h 1s

i:

Tn 5rh

Tn

X

Ilth

+ C

given by the fornula:

(3.12.86) k(E,h,E;, n ) = -S

=

(E+h) 1[<s2<(E+h)c5>’

3

- s2 <(Eth)s313h(5-n.~,h,hn),

-h where R (S,E,h,hn) = Fx+5,hR(x,~,h,hn). -h with the kernels Introduce the family of integral operators S

s

:

Tn Srh

X

Tn

Qrh

-L

C, given by the formula: 1-s

(3.12.87) s(E,h,S,rl) = ih,E(~,h,5,n)<< n> where k is defined by (3.12.86).

-’2

2

<(E+h)Cn>

-s

-’3

3

3 . Singular Perturbations on Smooth Manifolds without Boundary

398

Obviously, (3.12.85) will follow if one shows that uniformly with respect to (E,h) E (3.12.88)

-h

Sc

: L

2

(0,tI

n (T5,h)

X

-f

(0,tl

holds:

2 n L (T5,h).

1-1

Since (Eth) 1<5>-v2<~5>-’3R, as a function of x, belongs to S (R:) -h uniformly in c,h and 5 , one has for R = Fx-t(<-n),hR with any integer N > 0 the following inequality (see (1.3.10)):

where the constant CN may depend only on N. Now, (3.12.87), Lemma 3.12.51 and (3.12.89) with N = N0+n+2 (No being the same as in Lemma 3.12.51), yield:

where the constant C does not depend on E,h,S and rl. Hence, Lemma 3.12.8 and (3.12.90) yield (3.12.88), and that proves (3.12.85). Lemma 3.12.53.

-

(mn)be such (s),E,~ h = {x E mn 1 /x-xol < 61, V (e,h) E ( 0 . ~ ~X 1 (O,hol. t h a t supp u c B 6 txo Then f o r any s ‘ E a v and for 6,e0,h0 s u f f i c i e n t l y s m a l l , t h e r e e x i s t s a 1 s c o n s t a n t c which m y depend o n l y on s , s ’ , 6 , ~ ~ , such h~, that L e t a E Fv(mn) be e l l i p t i c o f order v and l e t u E H

(3.12.91)

1 i u l 1 ( s ) ,E,h 5 V

E

E

C(l lOp(a)ul 1 (s-v),E,h t

(O,E~], V h

=

1

(s‘)

,~,h)~

E (O,hol, V u E H(S),E,h (E$,

__ Proof. Introduce the notation: ao(x) r0 (x) = a0 (x)-a0 (x0) , r(x)

lul

=

SUPP U = B

ao(x,E,h.h6), a(x) = a(x.E.h.hE),

a(x)-a(xo), so that one has:

(3.12.92) a(x) = a (x )+rO(x)+r(x).

0 0 Since r(x) E F’(mn) , Proposition 3.1 .34 yields:

=< (3.12.93) )/Op(r)u// (s-V),E,h

CJ/uJ

(s+v-v), ~ , h .

m

Now, let + & E CO(B26,xO) be such that $6(x) 6 Denote r0 (x)

=

6,xo

5

1, V x E B

+6(x)ro(x). Lemma 3.12. 52 yields for each

(36)/2 ,xo

3.12. Elliptic Singular Perturbations

=

Op(r6)(E+h)s - v 2<(c+h)Dx.h >

s

399

-v 3u+K:u,

with some family of operators Kh which satisfies the inequality

where the constant

C6

may depend only on 6 and, as usual, e2 = (O,l,O).

6

Since r (x) is a smooth function of x with its support in B

6

26,x and 0

r (xo) E 0, one gets immediately, that 6

(3.12.96)

s

>

v - s '
I lOp(r )(E+h)

-v

x.h

2<(E+h)D > x,h

s -v

ulI

(O),E,h

'

where C does not depend on 6,E.h and u. By Proposition 3.12.49, one has

where C may depend only on s , s ' , ~ ~ , V h ~s' , E

a

V

1 s' Hence, (3.12.92)-(3.12.97) yield for 6 sufficiently small:

Since li <

ul+u2

V,

< v1+v2, one can use the interpolation inequality

(2.8.18) (with 6 1 sufficiently small instead of 6) in order to estimate

I I 'I Order

1 1 . 1 1 (s),E.h with a small coefficient (of 1 1 . 1 I ( ~ 1 ) , ~ ,with h a large one. Applying the same one, finally, gets (3.12.91). I estimate I 1 . 1 I (s-e,) ,E,h'

(s+~-v) ,E,h '1) and Of

inequality to

by the sum of

Remark 3.12.54. If a(x,s,h,hS) E

supp u

C

Rn\B

R,O'

v n F ( m ) is elliptic of order v and if u E H

(R;) , ( s ) ,E,h V E,h, with R sufficiently large, then the same argument

3 . Singular Perturbations on Smooth Manifolds without Boundary

400

as in the proof of Lemma 3.12.53, shows that with some constant C (which may depend on R ) holds the same inequality (3.12.91),V (E,h)E ( O , E ~X] (O,ho], V U€H(~),~,~(<) , supp u c Rn\BR,O.

Indeed, with 6 = R-l and

ro(x) = ao(x)-ao("), (3.12.96) is still valid, since a0 (x)-ao(m), as a

.

function of x E Rn , belongs to S(IRn)

Now, we are in a position to prove the following statement: Theorem 3.12.55. L e t a(x,s,h,hE) E FV(mn) be gZobaZZy e l l i p t i c of order v (i.e. (3.12.72)

holds w i t h some q > 0 , which does not depend on x E mn). Then f o r each S,S'

with

(3.12.99)

< s,

s'

a constant

C,

I

IUI

s' E

alvs, and

for ~

~ s u f,f i c ihe n t l~y s m a l l , t h e r e e x i s t s

which may depend on s , s ' , E ~ , ~such ~ , that

I ( s ),E,h 5

C(l

I

/Op(a)ul ( s - v ) ,€,h +

V E E (0,c01,

n Proof. Let Uo = R \Bl,s,o,

I lul I ( s ' ) &,h),

V h E (O,hol, V u E H(S),E,h

.

U j = Bs,x,, 1 Ij 2 J, { UJ, }0-1< , < J being a finite 3

covering of R n , where the parameter 6 will be chosen later on. Let

{ $J, IO=J= < . < J be the partition of unity subbordinate to the covering

{U,} 3 O<j<J'

so that one has:

u

x

=

$ju,

v

u.

OSjSJ

If 6 is sufficiently small, then by Lemma 3.12.53 and Remark 3.12.54, one has:

Applying Lemma 3.12.52 and the interpolation inequality (2.8.18), one can write down:

Furthermore, using Proposition 3.11.34 (or again Lemma 3.12.52), one gets the estimate:

3.12. Elliptic Singular Perturbations Now, (3.12.100)- (3.12.102) yield (3.12.99) .

40 1

I

Now, we prove the necessity of the ellipticity condition in order to have the estimate (3.12.99) for a E FU(zn)

.

Theorem 3.12.56.

-

If f o r a E FU(nn) t h e a p r i o r i e s t i m a t e (3.12.99) h o l d s w i t h some s'

s ' is ,

E a1vs, t h e n a(x,E,h,hS) is g l o b a l l y e l l i p t i c o f order v.

Proof. Let

ri

E

'{O})

(T:

.

and let $ E C;(Bn)

We are goinq to substitute into (3.12.99) the functions (3.12.103) u = $(x) exp(ih-l<x,rp and to let

0, h

E -t

0 under the condition that the ratio p = h/E is a

-f

given constant, p E R+. Applying (3.11.24) to Op(a)u with u given by (3.12.103) and with p =

h/E a given positive constant, one gets the following asymptotic

formula: (3.12.104) a(x,e,h,hD)u = -(U

= h where a .

+v

1

)

2

-V

'a (x,p,ri)$+hy$(h,x)1 , h 0

exp(ih-l<x,n>)[ (l+p-')

is the principal symbol of a, y = v 1+U 2- ( ! J 1+ p 2

being the same as in (3.12.67), and where

[ l$(h, - )

I lo

>

+

0,

0, p = (p1,p2,u3)

is uniformly bounded

with respect to h E (O,hol. Furthermore, a straightforward computation (or one more application of (3.11.24)) shows that with any u E R 3 the following asymptotic formula holds f o r u given by ( 3 . 1 2 . 1 0 3 ) :

Therefore, (3.12.99), (3.12.104), (3.12.105) yield: (3.12.106)

Iw(ri)

I " 2"

3 1181 ! o 5

CI

lao(-,P,ri)qIIo+C+h6 ,

where C is the same constant as in (3.12.99), may depend on $ and 6 = min{y,sl+s2-s;-s;,l} s'

E alVs, i.e. s1+s 2

> s'+s'

1 2 '

>

C

$

is some constant, which

0, given that s ' < s ,

3 . Singular Perturbations on Smooth Manifolds without Bouliday

402

Now, letting

0, h

E -+

-f

0, p = h/E being a given positive constant,

(3.12.106) yields:

the constant

C

here being the same, as on the right-hand side of (3.12.99).

Thus, (3.12.107) yields (3.12.72) with q(x)

S

C-l > 0, i.e. a is

globally elliptic. Examples 3.12.57. 1”. Let q(x)

E

m

C

(U), U _C R1 and let q(x) > 0, V x E U.

The symbol b(x,h/E,hE),

is elliptic of order

I ,

(0,0,1) and it is globally elliptic of the same

order if q(x) 2 qo

0, V x E

=

(in the case U = R1 , in addition, it is 1

assumed that q(x) = q_+q’(x) with q‘(x) E S(R

))

.

The corresponding difference singular perturbation b(x,h/E,hD) is an approximation by finite differences of the elliptic singular perturbation (l+iEq(x)Dx) E op s”(u), w = (o,o,I). Denote by b*(x,h/E,hD) the formal adjoint of b(x,h/e,hD).

The difference

singular perturbation (3.12.109) a(x,E,h,hD) = b*(x,h/E,hD)

0

b(x,h/E,hD)

is elliptic of order u = ( 0 , 0 , 2 ) , a E Op F W ( u ) , its principal symbol being ao(x,P,n)

=

l2

(b(x,p,q)

with b(x,p,n) given by (3.12.108); furthermore, the

difference singular perturbation (3.12.109) is a three point approximation by finite differences of the formally self-adjoint differential singular perturbation a(x,E,D) = (1-iED q(x)) (l+ieD q(x)) E Op S ( o ’ 0 ’ 2 ) ( U ) .

Besides,

if U is a finite interval, say U = (O,l), then the equations a(x,s,D)u = 0, 1 x E U and a(x,E,h,hD)u = 0, x E Uh = &$,fl U, have the same asymptotic x ’ E a U , in the following sense: solutions uo = exp(-jx-x’//(q(x’)~)), supla(x,~,D)u x o XEU with some constant C

I

S CE,

sup[a(x,E,h,D)u,-l6 C(E+h) XEU

0 which depends only on q(x).

Notice that a(x,y,~,h,hS)= b(y,h/E,hS)* b(x,h/E,hE) with b(y,p,n)* the complex conjugate of b ( y , p , n )

is the amplitude of the difference

sinqular perturbation (3.12.109) (see Remark 3.11.20).

3.12. Elliptic Singular Perturbations

403

The difference singular perturbation b+(h/E,hD) with the symbol -1 = l+p (exp(in)-l), b+ E F(of081)(~) , is a non-elliptic

b+(p,n)

approximation by finite differences of the singular perturbation l+i-ED. Indeed, b+(p,n) being also the principal symbol of b+(h/E,hD), one has: b+(2,1~)= 0 . Notice that b-(h/E,hD) = l+p-'(l-CI-') operator to the left) whose symbol is b- ( p , r l )

=

(with 0-1 the shift l+p-' (l-exp(-iq)) is an

elliptic approximation of l+iED. Consider a family of difference singular perturbations: t [0,1] 3 t + a (~,h,hD)defined by their symbols

2'.

(3.12.110) at (~,h,h5)= -~l
*

0

It is quite obvious that a (&,h,hS) = ir; (l+isr;): 1 E i<*(l+(E/h)(exp(ihE)-l)) is not elliptic, while a ( ~ , h , h < ) = i<(l+iEr;*)f 5

ir;(l+(-E/h) (l-exp(-ihE))) is elliptic of order (O,l,l),

It is easily seen, that for each t E (f,1] the difference singular perturbation at (~,h,hD)with the symbol (3.12.110) is an elliptic approximation of -ED2+iD, while for t E [O,f1 it is not elliptic. However, the approximation a t (E,h,hD) = --ED D* +(i/2) (Dx,h+D:,h) deserves x,h x,h some attention,since this is the only one of the family at which has the 2

accuracy O(h

),

as h

+

0. Besides, the ellipticity condition (3.12.72)

(with q(x) E q > 0) is only violated at p find a constant q ( p

0

t

t

q(po)

I w ( n ) I,

where ao(p,rl) = a (l,p,n) = -p of a f

m,

i.e. for each

po >

0 one can

such that

)

(3.12.111) lao(p,n)I 2

t

=

-1

t/ rl

E

1

T

ri'

V p E (O,pol,

lw(n)I*+iRe w ( n ) is the principal symbol

. Indeed, since ao(p,n) t # 0, v

p

E (O,pol

'd

n

E (T:{O})

and -1

1

lao(p,n) f I 2 Cn for n 0 , so that infimum of lu(rl)1 /aT(p,n) I 0 1 over n E (T \ { o } ) , p E (0,p 1 is a strictly positive constant q ( p 1 , 0 0 (3.12.111) holds with any given p o < m. -f

1

since a-(=,n) = iRe w ( n ) = 0 = sin n and the corresponding difference operator (i/2)(D +D* ) is a x,h x,h non-elliptic approximation of iDx, the ellipticity condition being violated On the other hand, it can't hold for

at n

po =

m,

= T.

The last example motivates the following

3. Singular Perturbations on Smooth Manifolds without Boundary

404

Definition 3.12.58.

A symbol a E F v ( u ) and t h e corresponding d i f f e r e n c e s i n g u l a r p e r t u r b a t i o n m there o p ( a ) a r e s a i d t o be weakZy e l l i p t i c of order v i f for each p o e x i s t s a continuous s t r i c t l y p o s i t i v e f u n c t i o n q(x), which may depend on po,

such t h a t t h e p r i n c i p a l symbol ao(x,p,rl) of a s a t i s f i e s t h e c o n d i t i o n

(3.12.72) w i t h

pE (O,pol.

A symbol a E Fv (mn)

(and t h e corresponding d i f f e r e n c e s i n g u l a r p e r t u r b a t i o n )

i s s a i d t o be g l o b a l l y weakly e l l i p t i c , i f i n (3.12.72) one can choose as q(x) a p o s i t i v e c o n s t a n t , which may depend on p o .

Using the previous argument, one proves in exactly the same way the following statement. Theorem 3.12.59.

v n The a p r i o r i e s t i m a t e (3.12.99) holds f o r a E op F ( R ) and for p E ( O , p o l , V p 0 < m , w i t h a c o n s t a n t C, which may depend on po, i f f a i s g l o b a l l y weakly e l l i p t i c . For elliptic difference singular perturbations the statements, similar to Proposition 3.12.16 and Corollary 3.12.17 are valid, the recurrence procedure for constructing the symbol of the left and right parametrix being exactly the same as in (3.12.341, (3.12.35). Furthermore, a statement similar to Theorem 3.6.2 (or Theorem 3.11.5) can be used in order to produce a symbol, having the asymptotic expansion defined by the recurrence procedure as in (3.12.34), (3.12.35). One can also use (3.11.49) as a definition (asymptotically equivalent to (3.11.47)) of the product of two symbols in F

(U) = U Fv (U), which 1#O v 1,o leads to a slightly different recurrence procedure for constructing the

symbol of the left and right parametrix for the elliptic difference singular

the right parametrix the terms

-

-1 Z: ak (x,~,h,hg)where for k>O of this asymptotic expansion are defined

perturbations. Namely one seeks a(x,E,h,hg)-l

recursively as follows (for 15 1 sufficiently large) : -1 . a (x,E,h,hS) = a(x,E,h,hE)-',

(3.12.112)

and for the left parametrix one has to switch a and a-l k-lal

on the right-hand

3.12. Elliptic Singular Perturbations

405

side of the second line in (3.12.112). but one can also use the same recurrence procedure for both parametrices, since all the symbols are, anyway, defined modulo a symbol of order (v1,-m,v3)with some vl,v3 E R . An example will be considered in order to illustrate the paranetrix construction for elliptic difference singular perturbations just sketched above. Example 3.12.60. 2

Let a(x,E,h,hS) = 1 + ~, where A(x) =

1 lakj (x)1 1 l ~ kj,~

is symmetric n

.

m

positive definite matrix with C -entries a (x), x E U _C Rn Since kj (u)~ isf a ~polynomial ' in ( c , c * ) it is more convenient to use a E ~ ( ~ f (3.12.112) for constructing a parametrix for a(x,~,h,hD)E Op

F(0,0,2)

(u).

However, still making use of (3.11.49), we shall proceed differently in order to produce a parametrix for the difference singular perturbation 2 -1 a(x,~,h,hD)= 1 + ~ . We seek a parametrix a (x,E,p,hD) for x ,h- 1x,h E F(o'or-2)(u) can be represented a(x,~,h,hD)whose symbol a (x,~,p,hS) asymptotically as a sum (3.12.113) a-l(x,E,p,n)

-

C

E

k -1 ak (x,p,n),

E ' O

k2O

uniformly with respect to x on each compact K C C U, p 2 p 0 > 0 with any given p o and n E TnWrite the ker:el

of the integral o2erator a-'(x,E,p,hD)

in the for-:

Using (3.11.49) (or a straightforward computation), one finds:

Now the coefficients in the asymptotic expansion (3.12.113) will be determined by requiring r(x,E,p,pq) defined by (3.12.114) to satisfy the asymptotic relation: r(x,E,p,pn)-l

N

= O(E )

for any integer N > 0.

The latter requirement leads to the following recurrence formulae for the coefficients a;'(x,o,q)

in (3.12.113):

3 . Singular Perturbations on Smooth Manifolds without Boundary

406

-1 where a-l = 0.

Now, choosing a cut-off function $(t) and a sequence 6 . 7

J.

0 as in the

proof of Theorem 3.6.1, we can write the following formula for the symbol of a parametrix for a(x,~,h,hD): (3.12.116) a-'(x,c,h,hS)

C

=

-1 k -1 + ( 6 k ~ )E ak (x,h/~,hS),

k>O with a ; '

defined by recurrence formulae (3.12.115).

It is left to the reader to check that for any integer N > 0 one has:

-1 (3.12.117) a(x,~,h,hD)o a (x,E,h,hD) - Id = E ~ R ~ ' ~ N '

(i.e., if A(x) = Am+A'(x) with A'(x) E S(lRn)) and is globally elliptic (of -1

order v = (0,0,2)),then a

(x,~,h,hD) constructed above is, in fact, a m

quasi-inverse operator for a with accuracy O ( E

),

that is (3.12.17) holds

for any integer N > 0 with R E P h which is uniformly bounded with respect to N E ( 0 .~ ~ 1 h ,E (O,hol fromH(s),E,h(<) into H (S-(O,O,-N)) ,E,h VsEz33.

'<'

E

I

The parametrix construction discussed in Example 3.12.60 can be extended to the class of elliptic singular perturbations of order ( O , O , v ) , whose symbol a E F(oronv) (U) has the property r := (a-ao) E F(-l'o'v-l) (U) ,

a (x,h/~,hS)being the principal symbol of a. 0

Indeed, one can use (3.11.47) (or, equivalently, (3.11.49)) for defining the (non-commutative) product a'b of two symbols a and b in F1,o(U) = U v

Fv1,o (U),

as a symbol a * b with the asymptotic expansion (3.11.47)

(or, equivalently, (3.11.49)). Now, using again recurrence procedure (3.12.115) with a (x,p,rl) instead of l+ , 0

-

-1

one can find a

symbol ai'(x,E,p,n) such that 1-a (x,h/~,hE) . a (x,~,h/~,hS) = 0 N = O ( E -~) with any integer N > 0. Therefore, one can write (modulo ~~-~ with V N > 0):

3.12. Elliptic Singular Perturbations

a(x,E,h,hS)

-1 -1 a . (x,~,h/~,hS) = l+r(x,~,h,hS) - a o (x,~,p,hS)

*

so that modulo an operator in Op F(-"'o'-'")

( U ) one has:

1 r0

(3.12.118) Op a

o

407

Op a ; '

=

Id + Op r

-1 Op . a

o

.

Since ~~l~ := op r op a;1 E op F(-lr0r-l) ( U ) , the operator (1+KEfh)-l 1 ,o exists and can be represented as a convergent Neumann series, so that E Op

(I+K'lh)-'

Fo

1 to

Let l+k-'(x,e,h,hS)

(U).

be the symbol of (I+KEfh)-l.

Since (3.12.119) Op a

-1

o

Op a .

1 Op(l+k- ) = Id

o

and there is in this case a one to one correspondence between the symbols and the operators in (3.12.119) (modulo a symbol of order ( - - , O , - - )

),

one

gets the conclusion that the singular difference perturbation with the -1 (1+k (x,e,h,hS))is a parametrix and at the same symbol a-'(x,E,h/E,hS)

-

0

time a quasi-inverse with accuracy

N

O(E ),

V N

> 0, for the elliptic

singular perturbation a(x,E,h,hD) above. Finally, we outline the extension of the elliptic theory Of singular perturbations in spaces ff

( s ) .E

to elliptic differential singularperturbations

in analoguous spaces with L -structure, 1 < p <

P

-,

which will be defined

below. We start with a result on Fourier multipliers in L (El") due to Hsrmander.

P

Definition 3.12.61. A tempered d i s t r i b u t i o n f E L

P

S'

( R ~ )is s a i d t o be a Fourier m u l t i p l i e r i n

(mn) if F ~ : ~ ~ F E~'L~ uand t h e r e e x i s t s a c o n s t a n t M such t h a t

The l e a s t c o n s t a n t x such t h a t (3.12.120) holds f o r a g i v e n f E s ' (Rn) and p E (1,~) is denoted by M (f) and t h e s e t of a l l Fourier m u l t i p l i e r s i n

P

L (mn)

P

is denoted by M

P

.

Our aim is to present the proof of the following result of HcSrmander: Theorem 3.12.62.

L e t f E L _ ( R ~ ) and l e t B > 0 be a c o n s t a n t such t h a t (3.12.121)

1

fRS15 I <2R

iRla'Daff~)fdS5 B2Rn,VR > 0 , V a with la1 5 [n/21+1,

5

3. Singular Perturbations on Smooth Manifolds without Bounda y

408

where [n/2] i s t h e i n t e g e r p a r t of n/2. Then f E M , V p E ( 0 , ~ )and, moreover, P (3.12.122) M ( f ) 5 C B, P P,n

where t h e c o n s t a n t c

depends only on i t s s u b s c r i p t s .

P,n The following consequence of Theorem 3.12.62 will be used

consistently later on. Corollary 3.12.63.

L e t f E L_(B~) and l e t

Bo > 0

be a c o n s t a n t such t h a t f o r a l l m u l t i -

i n d i c e s a w i t h l a / 6 [n/21+1 h o l d s : (3.12.123) IDaf(S) I 5 B o / t I -

v 5 E Rn\IO}.

5

Then f E M

P'

V p E (1,m) and, moreover,

(3.12.124) M (f) 5 C' B P p,n 0 '

where t h e c o n s t a n t C'

depends o n l y on i t s s u b s c r i p t s .

P#n Proof of Corollary 3.12.63.

It is easily seen that if f E Lw(Rn) satisfies (3.12.1231, then it satisfies (3.12.121), as well. Indeed,

the right-hand side of the last inequality being defined for n the limit for (n-2lal)+ 0.

=

21al as

I

With f E L_(Rn) satisfying (3.12.123) for la1 6 n the result stated in Corollary 3.12.63 was first established in [Mich., 1

, 21.

The proof of Theorem 3.12.62 requires a considerable effort and will be split into several steps. We start with the following statement: Theorem 3.12.64.

L e t p E ( 1 , ~ )and l e t p-'+p'-' (3.12.125) M ,(f) = M (f). P P

=

and 1. If f E M then f E M P P'

3.12. Elliptic Singular Perturbations

409

Proof of Theorem 3.12.64.

,

With u E S(lRn)

v E S ( R n ) , Parseval's identity and Holder's inequality

yield:

1

6 M (f) Iul

P

1 L (Rn)

P where, as usual, the upper

*

1 IVI I

(Rn) ' P' stands for the complex conjugate. L

Since S(lRn) is dense in L (Rn) , one can choose a sequence

P

{uklkZl, uk E S(Rn) which converges in L (Rn) to

P

1

/Fg,xf

p'-2 -1 (FS+xf Fx+Sv) * E L (Rn) as k Fx+SVI F Furthermore, one finds

-f

-.

Hence using (3.12.126) (with uk chosen as above) and the last two formulas, one gets

so that f

E M

and M P'

P'

(f) <= M ( f ) .

P

I

Now, by exchanging p and p ' , one gets (3.12.125).

Next we shall prove an interpolation result due to Riesz and Thorin (see [Zygm.,

1

1.

Theorem 3.12.65.

Let 1 Let T

<

p < r < q <

:

L (W") P

-f

m

and Zet

e E

L ( I R ~ ), T : L

suck c o n s t a n t s t h a t

P

9

-1

(0,1) be such t h a t r-l = ep-1+(1-8)q (xn)

+

L

4

(w") and l e t c

P

(T),

c

9

(T)

be

.

3 . Singular Perturbations on Smooth Manifolds without Boundary

410

Then

the following version of Phragmen-LindelBf's

For proving Theorem 3 . 1 2 . 6 5 maximum principle is needed: Theorem 3 . 1 2 . 6 6 .

E c 1 a- 5 Re z S a } and l e t f : Z + c1 be continuous and bounded on 2 and a n a l y t i c on t h e i n t e r i o r of 5. If

Let Z

= { z

t h e n one has: (3.12.1301

/f(z)( 5 M ,

V z

E 2.

Proof of Theorem 3 . 1 2 . 6 6 . If lim y+im

max a 5xSa

-

I

/f(x+iy) = 0 ,

+

then one gets ( 3 . 1 2 . 1 3 0 )

by applying the classical maximum principle to

f(z) in a rectangle { a - 5 Re z 6 a + , 11m z I 5 A} with A sufficiently large. Otherwise, one applies the classical maximum principle to the function f(z) exp(6z2), 6 > 0, and let 6

+

0.

I

Corollary 3 . 1 2 . 6 7 .

L e t Z and f be a s i n Theorem 3.12.66 and assume t h a t

Then for z, = ea- + ( 1 - 6 ) a + + iy,

e E

( 0 , 1 ) , holds:

Indeed, one has only to apply Theorem 3 . 1 2 . 6 6

to the function

3.12. Elliptic Singular Perturbations

41 1

i n order t o g e t (3.12.131). Proof o f Theorem 3.12.65. Denote by S t ( i R n ) t h e s e t of f u n c t i o n s on R n which t a k e o n l y a f i n i t e number of complex v a l u e s and v a n i s h o u t s i d e a b a l l w i t h s u f f i c i e n t l y l a r g e r a d i u s . A s a consequence o f t h e d e f i n i t i o n of L e b e s g u e ' s i n t e g r a t i o n , with 1 5 r <

S t ( i R n ) i s d e n s e i n any L ( W n )

m.

F u r t h e r , s i n c e t h e d u a l s p a c e ( L ( E n ) )' Lr, (Rn)

,

where r-'+r'-'

(3.12.133)

IIfl

IL

= 1 , one h a s r f o r e a c h f ErL

sup gESt(R

=

r

of L ( R n ) c o i n c i d e s w i t h

,I

IL

:

11 f ( x ) g ( x ) * d x l . mRn

=I

[gl

(Rn)

r' For z E C (3.12.134)

, 0 ~

5 Re z 5 1 , d e f i n e t h e f u n c t i o n r ( z ) by t h e e q u a l i t y :

+ -1-2

2 1 _- r(z) p

q

and t h e f u n c t i o n r ' ( z ) as f o l l o w s : (3.12.135)

1 +-= r'(z) r(z)

For f E S t ( R n )

,

g

1.

E

S t ( R n ) with j l f l l L

the functions: fz =

= 1,

r l f l r / r ( z ) ei a r g f I

g2

=

lgl

]

= 1 introduce /glIL r'

r'/r'(z)

i arg g

(3.12.136) .O(z) =

1

( T f Z )(x). g Z ( x ) d x .

Rn Notice t h a t (3.12.137)

O(8) =

1

(Tf) (x).g(x)dx. RRn

C Rn Denoting by 1 ( x ) t h e c h a r a c t e r i s t i c f u n c t i o n o f a s e t A A

w r i t e f o r f and g i n S t ( l R n ) :

Hence,

,

one c a n

3. Singular Perturbations on Smooth Manifolds without Bounda y

412

where Xk E C

1

and a > 0 are some numbers, which are well-defined by f and k

9.

As a consequence of the last representation for @ z ) , it is bounded and continuous on the strip ? = { z E C1

1

0 5 Re z 5 1

Further for Re z = 0 one finds that f (x)Irdx) ''l

=

1,

and, in the same way, that 119iy/lL = 1 9'

Now, applying HBlder's inequality and using the second line in (3.12.127), one finds:

Using the same argument, one gets the inequality: (3.12.139) l@(l+iy)

I

5

c

P

v y E

(T),

W.

Applying Corollary 3.12.67 and using (3.12.1381, (3.12.139), one gets:

1

(3.12.140) l @ ( z ) 6 C (T)Re Z(Cq(T))l-Re

P

,

1 V z E C , O S R e z S l .

The definition of @ ( z ) by (3.12.136), (3.12.137) and (3.12.140) yield: (3.12.141) 11

En

(Tf)(x)g(x)dxl 5

C

P

e

(T) (Cq(T))l-e, V f,g E St(Rn)

.

Now, with V f E St(Rn) and using (3.12.133), (3.12.141), one gets (after taking the supremum on the left-hand side of (3.12.141) over all

and that completes the proof, since St(Rn) is dense in L (Wn) .

I

Corollary 3.12.68.

L e t p E (1,m) and l e t p-'+p'-'

=

1. If f E M

P'

then f E

M

q'

V q

such t h a t

min(p,p') 2 q I max(p,p'). Indeed, this is an immediate consequence of Theorems 3.12.64 and 3.12.65.

I

3.12. Elliptic Singular Perturbations

413

Next, i-'ormander's q e n e r a l resu1.t c o r x e r n i n a _ F o u r i e r m u l t i p l i e r s ( i n

L (lRn)) P proved.

,

which a r e l o c a l l y i n t e g r a b l e f u n c t i o n s , w i l l b e s t a t e d and

k f o r t h e s e t o f a l l FE:,f, 'd f P (F-l f ) = M ( f ) . I t i s q u i t e o b v i o u s , t h a t coincides p e x P P s e t of d i r e c t F o u r i e r t r a n s f o r m s o f f E M as w e l l . P' 1 n I n t r o d u c e t h e f o l l o w i n g c l a s s K o f f u n c t i o n s i n L loc(R) 1 k E Lloc(Rn) i s s a i d t o b e l o n g t o t h e c l a s s K i f t h e r e e x i s t a We s h a l l u s e t h e n o t a t i o n

write

E M and P with t h e

.A

s e t A C IRn

,

a neighbourhood

0 of zero i n

function

bounded

and a c o n s t a n t C > 0 s u c h

lRn

t h a t f o r a l l s > 0 holds:

i

(3.12.142)

/ k ( x - y ) - k ( x ) Idx 5 C ,

v

y E so,

s(Rn\A) where by sB i s d e n o t e d t h e s e t : SB = { y E En Let

1

I

y = sx, x

b e c o l l e c t i o n o f a l l open c u b e s i n lRn

p a r a l l e l t o t h e c o o r d i n a t e a x e s . Choosing I. a t t h e o r i g i n and s u c h t h a t I.

C

0,

1; 3 A ,

C

E

B}.

with t h e i r edges,

1 , I:

C

I

with t h e i r c e n t r e s

w e d e f i n e f o r each I

C

I

a n o t h e r cube I * c 1 w i t h t h e same c e n t e r as I s u c h t h a t u(I*)

=

) L I ( I )where ,

(p(I;)/u(I0)

u

i s t h e Lebesgue measure on IRE.

Lemma 3.12.69.

1

n

L e t k E K and l e t I c I and I* a s d e f i n e d above. Then f o r V u E L ( l ~) , supp u c I , i u ( x ) d x

=

0, t h e f o l l o w i n g i n e q u a l i t y h o l d s :

I

where t h e c o n s t a n t

c i s t h e same a s i n

(3.12.142),

and where :c*u stands

f o r t h e convolution of k and u. Proof o f Lemma 3.12.69. L e t z b e t h e c e n t r e o f I and d e n o t e s = u ( I ) / p ( I 0 ) .

J u(x)dx

Since supp u

= 0 , one f i n d s u s i n g t h e d e f i n i t i o n o f t h e c l a s s

I (k*u)( x ) / d x =

(3.12.144)

IR \ I *

K:

C

I and

3. Singular Perturbations on Smooth Manifolds without Boundary

414

f

2 C

1-2

ju(y+z))dy = C J ju(x)]dx. I for the shifts of the corresponding sets by

Here (Rn\I*)-z and 1-2 stand

I

vector z .

Next, we prove the following Theorem 3.12.70.

L e t k E K fl

f o r some p E (1,-I. Then t h e r e e x i s t s a c o n s t a n t P t h a t f o r aZZ u E L ~ ( R ~w)i t h compact support holds:

c1 such

One needs for proving the last theorem, the following Lemma due to Calderon and Zygmund (see, for instance, [Her. 1 1 ) : Lemma 3.12.71.

L e t u E L ~ ( R ,~ )1 IuI I L1 ( Rn) a s a sum: (3.12.146) u

=

> 0,

and Zet s > 0 . Then u can be represented

v+ Z w k' kt 1

where v and wk s a t i s f y t h e foZZowing c o n d i t i o n s : (i) v E L (Rn) , wk E L1(Rn) 1

(ii)

,

k = 1.2 ,...,

1

/v( + IIWkll 5 31lull L1(Rn) k2l L1 (Rn) L1(R

I

)

(iii) Iv(x)] 5 2"s azmost everywhere, (iv) f o r c e r t a i n d i s j o i n t cubes

c I holds: supp w

Moreover, i f u has compact support, then v and w

k

c

and

k 2 1, can be chosen

t o have t h e i r supports contained i n some f i x e d compact s e t . Proof of Lemma 3.12.71. Divide Rn into a mesh of cubes from >

.-lll~ll

, so

and denote by I s.

with volume of each cube

1

over every cube of

L1(R 1

this mesh becomes

t

I

that the mean value of ju(x)

c

S.

Further, divide each cube ofthe mesh into 2

. . those of them, over which

,Il2,.

Obviously, one has:

n

equal Cuke

1.1

the mean value of

is

3.12. Elliptic Singular Perturbations

415

1 lu(x) /dx < 2nsui(Ilk).

(3.12.147) sv(Ilk) 5

Ilk Indeed, if Ilk is obtained as a result of the subdivision of a cube I' from the mesh above, then, according to the construction, one has: SU(Ilk)

5

J Ilk

lU(x) Idx 2 1 lu(x) [ d x <su(1') I'

=

Znsu(Ilk).

Define

wherexI V x

(XI is the characteristic function of Ilk, i.e. xlk(x) E 1, lk

E Ilk and

xI

(x) 0 elsewhere. lk Next, making again the same subdivision of the cubes from the mesh

above, which are not among I ll,I12,..., one singles out those new cubes over which the mean value of lu(x) I is L s and extends the

..

I Z 1 ,I22,.

definition by (3.12.148) of w ~ ~ ( xto) these cubes. Continuation of this process leads to theconstruction of disjoint cubes I . and corresponding 3k

functions w . ~ ( x ) ;one rearranges them, for convenience, as a sequepce, 3

denoted again by Ik, wk(x), and defines v(x) by setting v(x) V x f! Q

=

=

u(x),

U I . It is clear, that (3.12.146) holds and (i) is valid, as k

k

well. For showing (ii), first notice that

1 (Iv(x)I+lwk(x) 1 )dx

5 31 /u(x)ldx.

I

k

k '

Indeed, the last inequality is an immediate consequence of (3.12.148) and the definition of v(x). Since the cubes Ik, k = 1,2,..., are disjoint, supp wk

J Rn\Q

Iv(x) [dx =

In R ' 9

one gets immediately (ii):

lu(x) ldx,

5 Ik

and

3 . Singular Perturbations on Smooth Manifolds without Bounda y

416

Further, (iii) follows from (3.12.147) if x E Q. On the other hand, if x $! Q, then there are arbitrary small cubes containing x, over which the mean value of lu(x) 1 is <

so that lu(x) I 5 s almost everywhere on mny?,

s,

and that proves (iii). Further, as a consequence of (3.12.148), one has:

1 wk(x)dx

= 0,

...,

k = 1,2,

I k

and summing up in (3.12.147) with respect to k, one gets that

since I ~ k,

=

1,2,..., are disjoint.

1

Proof of Theorem 3.12.70. N

Denoting

U(X) =

(k*u)(x) and using the decomposition (3.12.146) of u(x), one

finds that almost everywhere

and, thus, almost everywhere holds:

Applying Lemma 3.12.69, one gets

*

*

Further, denoting Q* = U I with Ik defined by Ik as above and using (iv) k from Lemma 3.12.71, one gets

N

Introducing w(x)

Z Iwk(x) 1 , one finds, using again Lemma 3.12.69 N

=

k> 1 and (ii) from Lemma 3.12.71:

3.12. Elliptic Singular Perturbations

417

This last inequality yields:

1

(3.12.150) u{x E Rn\Q*

I

1 w(x) t (4)s) 5 6Cs- /uII

Hence, it follows from (3.12.149), (3.12.150) that w(x) < ( f ) sexcept on a set of measure at most

1

The assumption k E

P

w th some p E (1,m) and (ii), (iii) from Lemma

3.12.71 yield:

Therefore, one gets,using the last inequality:

N

Since the measure of the set, where w(x) 2 s/2 is bounded by constant (3.12.151). one findsrusing the last inequality, that UIX E Rn

I

jc;(x)

I

> s

and that is precisely (3.12.145) with a = s . with C3 = (2C2)p+6C+u(I~)/~(10),

I Next, using Theorem 3.12.70

an argument of Marcinkiewicz (see [ Z , l ] )

and Corollary 3.12.68, the following statement will be proved: Theorem 3.12.72.

f o r a l l p E ( l , m ) , o r e l s e f o r no such p. P for some p E (1,m). It will be shown that k E fir,

L e t k E K. Then e i t h e r k E Proof. Let k E __

v

k

P

r E (1,pl. For u E L (Rn) define v

E Lr(Rn) , ws E L (Rn) such that u

I

u(x) = ws(x) if /u(x) > s and u(x) = vs(x) if lu(x)

I

u +v s s' 6 s, where s > 0 is a =

given fixed number. Obviously, w

has compact support, ws E L1(Rn) ,

I lwsl

5

I1uI ILr(Rn)

3. Singular Perturbations on Smooth Manifolds without Bounda y

418

and, moreover, one has:

E Lp(Rn), j lvs 1 ]

By a similar argument one finds:v

5

ll~ll

and moreover,

For a given Lebesque-measurable function f ( x ) denote by m(t) the measure of the set where I f ( x )

I

>

t. Then for each f E L (Rn) with q E 1 one has: 9

m

m

(3.12.152)

1 If ( x ) Iqdx

=

Rn

-1 tqdm(t) 0

=

q

tq-'m(t)dt. 0

Further, one has for each f E L (Rn) : 9

so that

(3.12.153) m(t) = u{x E Rn

I

I

/f(x) > t} S I ' t

If1

1'

L q The last inequality will be used for estimating Introduce

!Jaw = (3.12.154) uls(t)

VIX =

E Rn 1

V{x E

JZ(X) 1 > Rn 1 ] G s ( X ) 1

N

N

where, as previously, u = k*u, w

t), >

t],

N

=

k*ws, v

=

k*vc. I

Applying Theorem 3.12.70 to k E

Since by assumption k E M

bSlI

P'

p, ws E L 1 ( R n ) , one finds:

one has with some constant C

0:

, so that using (3.12.153) one finds for L (mn) P ! ~ ~ ~ defined (t) by (3.12.154), the following inequality: 5 CI

L (R")

P

3.12. Elliptic Singiilar Perturbations

419

N

Now applying (3.12.152) with f = u , one finds: m

Further, since Iw (x)[ 5 t, Iys(x) 1 5 t implies lu(x) 1 5 2t. one has: N

N

p0(2t) 9 ~ ~ ~ ( t ) + p ~ so ~ (that t ) ,the following inequality holds: m

(3.12.156) 1

Iy(x)lrdx

Eln

=

2rr 1 tr-lp0(2t)dt 9 0 m

5 r2r(7 tr-'uls(t)dt+ 1 tr-1u2s(t)dt). 0 0

These two integrals on the right-hand side of the last inequality will be estimated by setting s = t in the definition of w

and v

Using again (3.12.152) and (3.12.155). one finds:

By definition of w (x t m

1 0 m

p t x E IRn

.

3. Singular Perturbations on Smooth Manifolds without Boundary

420

Combining (3.12.156)-(3.12.158), one gets finally, that

with some constant C > 0 , and that proves k E Now, Corollary 3.12.68 yields: k E

fir, V

Mr,

V r E (l,p].

r E (1,m).

I

Now, one needs only two more lemmas in order to be able to prove Theorem 3.12.62. Lemma 3.12.73.

There e x i s t s a f u n c t i o n

+ E Cm(Rn) , supp $ c 15 E 0

Rn 1 0) 6 IS1 < 21,

such t h a t

c

(3.12.159)

+(2-k0 E 1 ,

v

5 E Rn\{Ol.

kEZ

Proof of Lemma 3.12.73. Let 0 2 0 be a function in Cm(R+) , supp 0 0 O(r) # 0, V r E [1//2,/2]. Defining

+ ( S ) = @ ( / 5 /( )

z

r

C

Q(2-klcl))-1,

v

I

f < r < 2), and let

5 E Rn\t01,

+(O)

=

0,

kEZ

it is easily seen, that + ( 5 ) satisfies (3.12.159).

a

Lemma 3.12.74.

L e t T be c d i s t r i b u t i o n i n S'(xn) and l e t p E (1,m). Then t h e following t w o c o n d i t i o n s a r e equivaZent:

(3.12.160)

and (3.12.161)

3.12. Elliptic Singular Perturbations where

c is a constant and, a s u s u a l ,

p-'+p'-'

=

1.

proof of Lemma 3.12.74. HGlder's inequality yields:

Proof of Theorem 3.12.62, The proof consists of three steps. First it will be shown that an approximation of f(6) with compact support is in M2. Next, it will be proved that this approximation belongs to the class K defined above by (3.12.142). Finally, one shows that it is allowed to take the limit of these approximations. Step 1. Let $ ( 5 ) be the function constructed in Lemma 3.12.73 and let fk(F)

=

f(5)$(2-kg). Using Leibnltz forxula, one can estkate the

derivatives of f ( 5 ) by those of f(5) : k

Hence, (3.12.1211, (3.12.162) yield:

0 5 j 5 2, are some constants which depend only on n. where C n,j'

Introducing gk(x) = (Fgixfk)(x), Parceval's identity along with (3.12.163) yields:

42 1

3. Singular Perturbations on Smooth Manifolds without Boundary

422

(3.12.164)

1 (1+22k 1x1 2) X Igk(x) I 2dx

5

Rn

wiiere we have denoted X = [n/2]+1 and where C n,j' which depend only on n .

j

=

3,4, are some constants,

Using Cauchy-Schwarz' inequality and (3.12.164), one finds: (3.12.165)

lgk(x) Idx 5

mn

where, as previously, Since fk(5)

=

X = [n/21+1 and C depends only on n. n,5 (F g ( 5 1 , one can estimate f ( 5 ) in the following x+S k k

fashion:

almost everywhere on IRn. Introduce

Since at each point 5 E Rn at most two functions f ( 5 ) do not vanish, k one gets, using 3.12.1661,the following estimate almost everywhere on Rn : IFN(<

I

5 2Cn,5B = Cn,6B.

-1 Hence, for GN(x) = (FS+xFN) (x), one finds

that is to say that G

E M2, FN E M2, V N

=

O,l,

:

L2(Rn)

... and, moreover, the

L (Rn) is bounded 2 B, where C depends only on n. uniformly with respect to N by C n,6 n,6 Step 2. We show that G E K , V N, by estimating the integral norm of the convolution operator GN

+

N

1 IGN(x-y)-GN(x)Idx, (x(>2t

for Iyi 5 t.

First, we esti-nate such ar. integral for each gk(x) in the definition

3.12. Elliptic Singular Perturbations

423

of GN(x). Dropping the additive term 1 between the paranthesis on the left-hand side of (3.12.164), and using again Cauchy-Schwarz' inequality, one finds:

with the same X = [n/2]+1 and some constant

which depends only on n. n.7 Further, using (3.12.167), one gets for IyI < t the estimate: C

k Since n/2-X = n/2-[n/2]-1 5 -f, (3.12.168) can be used when 2 t t 1. k If 2 t < 1 , then one uses Cauchy-Schwarz' inequality, Parceval's identity and (3.12.162), respectively, in order to estimate the integrals on the left-hand side of (3.12.168). One has, using Cauchy-Schwarz' inequality with X = [n/2]+1:

Further, Parceval's identity and (3.12.162) yield:

Hence, using (3.12.169) and the last inequality, one gets:

3. Singular Perturbations on Smooth Manifolds without Bounda y

424

N

where t h e c o n s t a n t C

depends o n l y on n.

Combining ( 3 . 1 2 . 1 6 8 ) ,

( 3 . 1 2 . 1 7 0 ) , one f i n d s f o r G

I:

=

Ikl6N

I

/ G (x-y)-GN(x) IdxSCnB

)x/>2t

= {y

E Rn

1

min{t2k,(t2k)n'2-X1

C

= C(l)B.

-m
The l a s t i n e q u a l i t y shows t h a t GN E

0

gk :

K with

A = { x E Rn

~

1x1 6 2 1 and

/ y / < 11 i n t h e d e f i n i t i o n of K . y i e l d s GN E

N o w , Theorem 3.12.72

fip,

V p E (1,m) a n d , moreover,

where t h e c o n s t a n t C may depend o n l y on p and n. Ptn T h e r e f o r e , FN E M V p E (1,m) and M (FN) = fi ( G ) . P' P P N S t e p 3. W e have t o show t h a t one may l e t N + S i n c e FN(C) + f ( < ) , V 5

m.

E Rn\{O}, p o i n t w i s e and

IFN(:)

I

i s bounded, one

.

h a s : F + f i n S ' ( R n ) F u r t h e r , s i n c e t h e F o u r i e r t r a n s f o r m i s an isomorN phism o f S ' ( R n ) o n t o i t s e l f , one h a s : GN + G = F-' f i n S ' ( R n ) , t o o . W e S+X

a r e g o i n g t o a p p l y Lemma 3.12.74 i n o r d e r t o show t h a t ( 3 . 1 2 . 1 7 1 ) h o l d s for N

+ m,

i . e . Lor G .

Consider t h e s e t S '

B

C

S'(Rn)

d e f i n e d as f o l l o w s :

v Obviously, S ; ' I '

-f

is a closed set i n S ' ( R n )

,

. Hence,

then

I (GN*u*v)( 0 ) I

+

I (G*u*v) ( 0 ) 1 ,

v

u , v E S(7Rn)

T h e r e f o r e , one h a s f o r a l l u , v E S ( R n )

(3.12.172)

I.

, s i n c e f o r each given u , v E S ( R n ) ,

(T*U*v) ( 0 ) i s a c o n t i n u o u s l i n e a r form on S ' ( W n )

S'(Rn)

Now,

u,v E S(Rn)

:

and Lemma 3.12.74 y i e l d :

and t h a t ends t h e p r o o f o f Theorem 3.12.64.

I

.

if GN

+

G in

3.12. Elliptic Singular Perturbations

425

The proof presented here is due to Hormander (see [Her, 11). Now, we introduce analogues H (IR~), p E (1, - I , of spaces ( s ) ,P.E H ( s ) ,€(Rn) with L -structure and point out their properties which are

P necessary for the development of the corresponding L -theory of elliptic P singular perturbations. LogicalLy this material belongs to Chapter 2, but,

since the derivation of the basic properties of the spaces

H

( s ) ,PIE

is

based upon the use of the Fourier-multipliers result stated in Corollary 3.12.63, it is included in this section. Definition 3.12.75.

For each

s =

(s1,s2,s3)E n3 and each p E (I,-) d e f i n e

and denote (3.12.174) H

3 d ( (s),P,E

=

1

Iu

u

A f a m i l y o f d i s t r i b u t i o n s U(E,. )

space

:

(O,Eo1

-f

: (O,E~] +

f l l u l I ( s ) ,P,E <

S'(RRn)

-1.

s'(nn) i s s a i d t o belong t o t h e

H ( s ) ,P( R ~ ), if

Remark 3.12.76. With s

=

(O,s,O)

the spaces H

( s ) rP

(Rn) are nothing else but the classical

Bessel-potential spaces, denoted usually HS (IRn) . They have the following

P

duality property: (Hs(Rn))

=

Hz,(mn) where p-lcp * - l =1 (cf., for instance,

P

[Tr, 11). Remark 3.12.77. It is easily seen that for each

s

E R n , p E ( 1 , ~ )the family

( 0 . ~ ~3 1E + H (an)is a family of Banach spaces, (s), P . E of course, a Banach space, as well.

f f

( s ),P

(Rn) being,

Now, an analogue of Proposition 2.1.8 will be stated and proved using the Fourier-multiplier result in Corollary 3.12.63. Proposition 3.12.78.

L e t s' 6 s , p E (I,-). Then t h e r e e x i s t s a c o n s t a n t c only on i t s s u b s c r i p t s , such t h a t

s ' ,s ,P'

which depends

3. Singular Perturbations on Smooth Manifolds without Boundary

426

Proof. First, we check that <ES>-' in L (mn) ,

P

v

<E<>'<S>-'

and

are Fourier-multipliers

a 2 0.

One shows easily,using the induction argument, that for each multiindex a holds:

where Q ( 5 ) is some polynomial whose degree is at most

[ a1,

so that one

has

Now, using (3.12.177), one finds

I

(3.12.178) D"<E~>-'

s

5

I

= E

I a I Da-' 1 n

S

s c 1 I 5 l - ~ a ~ ,v 5

C€lal<,S> -u-/ul

E Rn-.IO},

provided that u 2 0. Further, the Leibnitz formula yields: (3.12.179) /Da(<€5><5>-')'l

5

=

1

I: $53

(i) (DB<E<>') 5

(Da-'<S>-'

5

)I s

s c

v 5 #

0,

and the same computation shows that (3.12.180) (Da(E<6><E5>-1)u/2 C2)5)-Ia', v 5 # 0, v a .

5

Hence, as a consequence of Corollary 3.12.63, both <EE'-' are Fourier-multipliers in L (Rn) ,

<EC>'<~>-'

P

Therefore, one has with u = s - s ' u2+rJ3 2 0:

>

v

U

and

2 0.

0 , i.e. U 1 2 0 , a +u 2 0 , 1 2 -

v a,

3.12. Elliptic Singular Perturbations

421

Corollary 3.12.79.

If

s

> s'

then H

H (s).P(Bn) 5

(IRn)5 H ( s ' ) ,P,E(mn), V E

(s),P,E H ( s ' ,p ) (R").

E ( 0 , ~ and ~l

Lemma 3.12.80.

Let s

c

=

3

(sl,s2,s3)E IR , s2 > 0 , p E ( 1 , m ) .

Then t h e r e e x i s t s a c o n s t a n t

> 0 such t h a t holds:

Proof. Without restriction of generality, one can assume that -

s1 = 0.

Further, as a consequence of (3.12.177) and Corollary 3.12.63,

fl(c)

-S

=

<5>

is a Fourier-multiplier in L (IR")

using again the Leibnitz f2(5) = <5>s2(1+151s2)-1

P

, t/

s2 2 0 . Furthermore,

formula, one gets immediately that and f3(<) =

151s2-s2 satisfy (3.12.123), so

that f ( 5 ) and f ( 5 ) are Fourier-multipliers, as well, by virtue of the 2 3 same Corollary 3.12.63. Hence, one has with s = (0,s2,s3), s2 > 0:

On the other hand, one finds:

3 . Singular Perturbations on Smooth Manifolds without Boundary

428

and

Lemma 3.12.81.

Let

c

s = (sl,s2,s3)E R

> 0

3

,

s 3 > 0 , p E (I,-).

-S

Proof. (E

to

One checks easily that <ES>

15 I <EC>-') E

Then t h e r e e x i s t s a c o n s t a n t

such t h a t h o l d s :

E (O,E

3

, <E<>'~ (1+( E 15 I ) s 3 )

-'

and

s3 are Fourier-multipliers in L ( R n ) uniformly with respect 0

1

Lemma 3.12.80.

P

and uses the same argument as previously in the proof of

I

For s = (sl,s2,s3)E R

X

R+

X

R + , p E ( 1 , ~ )and u E S ( R n ) introduce

the family of norms

Corollary. L e t s E R x R+ x R + , p E (I,-). Then t h e f a m i l i e s of norms

-

and I I I I I I ( s ) there e x i s t s a constant c >

a r e e q u i v a l e n t uniformly w i t h r e s p e c t t o

o

E

I I .I 1 ( S ) , P , E E (O,E

0

I i.e.

such t h a t

Indeed, using the definition of

1 1 . I 1 ( s ) .P,E

and the previous two

3.72. Elliptic Singular Perturbations

429

lemmas, one finds:

Therefore, in order to prove (3.12.184). one has only to show, that with some constant C > 0 one has:

is defined by (3.12.183). For that purpose, it suffices to show that f ( 5 ) = 15

I s3 (I+ 15 I

s +s

2

3 -1 )

with V s2 2 0, s 3 2 0 , is a Fourier-multiplier in L (Rn) . One checks easily, using the Leibniz

formula, that the condition (3.12.123) for

fs(c) is satisfied, so that Corollary 3.12.63 applies. This gives:

1 Corollary 3.12.83.

L e t s E R+

z+, p E (I,-). Then t h e r e e x i s t s a c o n s t a n t C

x Z+ x

> 0

such

t h a t one has: -S

(3.12.186) C-lI lul s I I E

+

I

5

E

( s ) ,P,E s +s

3D 2 3 x , UI

I

1

(1

/u/

I

s2

I

+ 1 (/IDx,UI L (IR*') l < = j s n 2 L

P

P

I

) )

I

5 CI lul (.-)

(w")

+

,P,EI

L (Rn)

P

v

E

E

(o,Eo],

v

U

E H

( s ) rP,E

(Rn) .

Indeed, one uses Corollary 3.12.82 and Corollary 3.12.63 for showing (3.12.186). For doing that, one has to show that for each function f

.(F)

or1

=

<"SI-" 1

0

>

0 the

satisfies (3.12.123). This can be easily seen by

3 . Singular Perturbations on Smooth Manifolds without Boundary

430

induction argument. Therefore, one has for any positive integer a:

On the other hand, with any positive even integer a holds:

and for a positive odd integer one has:

using (3.12.187)-(3.12.189), one gets immediately (3.12.186).

I

Remark 3.12.84. For s = (sl,s2,s3)E R families of norms 1 1 1 . 1 are equivalent with to

E

x

lR+

x

R+

,

s

,

1 I I ( s ) ,E,P without

1 1 . I 1 ( s ) ,P.E

E Z+, j = 2,3, one can define

using the Fourier-transform, which

introduced above, uniformly with respect

I

E (O,Eol-

-1 Let u E (0,l) and let 2n(n+a) < p <

m.

equivalence (see, for instance, [Ste, 1,21):

where

Replacing

U ;l)u

by

Then one has the following

3.12. Elliptic Singular Perturbations

43 1

the equivalency (3.12.190) holds for u E ( 0 , Z ) . NOW, if s

=

m+u with some integer m > 0 and some u E (O,l), then one

has the following equivalency (see [Ste, 1,2])

Now, let u

=

(0,u2,u3) with u , E 3

(O,l), j

=

2,3. Then holds:

-

\-.

I

and it is quite obvious, how the last equivalency extends to the case when s = (sl,s2,s3),s1 E I R , s . = m.+u. j = 2,3 with integer m . 2 0 , m 2+m3 > 0

1

and

U.

E (O,l), j

=

3

2,3.

I' I

3

3

Now we are in a position to outline the proof of the two-sided a priori estimates in spaces H

(s),P,E

(Rn) for elliptic singular perturbations

We need the following technical Lemma 3.12.85.

L e t a E L'(R")

c and

be e l l i p t i c of order v . Then t h e r e e x i s t p o s i t i v e c o n s t a n t s

R such t h a t

(3.12.192) la(x,E,E)I 2 CE

-ul

151 "2<65>'3,

v

x E

Rn, V

provided t h a t E~ i s s u f f i c i e n t l y s m a l l . Moreover w i t h xR(S) E Cm(Rn) , xR t 0, f o r 1 5 ) 2 ZR, t h e symbo2 xR(S)a(x,E,5)-'

couple of m u l t i - i n d i c e s

a,B

holds:

xR

E

o

v

E

E

(o,EO1.

5 E R n , 151

f o r 151 5 R,

beZongs t o L-~(R")

2 R,

xR

, i.e.

:1

for each

3. Singular Perturbations on Smooth Manifolds without Boundary

432 Proof. Since -

v n (see D e f i n i t i o n 3 . 1 2 . 1 ) , one h a s a E S ( R )

a E L'(R")

.

F u r t h e r m o r e , a ( x , E , S ) b e i n g e l l i p t i c , i t s p r i n c i p a l symbol a 0 ( x , E , S ) s a t i s f i e s ( 3 . 1 2 . 2 ) , so t h a t one g e t s u s i n g ( 3 . 1 2 . 2 ) and ( 3 . 3 . 4 ) w i t h a = B = O :

la(x,E,c)

1

-v 2 (C/~)E

2 lao(x,c,S)

1

1-1

(a-ao) ( x , E , < )

1

2

-vl

151

CE

2<E<>

v3

( 1 - c l ( E + < ~ > - l ) )2

v

151 2<EF,>v3, V x E

V

IRn,

E

E

(o,Eo],

v 5 E

En,

151

b R,

( E +RP1) S f. 1 0 N o w , w e check (3.12.193) u s i n g i n d u c t i o n w i t h r e s p e c t t o a and 5. With-

provided t h a t C

o u t r e s t r i c t i o n o f g e n e r a l i t y , one c a n assume t h a t v1 = 0 . F o r a = B = 0 (3.12.193)

i s an immediate consequence o f

(3.12.192). W e use i n d u c t i o n i n

o r d e r t o show, t h a t f o r 151 > 2R h o l d s :

I n d e e d , assuming t h a t ( 3 . 1 2 . 1 9 4 ) ,

(3.12.195)

hold for l a l + ( B (

=

m, one

finds that

where b a + e k , B

=

aD

5b a , g - ( l + l a ~ + ~ B l ) b a , B D S ksaa t i s f i e s

i n s t e a d o f a , as it c a n e a s i l y b e s e e n . Bcek a -1 The Same argument a p p l i e s t o D D a

x

s

,

(3.12.195)

w i t h a+ek

t h e d e t a i l s being l e f t t o t h e

reader. Next, w e p r o v e t h e main r e s u l t , s t a t e d i n t h e f o l l o w i n g Theorem 3.12.86.

Let a E

L'(R~)

and p E

(l,m),

E

E

( 0 , ~ ]

0

(3.12.196)

be e l l i p t i c o f order v. Then f o r each s E I R ~ ,s ' E alvs t h e foZZowing equivalency holds uniformly w i t h r e s p e c t t o

w i t h c o n s t a n t s , which may depend on s , s ' , p and

1 / u / 1 (s),P,E

provided t h a t

E~

- j lop

I

( a ) u l ( s - v ) ,P,E

i s s u f f i c i e n t l y small.

+

E

~

:

I l u l I ( s ' ) ,P,E'

3.12. Elliptic Singular Perturbations

433

Proof. First, we prove (3.12.196) for a = a(E,S) which does not depend on ___ x. Since s' E a l V s , i.e. s; = sl, one can assume without restriction of generality that s ' = s1 = u1 = 0. With a E a(E,F,),s = (0,s2,s3), 1 v = (0,u2,u3) and x ( 5 ) as in Lemma 3.12.85, one can write R

Now, using Leibnitz' formula and (3.12.193), it is easily seen that f

(E,

5)

=

xR ( 5 ) a( E, 5 ) -' I < 1 v2<~S>u3satisfies the

condition (3.12.123)

uniformly with respect to E E ( O , E 01 , i.e. with a constant Bo on the righthand side of (3.12.123), which does not depend on E E ( O , E 01 . Hence, Corollary 3.12.63 appliy to fl(E,S). The same, obviously, holds for s -s s3-sj m , since f2(E,S) E C 0 (Rn) . Therefore, f2(E,S) = (l-xR)<S> '<ES> Corollary 3.12.63 yields:

where the constant C depends only on s,s',p and

E 0' On the other hand, Proposition 3.12.78 yields:

I lul I

( s t )

,P,E 5 CI

lul I ( s ) ,P,E-

Furthermore, one uses again the splitting

xR as in Lemma 3.12.85, and the previous argument,in order to show that as a consequence of Corollary 3.12.63, the following inequality holds:

with

I lo?(a) with a constant

C,

UI

I (s-") ,P,E 5

I

cl lul (s),P,E

which does not depend on

E

and u.

Now, if a=a(x,E,S), then one has to use a partition of unity argument, that is to prove analogues of Lemma 3.12.9

and Lemma 3.12.10 in

3 . Singular Perturbations on Smooth Manifolds without Boundary

434 spaces H

( s ) ,PrE

(Rn) , which can be shown again by means of Corollary

I

3.12.63, but will not be done here. Example 3.12.87.

Consider the singular perturbation which appears in the dislocation theory and whose symbol a(E,S) is

5, (3.12.197) a(E.5) = (l+exp(-~/Sl))sgn

E

> 0, 5 E R

(see also Example 3.2.6). This singular perturbation is elliptic of order zero but a f L0 (IR1 ) 1) , since it is not smooth at 5 = 0. However, with x6 E Cm (R x6 E 0 for

151

5 6,

x6

E 1 for I S / 2 26 the singular perturbation O p ( x (S)a(E,<))with

6

a ( € , < ) given by (3.12.1971, is an elliptic singular perturbation of order

0 1 zero in Op L (R ) , so that (3.12.196) holds with v = 0 . A

straightforward computation shows that (3.12.123) is satisfied for

a(E,S) given by (3.12.197) with Bo = 2 . Hence, uniformly with respect to E

holds:

(3.12.198) Op(a(E,<)) : L (R1

1

,

L (R )

+

P

P

V p E (1,~).

Furthermore, a straightforward computation shows that (3.12.123) holds for a(E,S)-'

=

(l+exp(-E:Sj))-lsgn5 , as well, with Bo = 1, so that

uniformly with respect to (3.12.199) Op(a(E,C)-l)

E

: L

P

E (O,E

1

(R )

1,

0

V

E~

1

+

L (IR )

P

<

m,

holds:

.

Hence, combining (3.12.198), (3.12.1991, one gets that the singular perturbation with symbol (3.12.197) is uniformly with respect to E E ( O , E 1 , 0 an isomorphic mapping of L (I71 ) onto itself, V p E (1,m). Notice that the

P

reduced operator

(E =

0) for Op a(E,S) given by (3.12.197) is the multiple

of the Hilbert transform (@(a

0

( S ) ) u ) (x)

=

2v.p.

i

7

J1 (x-y)-1 u(y)dy.

I

R

Example 3.12.88. Let k(x) be locally integrable function, which satisfies the following conditions: (i) k(x) E 0 for

1x1 < 1;

(ii) k E K , i.e. k satisfies (3.12.142);

3.12. Elliptic Singular Perturbations

(iii) there exists a function k (w), w E Rn, k 0

0

435 1

E L (R

)

such that with

some constants C > 0 , y > 0 holds a.e.:

(iv) the function k

0

J ko(w)dw

=

( 0 )has

zero mean value over R

:

0.

On We shall denote by k (x) the function, which is identically zero for 1x1 < 1 0 and coincides with k (w) for 1x1 2 1. 0

IxI-~

We are going to show that the family of convolution operators

E

-f

K

with kernels

is uniformly bounded in L

P

One can view K

(mn),

1 < p <

-,

as a singular perturbation of the convolution operator K 0

with the kernel ko(w)/ x / -for ~ 1x1 2 1 and identically zero for 1x1 < 1, this convolution operator having been considered previously in [Cal-Zig, 1 1 and [Hbr, 11. We have to show that GE(€,) = F

X-fE5

belongs to K

(3.12.142) with a constant, which does not depend on

n k2 and E

satisfies

E (0,lI. Then

(3.12.200) will follow from Theorem 3.12.72. First, we show that 1; ( 5 ) E Lm(Rn) uniformly with respect to so that uniformly with respect to E E (0,1] will hold:

E

E (0,11,

E E fi2. Notice that

if y E 0, then kE(x-y)-kE(x) E L1(Rn) , as a function of x E IRn, since k

E K , for each

E

E (0,1]. Therefore its Fourier transform

) (exp(-i)-l)E(E5) is continuous in 5 for all y E 0. Thus, ~ ( E S is continuous for 5 # 0 and bounded for 5 E

-f

m

(uniformly with respect to

E (0,ll). Furthermore, obviously one has (as a consequence of (iii), (iv)) 1

k(tEE)-E(ES) =

1 1 ko(w)exp(-i)r-'drdw

+

[wj=l t

Denoting the first and second integrals on the right hand side of the last

3 . Singular Perturbations on Smooth Manifolds without Boundary

436

formula by Il(5,t) and 12(€,5,t), respectively, one finds that

On the other hand, if (iv) is satisfied, then

When 5 E Qn, t E (0,1], the right hand side of the last formula is bounded by the constant C

21~1 /k

=

0

1 I Limn)'

Hence, since I1(E,t) = io(t<)-co(6)

1

with i o= F k (x), one gets the conclusion that / g o ( < ) is bounded for xt5 0 5 # 0. Therefore, 1; ( 5 ) is the sum of a bounded on Rn function and, 0

possibly, a linear conbination of 6-function and its derivatives supported at the origin. It is easily seen that the latter singular part in the sum representing k0(5), must be zero. Indeed, it suffices to show that for each $ 6 ( x ) = $(x/6), with $ E C;(Rn)

1;0 ( 46 ) = k0(4&)

=

, holds:

0 for 6

+

+

0, where

J k (x)J(6x)6"dx.

wn

O

I

E S(Rn) , so that ]i(y) 5 C-' and the inequality

Using the fact that

1 Iko(rw)Idw

5 Cr-n,

n' one finds, introducing polar coordinates, that

lCo(b6)l

5 ~ 6 "ln(1+6-l).

Hence, indeed, kO(E) coincides, as a distribution, with some function in L,(Rn)

. Now, using (iii), we can estimate the second integral I2 ( ~ , < , t ) in

the following fashion m

1 J

l12(~,5,t)I2 C

-1 r (r/tE)-Ydrdw 5 C

Iw/=lt

v (E,E,t) E In particular, for the function k(x)

=

-1

1'

[0,1]

-2

F k (5)<5> s+x 0

x

, for

wn

x

w+.

1x1 2 1 , k(x)

5

0

for 1x1 < 1, where 1; ( 5 ) is homogeneous of order zero and satisfies (iii), 0

(iv), the conditions (i)-(iv) are fulfilled, and the corresponding family of convolution operators K

with kernels kE(x) = E-'k(x/E)

bounded in L (Rn) , 1 < p <

m,

P

V

E

E (0.11, i.e. iE(5)

=

are uniformly

];(ES) are

3.13. Girding's Inequality (uniformly with respect to

E

437

E (0,1]) Fourier-multipliers in L (En)

P

Notice that the convolution operator -1 io(:)<:>-2 k (x) = E-"k(x/E), k(x) = F X'C -1 1; ( 5 ) oEerators with kernels k (x) = F 0 X'[ 0 latter being the exponentially decreasing

.

with the distributional kernel is product of convolution and GE(x)

= F-l < E C > - ~ , the C-tx at infinity fundamental solution

for the elliptic singular perturbation l-cLA with A the Laplace operator.

I 3.13. Garding's inequality

This inequality, first proved in [Vish, 1

1

for strongly elliptic

differential operators in domains with small diameter and properly extended in [G&,

1

1

to strongly elliptic differential operators in

arbitrary domains and in [Cal-Zig, 1

1

to singular integral operators,

plays an important role in different fields of the theory of pseudodifferential operators. Its sharp form, first proved in [Lax-Nir, 1

]

for pseudodifferential operators and for one parameter families of difference operators appearing in the stable approximations of well posed evolution problems, is one of the central results in the linear theory of pseudodifferential operators. A simplified proof of the sharp form of G&ding's [Vai, 1

1

inequality was given in [Friedr, 1

1

and later on in

where the method introduced by Friedrichs (the use of a smooth

mollifier) was slightly simplified. For classes F"

of difference (s)

operators (see Definition 3.12.32) the sharp form of Giirding's inequality was first established in [Fr,8,91 in corresponding spaces H

(Rt) (s),h

,

using the method introduced by Friedrichs. We shall present here the proof of Gzrding's inequality for the classes of singular and difference singular perturbations introduced above and shall give several applications of this inequality in the elliptic theory and to some classes of singularly perturbed problems of evolutionary type. We start with the simplest form of Garding's inequality Concerning strongly elliptic singular perturbations. We consider here singular perturbations, whose symbols are pxp matrices, the corresponding symbol classes S" ( U ), S v ( U ) , K " ( U ) , H " ( u ) , 1 to FY,o(U), F v ( U ) being defined a s previously in sections 3.3, 3.6, 3.11, 3.12 with the only difference that in all these definitions the absolute values in the corresponding inequalities should be replaced by the norms of corresponding matrices in Hom(CP;CP).

3. Singular Perturbations on Smooth Manifolds without Boundary

438

Definition 3.13.1. A symbol a(x,E,S) E s v ( U ) , a

:

U

(O,E~]

X

lRn+ Hom(CP;CP), i s s a i d t o

X

be s t r o n g l y e l l i p t i c , i f i t s p r i n c i p a l symbol ao(x,c,S) s a t i s f i e s t h e condition -v

Re a (x,E,<) 2 y

(3.13.1)

0

E

v

'151 2v3 Id, V(X,E,<) E u x

where Y i s some p o s i t i v e c o n s t a n t , Re . a

=

( O , E ~ ~ (Wn\tO}), X

( 4 ) (ao+a;)) i s t h e ( s e l f - a d j o i n t )

and Id i s t h e i d e n t i t y i n Hom(CP;CP), r e a l p a r t of t h e m a t r i x a ( x , E , ~ ) 0

t h e i n e q u a l i t y B b 0 for any s e l f - a d j o i n t m a t r i x B i n Hom(CP;CP) being

E cp. op a E op sv(u) i s s a i d t o be s t r o n g l y

i n t e r p r e t e d i n t h e usual way: A singular perturbation

=

20, V z

e l l i p t i c o f order v, i f i t s symbol a(x,E,S) E sv(u) i s s t r o n g l y e l l i p t i c . Lemma 3.13.2.

L e t a E s V ( u ) b e s t r o n g l y e l l i p t i c . Then t h e r e e x i s t s a symbol b E Sv'2(u) such t h a t (3.13.2)

(Re

where b*(x,~,5)i s t h e conjugate symbol of t h e symbol b(x,s,<) whose asymptotic expansion i s g i v e n by (3.7.16),

i.e.

w i t h b(x,E,S)* t h e conjugate m a t r i x of b(x,E,S), and where

-

stands f o r

t h e (non-commutative) product of symbols, d e f i n e d by (3.12.311, i . e . b*(x,E,S).b(x,E,S) i s a symbol w i t h t h e asymptotic expansion

Without restriction of generality one can assume that v = 0, since one can v1

consider

E

-v

<S>

-V

*<ES>

a principal symbol .

3

a(x,E,5) instead of a(x,E,C). Furthermore, if the 0

of a E S (U) satisfies (3.13.11, then for I S 1

sufficiently large the same inequality holds for a(x,~,C)itself with y/2 instead of y. Hence, one can find a symbol r(x,s,5) E that

S(or-m'O)

1.0

(U) such

3.13. Girding's Inequality

439 2

Indeed, (3.13.5) is valid with r(x,E,S) constant

C >

=

Ce-lC1 , provided that the

0 is sufficiently large.

Therefore, one can assume from the very beginning that a E S0 (U) satisfies the inequality: (3.13.6)

Re a(x,E,S) 2 y Id, V 1

( x , E , ~ )

E ux (O,E~] x lRn,

with some constant y1 > 0 ; here Re a = ( 1 / 2 ) (a+a*). We shall define the symbol b(x,E,C) E So ( U ) recursively to satisfy 1#O (3.13.2). Let (3.13.7) where

r

1 i Xt(XI-Re a(x,~,S))-ldX, bo(x,E,S) = 2~ri ~

is any contour surrounding the spectrum of the matrix Re a(x,~,S)

and excluding the origin. Since a E (x,E,~).

S

0

(U) satisfies (3.12.6), one can choose

r

independent of

Therefore, bo(x,E,C) is again a smooth function of (x,S) with

values in Hom(CP;CP), and moreover, one has: (3.13.8)

bO(x,E,S)* = bO(x,E,S), bO(x,~,S)2 = Re a(x,~,<).

Further, a straightforward computation shows that b (x,s,S) E 0 formulae for the adjoint symbol bi(x,E,E) and the product

So

1 to

(U), the

bi(x,E,C)'bO(x,~,S)lead to the conclusion that

and that the principal symbol of r ( X , E , ~ ) is a self-adjoint matrix. 1 Using the induction argument, assume that we have already found bO(x,E,S), bl(x,E,S),...,bk(x,E,S) such that

As a consequence of (3.13.10), the principal symbol of r ( X , E , ~ ) is a selfk

adjoint matrix. We have to determine a matrix bk+f(x,~,e) in such a way that def (3.13.11) rk+l(x,E,S)

=

(Re a(x,E,S)-( Z b*(x,E,S)).( Z bj(x,E,S))) 05 j5k+l OSj
3 . Singular Perturbations on Smooth Manifolds without Bounda y

440

Inclusions (3.13.10), (3.13.11) yield the following equation for bk+l(x,~,S):

'

C b*(x,E,S)-bk+l(x,E,<). O<jZk+l

If bk+l(x,~,S)is chosen to satisfy the equat.ion

Moreover, since bO(x,E,<) and the principal symbol of r (x,E,S) are k both self-adjoint, so it is also for the principal symbol of bk+l(x,~,<). Furthermore, bO(x,E,5) t y Id, with some constant y o > 0. 0 The matrix b ( x , E , < ) , given by k+l (x,E, 5 ) --tbO(x,E,S) (3.13.14) bkil(x,€,S) = 1 e dt rk (x,E , 5 ) e 0

-tho

is well defined (since bo is positive definite uniformly with respect to x,E,S) and satisfies (3.13.13). Indeed, one has:

Furthermore, it is obvious that the principal symbol of bk+l(x,~,E)is self-adjoint. It is easily seen that bk+l(x,~,S)(given by (3.13.14)) is the only solution of the matrix-equation (3.13.13). Indeed, one has to show that the only matrix X E Hom(CP;CP), that satisfies the equation

0, provided that b (x,E,S) is positive definite. 0 Multiplying (3.13.15) by exp(-tbo(x,E,S)) from the left and from the right, one gets the equation, which can be rewritten in the following form: is X

- -(e dt

-tb -tb Oxe O)

=

0,

v

t 2 0.

3.13. Girding’s Inequality -tbo Hence, e-tboXe

X,

5

v

t 2 0 . Now letting t

-f

44 1

+-,

one gets the

conclusion that X E 0. Hence, bk+l(x,~,E) defined by (3.13.14) is the only solution of (3.13.13), so that the tefms b (x,E,S) in an asymptotic expansion of b(x,E,S) are

k

uniquely defined by the recursive procedure above. Therefore, the symbol b(x,E,S) which satisfies (3.13.2) is well defined modulo a symbol in (u, ,--,v3) I (U). S 1 ,o Remark 3.13.3. Using precisely the same argument, one can define recursively by the same formulae (3.13.7), (3.13.9), (3.13.lo), (3.13.131, (3.13.14) with a(x,E,S) itself instead of Re a(x,~,S),a symbol c(x,E,~)such that t (3.13.16) (a(x,E,S)-c(x,E,S) -c(x,E,S))E

(v1,-m,u3) (U),

s1 t o

t where C ( X , E , ~ ) is the transpose symbol of

C ( X , E , ~ ) ,

defined (asymptotic-

ally) by the first of formulae (3.7.16). Furthermore, c(x,E,~)is well defined modulo a symbol in

(V1?,V3) S

:u).

1 ,O

Corollary 3.13.4.

L e t a(x,E,S) E Sv(U) be s t r o n g l y e l l i p t i c i n t h e sense of Definition 3.13.1 and l e t A~ = op a be t h e corresponding s t r o n g l y e Z l i p t i c s i n g u l a r p e r t u r b a t i o n . Then t h e r e e x i s t s a s t r o n g l y e l l i p t i c s i n g u l a r p e r t u r b a t i o n B

E op S~’~(U)such t h a t h o l d s :

(vl,-,,v3)

(3.13.17) (Re AE - B*B E

where

BZ

E Op S

) E

i s t h e a d j o i n t of m

B~

1 to

with respect t o the duality relation <

1

between CE(u) and (co(u)) and Re A E Indeed, with b(x.E.6) E Sv’2(U) 3.13.2 one gets for B

=

=

,

>

(1/2)(AE+A:).

as defined in the proof of Lemma

Op b immediately (3.13.17),as a consequence of

this lemma. Further, with

=

C

Op c(x,~,S)with C ( X , E , ~ ) constructed as discussed

in Remark 3.13.3, one has the inclusion (u1,-m,u3) (3.13.18) (AE ) -,)C

E

OP S

1 ,o

(U),

L

where C L is the transpose of the family of singular perturbations CE E opEsv/2(u), CE

:

CpJ)

+

Cm(U).

3 . Singular Perturbations on Smooth Manifolds without Bounda y

442

Theorem 3.13.5. 3

be strongly e l l i p t i c . Then f o r any s E R , f o r any c : ~ c # cK tc u there e x i s t constants c0 > 0 and c1 > 0 such t h a t Let a ( x , ~ , ~E) op s'(u)

(3.13.19) Re(a(x.E,D)u,u) 2 coI ( U I I 2(v/2) ,E-c11

where

(

,

)

I u I 1 ( s ) , E , . v ~ c:(K),vEE E

(o,E01,

2

stands for the inner product in L (K).

Proof. __ Corollary 3.13.4 guarantees the existence of a singular perturbation B

=

Op b(x,E,S) E Op S"/2(U) such that (3.13.7)

is valid with

AE = Op(a(x,E,~)-(y/2)E-"1<S>u2<ES>"3 Id), where the constant y is the one, appearing in (3.13.1). Denoting Rr = Re Ac-(y/2)€ -"1u2<ED>u3 Id B*B E

E'

one can write for

v

u E

m

c0 (K):

I IUI (Y/2) I l u !

Re(AEu,u) = (Y/2) 2

(V/2),E

+(B u,B u)+(REu,u) k E

E

3

V s E l R ,

I

where c1 may depend on s and K. Remark 3.13.6.

Definition 3.13.1 carries over in an obvious way to singular perturbations ona smooth manifold M without boundary, their principal symbols being well defined for V

E

E R+ as a mapping from the cotangent bundle T*(M)

into Hom(CP;CP) in the case of matrix-valued symbols. Hence for a strongly 3 elliptic singular perturbation a(x,E,D) : Cm(M) + Cm(M) of order u E R 3 one has for each s E R :

where the constant co depends only on y in (3.13.1) and the constant C

1

does not depend on u E Cm(M). Example 3.13.7. Consider again the singular perturbation A

from Example 3.10.15 defined

by (3.10.54) or, equivalently by (3.10.58) on the unit circle R

1

C

C.

Its principal symbol a0 ( x , E , S ) = 1 + E / < 1 satisfies (3.13.1) with u = (0,0,1) is strongly elliptic of order (O,O,l). Defining

and yo = 1, so that A

the singular perturbation B

=

F-l(l+Elnl)fF with F the Fourier series E

one has in this case (A u,u)

=

(B u,B u) 2 c E

E

0

1

2

and AE = BE. Besides, 2 with some /U/ 1

expansion operator, it is easily seen that B* = B

E

(0,Ort)

,E

3.13. Gdrding's Inequality

443

constant co > 0 which may depend upon the partition of unity in the definition of the norms 1 if3 E c I I e l 5 TI. fil = Ie

1. I [

on the circle

(O,O,!)

,E

Singular perturbation (3.10.66) is strongly elliptic of the same order v = (0,0,1) BE

iff Re a(z) t . a

having ( I + € Re a(z)

/
>

r,

0, V z E

the corresponding operator

as its principal symbol.

It will be shown later that for the singular perturbation A

defined

by (3.10.66) the following inequality holds:

where the constant C does not depend on u and

E.

Now several forms of sharp Gzrding's type inequalities will be considered and we start with the one for one parameter families of difference operators introduced in 3.11. We follow essentially [Friedr, 1 1

1

and [Vai, 1

for proving this form of sharp Gzrdinq's inequality.

The symbols considered are valued in Hom(CP;CP), i.e. are pxp matrixsymbols.

As usual, we denote by a,(x,n)

the principal symbol of a symbol

a E F"(U), ao(x,n) = lim h"l:(x,h,n). h+O Theorem 3.13.8. O-ii

Let p E F ( n

)

Assume t h a t t h e primcipal symboZ pO(x,ri) E n e g a t i v e hermitian m a t r i x :

1,I?

)

and is a non-

(3.13.22) Po(X,I1) 2 0 , v (x,r?)E lRnxTy,n.

Moreover, assume t h a t t h e r e e x i s t s a c o n s t a n t

B

c such t h a t 1

I

(3.13.23)<~>~*~lD~p(x,h,ri)-D~p,(x,n) B 6 Clh, V (x,h,n) E Rn x(O,ho]xTy,ri,

IB/

v 8, Then f o r ph

=

op(p) one has:

2

(3.13.24) Re(Phu,u)t -C2h/l u l

lo,

V u

m

n

E CO(R

),

V h

E (O,hOl

where c 2 i s some c o n s t a n t , which may depend on ho, and where

I 1. I 1

5 n+l

2

are t h e i n n e r prod-uct and t h e norm in L ( n n)

.

(

,

)

and

3 . Singular Perturbations on Smooth Manifolds without Boundavy

444

Proof. As

a consequence of assumption (3.13.23),it suffices to prove (3.13.24)

for the difference operator Ph,D

-

Op po, whose acting on a function

u E Cm(IRn) is given by the formula: 0

Indeed, if p

pm(h,hE) then, as a consequence of (3.13.23) and Parceval's

identity, one has:

I I Ph-Ph,ol lL2,L2

5 C h. Thus, without restriction of 1

generality, one can assume, that pm(h,h5) x E IRn , belongs to S(IR:)

5

0, so that p, as a function of

and so it is for po, as well. We use again the

same argument, as in the proof ot Theorem 3.11.15. Denote r(x,h,hE) One has for

=

V(5)

F

x+S

=

=

%".

u the following representation:

J

=

p(x,h,hS)-po(x,hC), Rh = Op r, v

G(E-n,h,hn)G(Il)dIl,

IRn where ;(C,h,hrl)

=

Fx+Er.

For any function 41 E S(Rn

)

and for each multi-index a one has the

following inequality, as a consequence of Parceval's identity:

where R

is the area of the unit spheere in IRn. n Applying the last inequality with 161 5 n+l to r(x,h,n) (as a function

of x E lRn

)

and using (3.13.23), one gets the conclusion that there exists

a constant C such that

Therefore, one has

I"(<) 1

6 Ch

f Rn

Applying the inequality:

3.13. Girding's Inequality one finds that

I / P -P

I

445

5 C h, where the constant C

h,o L2(Rn) +L2(iRn) depends only on n and C 1 in (3.13.23). Thus, it remains to prove (3.13.24) for P defined by (3.12.25). h,O If po did not depend on x, then, obviously, one would have: Re(Ph,Ou,u) 2 0 . Freezing the symbol p (x,n) at a point x = y and writing 0

the corresponding difference operator with the constant symbol p (y,hS) as 0

pO(y,hDx), one still has, of course, that (3.13.26) Re(po(y,hDx)u,u) = Re J n u(x)*po(y,hD )u(x)dx 2 0 , IR b u t , the latter quadratic form has very little to do with the original

one, which, using the notation Ph,o

=

po(x,hDx) can be written as follows:

(3.13.27) Re(po(x,hDx)u,u) = Re J u(x)*po(x,hDx)u(x)dx. iRn As previously, here u(x)

*

is the complex conjugate of u(x).

However, formally multiplying (3.13.26) by 6(x-y) and integrating it with respect to y E IRn, one gets (3.13.27). The idea of Friedrich's mollifier is to replace 6(x-y) by a specially chosen family of test functions $6 such that @

-t

6fx-y), as 6

-+

0.

Hence, the quadratic form Re

1

J

46(x-y)u(x)*po(y,hDx)u(x)dx dy

Rn IRn could be eventually a suitable candidate for approximating (3.13.27). Unfortunately, it is not defined by a symmetric kernel, so that trying to achieve the symmetry as one of the goals inapproximating (3.13.27), onecomes 2 with a, a family of non-negative test functions 6 and with the following quadratic form Qh as an approximation of (3.13.27). to the choice of 46 as

a6,

whose symmetry and non-negativity are obvious. Indeed, using the Parceval's identity and the notation v(c,y) = Fx+c($6 u), one can rewrite (3.13.28) in the following equivalent form:

6 = (21r)-" 1 (3.13.29) Qh

I v(5,y)*PO(y,h5)v(5,y)dSdy 2

nniRn

0.

3. Singular Perturbations on Smooth Manifolds without Boundary

446

6

Quadratic form.(3.13.28) can be rewritten in the form (Phu,u), where

6

(3.13.30) (Phu)(XI

Ji (x-Y)pO(Y,hDx)(Ji6(x-Y)u(x))dy.

=

Rn -n/2 a n We choose Ji (x) = 6 $(x/6), where $ E C ( R ) , Ji 2 0 , 6 0 2 supp $ C B 1 = {x E Rn 1x1 < 1 1 , J $ (x)dx = 1, V 6 > 0 .

1

Further requirements on Ji and 6 will later naturally come out. With this choice of Ji6 and using the substitution x-y = 62, one can rewrite (3.13.30) in the following fashion: 6 (3.13.31) (Phu)(X)

=

-1

/nJi(Z)Po(~-6~,hDx+h6Dz) (Ji(z)u(x))dz. R

Indeed, the Leibnitz' formula yields: pO(y,Dx)(Ji6(x-y)u(x))=

C

hI4 -(pAa)

(y,hDx)u(x))(D:Ji6(x-y))

=

a

=

6-n/2po(y,hDx+h6-1Dz) ($(z)u(x)).

6 Naturally, P can be viewed as a family of difference operators defined by h the following mollified symbol

6

(3.13.32)po (x,rl) = J

Ji ( z ) p o(x-6z,n+h6-1DZ)Ji(z)dz.

R

Representing p (x,n) as a Fourier series 0

Po(X.T1) =

POk(x) exp(i), kEZn

one can rewrite (3.13.32) as follows: 1 J Ji(y)Pok(x-h6-'y)Ji(y+h6-lk)dy exp(i), kEZn Rn 6 but we shall not use the latter representation of p (x,~).

P;(X,n)

=

0

Using (3.13.32) and Taylor's expansion up to the order two with remainder in integral form, one finds:

3.13. Girding’s Inequality

447

where

Here, as usual,

(

)x,

(

) 5 are gradients with respect to x and 5 ,

( )xx’ stand for the corresponding matrix of second derivatives and

( ) 5 5 ’ ( )xs < , > denotes the inner product in Rn.

Obviously, one has:

Further, choosing J, such that

/

2

$ ( z ) z dz = 0,

k

R

1 5 k 5 n

(for instance, choosing $ ( z ) to be even), one finds, that

6

R (u) and, as a consequence of (3.13.34), the optimal h choice of 6 is: 6 = h f . f 2 : L2(lRn) -t L ( R n ) in the same Now, estimating the norm of ((F’:-Ph,o)u)

(x)

=

<

) : L2(Rn) L2(Rn fashion as it has been done for the norm of (P -P h h,O one gets the conclusion, that there exists a constant C such that -f

I

hf 2 (Rh u,u) I 5 Chi IuI l o ,

V u E C:(Rn

),

V h

)

,

E (O,hol.

h’ Therefore, given that (Ph u,u) b 0, one finds: hf hf (P u,u) +Re((Ph,O-Ph ) u , u ) +Re((Ph-Ph,O)u,u) 2 h 2 I t -Chi lu/1 0’

Re(P u,u) h

=

Remark 3.13.9. As

a consequence of the assumption p

0

E C2 ( R n X TY

) r n

and (3.13.13) , the

difference operators in Theorem 3.13.8 are, in fact, finite difference 0 0

approximations of the multiplication operator by the matrix p (x) = po(x,O),

3. Singular Perturbations on Smooth Manifolds without Boundary

448

which is the reduced operator for the family Ph. That explains the presence of a small parameter h on the right hand side of (3.13.241, the latter leading for h

0

-f

0 to the obvious inequality: (p u , u ) 2 0, V u E CO(Rn) 0

.

This kind of operators appears in stable discretizations of problems of 0 being in the latter case the identity. 0

evolutionary type, p Remark 3.13.10.

P hrP belonging to some closed parameter set E (i.e.

Obviously, (3.13.24) holds for any family of difference operators p uniformly with respect to

p

-f

with C 2 which does not depend on p either) if the conditions of Theorem 3.13.9 are satisfied uniformly with respect to p E E. Corollary 3.13.11. 0-

op q, q E F 2 qo(x,q) E c ( x n x T~

L e t Q~

=

(fl) . Assume that i t s principaZ syrnbo2 ),

s a t i s f i e s (3.13.13) and has i t s norm i n

It11

Hom(CP;CP) 5 1: (3.13.35) lqo(x,rl)I

cp+cp

5 1,

v

(x,rl) E lRn

X

Tn

1,o-

Then t h e r e e x i s t s a c o n s t a n t c such t h a t

Indeed, (3.13.36) is equivalent with the inequality: 2 m n (Phu , u ) 2 -ChjlujjO, V u E C O ( R ) , where Ph = Id-Q;Qh

(with

Qi the adjoint in L

2

(W") of Qh) has as its

principal symbol p0(x,n) the following hermitian non-negative matrix: po(x,rl) = l-qo(x,q)*q (x,rl), the latter being a consequence of (3.11.471, 0

and the non-negativity of po following immediately from (3.13.35).

1

Remark 3.13.12. The conclusion of Theorem 3.13.8 is still true for the difference operators

(g)

is a family of meshf f ( o ) ,,(W:) where ff ( 0 ) ,h Ph : f f ( o ) ,h(lRi) function spaces with the inner products and norms defined by (2.7.1), -+

(2.7.2),

respectively. In other words, under the assumptions of Theorem

3.13.8 the following inequality holds with some constant C which does not depend on h and u:

3.13. Ga"rding"s Inequality

449

Indeed, one proves (3.13.37) using the same argument as in the proof of Theorem 3.13.8 and combining it with the one in the proof of Theorem 3.11.15 (with s = v = 0). Example 3.13.13. Let a(x)

:

IR

+

R be such that (a(x)-a(m)) E S ( R

Consider the difference operator Q h (3.13.38) qr(x,n)

=

Op q

)

, and let 0

5 a(x) 2 a

0'

with the symbol

l-ra(x)+ra(x)exp(irl),

=

where r E (0,r 1 is a given parameter. 0 The difference operator Qh with symbol (3.13.38) solves the following Cauchy problem

where, as previously introduced, D and D are forward difference t,T x.h approximations of Dt and Dx, respectively and r = T/h. Indeed, for each t = N? with integer N > 0 the solution of (3.13.39) is given by the formula

If ro satisfies the stability condition: rOao S 1, for each rl r = T/h E (0,r ] onehas Iqr(x,rl) I S 1, t/ (x,n) E R x T1. 0 Hence, as a consequence of Corollary 3.13.11, inequality (3.13.36)

holds for Qh, so that one has:

-_

and the L -norm of the solutions u of (3.13.39) are uniformly bounded with 2

respect to h E (0,h 1 on each finite time interval 0 4 t 4 T < 0

Further, consider the semi-discretized Cauchy problem -1

atu(x,t)+h (3.13.40)

(P u ) (x,t) = 0 , h

U(X,O) = uo(x) where Ph

=

Op p has as its symbol p(x,hE) = a(x) (l-exp(-ihC)), so that (Phv)(x)

=

a(x) (v(x)-v(x-h)).

3. Singular Perturbations on Smooth Manifolds without Boundary

450

One finds easily in this case

so that

v,v)+(~,v,v), Re(P v,v) = Re(P h h,O where we have denoted:

Obviously, one has:

where C

=

sup

axa(x) 1 .

xElR

a

E Cm(Wx T

Furthermore, p,,

) , po(x,n) t 0, V ( x , n ) E R x T l so 1 .rl ,n' one gets:

that applying Theorem 3.13.8, (3.13.42)

Re(P V,V) 2 -Chi h

I v / lo,2

with some constant C, which does not depend on h and v. On the other hand, an easy straightforward computation yields:

with

5, is

defined by the second line in ( 3 . 1 3 . 4 1 ) .

Using ( 3 . 1 3 . 4 3 )

and the fact that ( 1 - 0 ) *

2Re(P v,v) h

=

=

(1-0

-1

((P +P*)v,v) = (a(x)(1-0 -1 )v, h h

),

one finds:

(l-O-l)v)+(\O-lv,v).

Hence, yiven that a(x) 10, the last identity yields: (3.13.44)

(+)

(mah21ID:,h~I

2 < ( f ) (Mah /D:,hVI

I

where a'(x)

=

a

) ;1

Io+ma,h/ 2 /vI

l o2

6 Re(Phv,v) 4 2

+

M a t h /IVI l o )

a(x) and the notation mf and Mf stands for infinimum and

supremum, respectively, of a given function f(x) over W . Thus, since m

L

0, one finds that the best possible choice of the

constant C on the right hand side of ( 3 . 1 3 . 4 2 )

is:

3.13. Girding's Inequality C =

45 1

inf a a(x). XEIR

-(+)

2

Furthermore, taking the L (R)-inner product in (3.13.40) and using the first of inequalities (3.13.44), one finds:

so that Gronwall's lemma yields:

Example 3.13.14. Let A E Horn(CP;CP) have real eigenvalues A . 1 5 j 5 p , and be similar to 7' the diagonal matrix diag(A l,...,A ) .

P

Consider the following strictly hyperbolic Cauchy problem: u

(3.13.45)

{u(x,O) -AU

(x,t) E

=

0,

=

uo(x)

nXB+

.

We are qoint to exhibit a class of high order two-layer stable difference schemes suitable for the numerical treatment of (3.13.45). These schemes can be written as follows rh

T =

V(t+T,X) = (Qh,"V)(ttx),

where r > 0 is a given constant and where the difference operator Qh,v - O P v' has the symbol: (3.13.46) qv

=

C

Ikl
a

k

exp(ikhS)

with matrix-coefficients to be chosen in order to provide a high order approximation of (3.13.45). Since the solution of (3.13.45) has the form:

one has to approximate the exponent exp(irh5A) by the symbols sf fcrm (3.13.46) with given integer v > 0, i.e. the coefficients ak on the right han? side of (3.13.46) are to be chosen to satisfy the condition: (3.13.47) exp(irqA)-

1 ak exp(ikrl) (k(5v

=

o[n

2 -v

)

for

rl

+

0.

3 . Singular Perturbations on Smooth Manifolds without Boundary

452

Taylor‘s expansion for exponents in (3.13.47) yields the following system of equations for the matrix coefficients ak:

It is readily checked that ak are given by the formula: (-l)v-k (3.13.49) a = k (V-k) ! (V+k)!

n (rA-pI), IPlZV PZk where I is the identity matrix in Hom(CP;CP) Indeed, since ak are polynomials in A, it suffices to show the identities

It is sufficient to show (3.13.40) for any 2V+l distinct values of h ,

since the left hand side of (3.13.50) is a polynomial of degree at most 2v. Taking h

=

m/r, m = @,fl, ...,+v

(3.13.51) qv = q (rA;rl) =

‘

Ik(
,

one finds:

(-l)V-k (v-k)! (v+k)!

Il

(rA-pI)exp(ikq).

IPl=
Pfk We are going to show that for each r such that (3.13.52) IrX.1 5 1, 3

1 5 j Z p,

the difference approximation of (3.13.45) given by (3.13.51) is stable in 2

L (R) in von Neumann’s sense, i.e. for t = NT E [-T,T] where N E Z and

0 < T < -,one has:

where the constant C depends only on T,r and the matrix A. Since q

is a polynomial in A, it suffices to show that

(3.13.53) lqv(C;?) 1 S 1, V 5 E [-l,lI, V r~ E Denote

1

3.13. Girding's Inequality (3.13.54)

II

( 5 ) = 2115 II

453

(1-52/p 2) .

lSp5w One has:

As a consequence of (3.13.55), q v ( 5 ; r i ) is the solution of the following

initial value problem:

(3.13.56)

{dq /dn

=

igq -i(21~)-'f~(g;q)

qv(5;0) = 1 where

Hence,

where we have denoted:

In the last formula one can take 5 to be complex, as well. Taking Im 5 < 0 and letting lim

il

+

+-,

one has, as a consequence of (3.13.55):

exp(-ign)qw(g;il) = 0,

q++m

so that (3.13.58) and the last formula yields: m

icw(g)/(sin(8/2))2wexp(-igB)dB = 1.

0 Therefore, one finds for Im 5 < 0: m

q v ( 5 ; n ) = icv(g)/(sin(8/2))2vexp(iC(a-8))dO = a

= Cv(5)/(2

m

sin IIg)/(eing-e-iI15)e-i5e(sin -)e+a 2vdo = n

2

3 . Singular Perturbations on Smootn Manifolds without Boundary

454

The uniqueness of the analytic extension leads to the conclusion:

Given the representation of sin 1 ~ gas an infinite product

z

sin 1 ~ g= 711

" (l-<'/pL),

pL 1 one finds that n v ( c ) / ( 2 sin ~ g 5) I ,

t/

5 E [-1,11.

2v 2 Furthermore, one has 2 ( v ! ) / ( 2 v ) !

< 1 , so that the last two

inequalities and (3.13.59) yield (3.13.53).

If some of the eigenvalues of A are complex numbers, then the Cauchy problem is ill-posed. Nevertheless, the difference schemes with symbols qv(rA;hC) are stable in the sense, introduced in [Fr.-Ch,i], [Fr., 1 1 (see also [Fr., 2

1

where their stability is proved).

With v = 1, one gets the following difference scheme for solving numerically (3.13.45) : (3.13.60) v(x,t+T) = [ (+)rA(rA-I)@-'+(I-r

2 2

A )+(t)rA(rA+I)Olv(x,t)

2

which is stable in L (1R) for IrX.I 5 1, 1 5 j 5 p with A . E R the eigen3

1

values of A and has the viscosity term of the form:

T Dx,hD*x,hv(x,t),

i.e.

(3.13.50) can be rewritten as follows:

-(L)

[iD (D +D* )+?Dx,hD:,h]~(~,t) = 0 . t , ~2 x,h x,h Now, let A

=

A(x) E Hom(Cp;CP) be symmetric and assume that there

exists Am E Hom(CP;CP) such that (A(x)-A_) E S ( R )

. Consider

again

(3.13.60) where A depends on x and let Qh be the corresponding difference operator on the right hand side of (3.13.60), whose symbol q is given by the formula: +(I-rA(x)) (I+rA(x))+

3.13. Girding's Inequality

455

Since q(x,hS) = q(x,hS)*, the estimate [ql 5 1 holds, if for each c+c eigenvalue h(x,n) of q(x,n) one has: Ih(x,n) S 1; the latter is valid.

I

provided that there exists a constant r > 0 such that /rh, (x)1 5 1, 3 1 5 j 5 p, V x E R , where A . (x) E R , 1 5 j 5 p are the eigenvalues of 3 A(x). Since q(x,hE) coincides with its principal symbol and satisfies all the conditions of Corollary 3.13.11, one gets the conclusion that (3 13.36) holds for Qh defined by (3.13.60). Assume now that in A in (3.13.45) is hermitian: A = A* and let P+ and P

be projection operators in Cp on the subspaces corresponding to non-

negative and strictly negative eigenvalues of A respectively. Hence A has the following block-structure: A = A 8 (-Am) with A+ =P+A, A-=-P-A in a suitable basis in Cp. Denote by B+ the Cayley transformation of rA+, V r > 0, i.e. B+_= (rA+-I)(rA++I)=l, the latter being well-defined, since -

-

rA2 are non-negative hermitian operators in Cp. Consider the following finite difference approximation of the hyperbolic problem (3.13.45) with Hermitian A: [P+-B+Oh)O(O P -B )8 -(8 P -B ) @ ( P - - O B )]V(x,t) h - - T h + + h -

=

0,

where, as previously, 0 and 0 are the shift operators in x and t, h respectively, (8 v) (x,t) = v(x+h,t), ( 8 v) (x,t) = v(x,t+T), and r = T/h. h It is readily seen that the solution of this finite difference problem is given by the formula:

where Qh is the difference operator with the symbol q, qo (r;0 )

=

(exp(in)P+-B+) (P+-exp(in)B+)-'o (P--exp(iq)B-)(exp(in)P- -B- ) - ' ,

1 q0 (r;ri) being a unitary operator in Cp, V (r,n) E n + x T n (see also

Example 3.8.83) . Hence,

3. Singular Perturbations on Smooth Manifolds without Bounday

456 If A

=

A(x) depends on x

E

W

and i s s t i l l h e r m i t i a n , t h e n f o r t h e S a m e

a p p r o x i m a t i o n above w i t h A = A ( x ) , t h e p r i n c i p a l symbol q 0 ( r , x , n ) o f t h e c o r r e s p o n d i n g o p e r a t o r Q i s s t i l l a u n i t a r y matrix i n C p , s o t h a t , u s i n g h a g a i n C o r o l l a r y 3.13.11, one comes t o t h e c o n c l u s i o n t h a t (3.13.36) h o l d s

fo; t h i s Qh, \J r > 0 . Example 3.13.15. L e t a ( x , ~ D) = Op a ( x , E S ) , a

E L(o’ofu)

Consider t h e e q u a t i o n a(x,ED)u assumed t h a t f

E C;(Rn

)

= f ,

x

( R n ) b e e l l i p t i c of o r d e r (O,O,u).

E

Wn

, where, f o r s i m p l i c i t y

it i s

.

L e t a ( x , p , h D ) b e an e l l i p t i c f i n i t e d i f f e r e n c e a p p r o x i m a t i o n o f o r d e r (O,O,U)

of a ? x , t D x ) , whose symbol i s a ( x , p , h < ) , R n

p = h/E,

(3.13.61)

x R+ X Tn + Hom(CP;CP), hrS and c o n s i d e r t h e c o r r e s p o n d i n g d i f f e r e n c e e q u a t i o n on Rn x,h .

.

(a(x,p,hD ) v ) ( x ) = f ( x ) ,

x E

$.

One o f t h e problems a r i s i n g i n n u m e r i c a l t r e a t m e n t o f l i n e a r e q u a t i o n s (3.13.61)

i.e. thedependenceofthe relative

i s t h e one o f w e l l - c o n d i t i o n i n g ,

e r r o r i n t h e s o l u t i o n v upon t h e r e l a t i v e e r r o r i n t h e datum f . The l a t t e r c z n bc e x p r e s s e d i n terms o f t h e upper and l o w e r bounds o f t h e s p e c t r u m o f the operator b = a*

0

a . Indeed, i f y ( a ) 2 i s t h e r a t i o

o f t h e upper and

l o w e r bounds f o r t h e s p e c t r u m o f b , t h e n t h e r e l a t i v e e r r o r i n t h e s o l u t i o n 2 measured i n L (an ) i s bounded by y ( b ) t i m e s t h e r e l a t i v e e r r o r i n f a xrh measured i n LL (IRn ) (see f o r i n s t a n c e [ B e r . - Z h i , 1 1 ) . I n o t h e r words, i n X rh o r d e r t o estimate t h e c o n d i t i o n number y(a) one h a s t o estimate t h e u p p e r and lower bounds f o r t h e q u a d r a t i c form ( b v , v )h , where ( , ) h i s t h e i n n e r 2 product i n L (Rn x,h) Applying Theorem 3.13.8 and u s i n g Remark 3.13.12, one f i n d s t h e

.

f o l l o w i n g estimates f o r t h e u p p e r and lower bounds r e s p e c t i v e l y , o f t h e q u a d r a t i c form ( b v , v )

h

A

max

( p , h ) and Xmin(p,h),

:

where c and C a r e some c o n s t a n t s a n d , f o r a b b r e v i a t i o n , t h e n o t a t i o n

i s u s e d f o r t h e norm i n H o m ( C P ; C P ) . T h e r e f o r e , t h e r e e x i s t s a c o n s t a n t C1 s u c h t h a t

1 1

3.13. Girding's Inequality (3.13.62)

457

y ( a I 2 5 C h+ 1

[ a(x,P , n) *a (x,p ,n)l /inf I a(x.p .n)*a(x,p,n) 1. (x,n)EBnxTn ,n 1r n As a consequence of the ellipticity condition, for each p o > 0 there

+sup (x,n)ERnX T Y

exists a constant C

such that PO

< c

,

v p t p

PO

Indeed, ( 3 . 1 3 . 6 3 ) take v 2

=

0, v 3

=

0'

follows immediately from ( 3 . 1 2 . 7 2 ) where one should

v.

2 2

2

Consider, for instance, the singular perturbation 1 + q~ (x)Dx, x E B , 2

2

which is elliptic of order ( 0 , 0 , 2 ) if q (x) L qo > 0, V x E B . Consider 2 2

2

first the following finite difference approximation a of 1 + ~q (x)D

:

2 2

a(x,p,~~,~,~:, = ~I+€ ) q (X)Dx,hD* x,h' which is elliptic of the same order. Applying ( 3 . 1 3 . 6 2 ) , one finds the following estimate for the conditioning number for the corresponding equation ( 3 . 1 3 . 6 1 ) y(a) 5 Ch+sup (x,n)EIR x T where

saxXER sup =

(I+% 1r

-2 2

2

q (x)sin ( r l / 2 ) )

=

in this case:

Ch+ ( 4 p

-2

2

sax+' )

n

q(x).

NOW consider the singular perturbation a(x,E,D

)

=

(l+iEDxq(x))~(l-Eq(x)Dx), which is formally self-adjoint and

, > 0, elliptic of order ( 0 , 0 , 2 ) if q(x) 2 qmln

difference singular perturbation M (3.13.64)

Mq(x)

=

v

x E R . Introduce the

q(x) '

-1 (l-exp(-p/q(x)) (l-exp(-p/q(x))8), P = h/E,

whith 0 the shift operator (Ov)(x) = v(x+h). Difference operator ( 3 . 1 3 . 6 4 ) being a finite difference approximation of the singular perturbation l-iEq(x)D , its formal adjoint M* q'

M* 9

=

(1-0

-1

exp(-p/q(x)) (l-exp(-pq(x))) - '

is a finite difference approximation of the formal adjoint of (l-iEq(x)D which is (l+iED q(x)).

)

3 . Singular Perturbations on Smooth Manifolds without Boundary

458

Hence, the operator a(x,E,h,o), (3.13.65) a(x,s,h,O) = M*

4

M

4

is a finite difference approximation of a(x,c,D

)

(l+iED q(x))o(l-isq(x)Dx).

=

The principal symbol of a(x,s,h,0) being -2 2 a0(x,p,n) = (l-exp(-p/q(x))) Il-exp(in-p/q(x)) I , it is elliptic of order (0,0,2). Furthermore, (3.13.62) yields the following estimate for the conditioning number y(a) in this case:

Hence, for each p 0 > 0 the corresponding problem (3.13.61) is wellconditioned for the difference operator (3.13.65) uniformly with respect toptp

0'

Example 3.13.16. Let a(x,sD

)

=

Op a, a E !-(ororv)(Rn) be again as in Example 3.13.15 and

let again a(x,p,hD a(x,eD

)

be an elliptic finite difference approximation of

)

on the family of grids R:,h.

We consider again the equation

(3.13.61). which can be solved numerically by using the following iterative procedure: (3.13.66) uv+l = u

-T<ED

-2v > a(x,p,hDx)*(a(x,p,hDx)uv-f), x,h

>-2v is the difference singular perturbation with the symbol where <ED -2" xrh = ( 1 + ~ ~ ] < 1 ~and ) - ~T is a parameter to be chosen to guarantee the <E<> convergence of the iterations (3.13.66). As a consequence of ellipticity condition (3.12.721, one has:

def (3.13.67) 0 < m

=

inf (x,p,rl)xRn X

s sup

-2v a(x,p , n ) *a(x,p,n ) I 5 xTn + 1,rl def -2"la(x,p,n)*a(x,p,n) 1 = M < m. ~

R

(x,p,n)ERnx R + XTn 1,n Here, as previously, w ( n ) = C(h,E) with n = ht, h

= 1.

-1 Hence choosing a positive number r E ((M-m)(M+m) ,I) and T

E [(l-r)M,(l+r)M], one gets the conclusion that the principal symbol

q0(x,n) I

3.13. Girding's Inequality

459

of the difference operator Q h,E' Qh,€

=

Id-7<ED > x,h

-2v

a(x,p,hDx)* o a(x,p,hDx)

satisfies the inequality: lqo(x,q) I 5 r,

v

(x,q) E

xn

x T:,,.

Therefore, as a consequence of the corresponding version of Corollary

,

3.13.11 in L2(lRE)

one gets the conclusion that

where C is some constant, which may depend on p . It can be shown, but will not be done here, that (3-13-68)I IQh,Eil

5 r+c(h+E), V (h,E) E (O,ho] X (O,Eol,

L ( W h )L'

(Bh)

where the constant c does not depend on h and E. In fact, one establishes (3.13.68) by using the argument in the proof of Theorem 3.13.8 and estimating carefully the second derivatives of the symbol q of Q

by constants, which in the case considered can be chosen h,E independent on p. This analysis will be done later in connection with the

corresponding version of Theorem 3.13.8 for singular perturbations of order

(O,O,V)

-

For h and

sufficiently small and with the choice of y and

E

T

indicated above, (3.13.68) yields the converqence of the iterative procedure (3.13.66), the asymptotically (for E

-f

0, h

-+

0 ) best choice of

r being r

=

(M-m)(M+m)

-1

with m and M defined by (3.13.67) and the rate of convergence being rv as

v

+

w.

For a(x,ED

)

=

(l+iED q(x))

o

(l-iEq(x)D

)

from Example 3.13.15, one

can use the following iterative procedure: (1+E2Dx,hD:,h)UV+l with the choice of

T

=

[(I+€ 2Dx,hD~,h)-Ta(x,E,h,O)lu+Tf V

as indicated above.

3. Singular Perturbations on Smooth Manifolds without Bounda y

460

Remark 3.13.17. O n L e t p E S (37 )

, p b e i n g v a l u e d i n Hom(CP,CP), i . e .

P(X,E,~)

i s a pxp

m a t r i x . L e t t h e p r i n c i p a l symbol p ( x , & < ) o f p be a n o n - n e g a t i v e h e r m i t i a n 0

m a t r i x : po(x,I1) 2

po E C ( B n x R n )

=

p o ( x , n ) * 2 0, V (x,Tl) E R n x R n and assume t h a t

. Furthermore,

assume t h a t w i t h s o m e c o n s t a n t C > 0 t h e

following inequality holds: ( 3 . 1 3 . 6 9 ) <x>n + l < E ~ > ] D B ( p ( x , E , S ) - p O ( x r ~ , S ) )SI C E ,

v

(X,E,E)

E

mn

5 n+l,

x (O,Eol

x IRn

.

Then e x a c t l y t h e s a m e argument a s i n t h e p r o o f o f Theorem 3 . 1 3 . 8 y i e l d s t h e following inequality ( 3 . 1 3 . 7 0 ) R e ( P u , u ) 2 -cE/ ( u I

1 2(o,o,o)

V

E

E

(O,Eo],

V u E H

(O,O,O) , E

(R?,

= O p ( p ) a n d c i s s o m e c o n s t a n t , which d o e s n o t depend on u and E .

where P

I n e q u a l i t y (3.13.70)

c a n b e s h a r p e n e d : namely t h e norm

o n i t s r i g h t hand side c a n b e r e p l a c e d by a weaker norm

I I . 1 1 (o,o,o)

,€

11. I

(O.O,-f) , E ' However t o show t h i s r e q u i r e s a d i f f e r e n t k i n d of m o l l i f y i n g t h a n t h e one u s e d i n t h e p r o o f o f Theorem 3.13.8. C o r o l l a r y 3.13.18.

Let a E

S ( o t o r v )( R n )

constant, i . e .

be s t r o n g l y e l l i p t i c w i t h y a s t h e strong e l l i p t i c i t y

R e a (x,n)

0

2 y , V ( x , q ) E I R ~ ~ I where R ~ , a O ( x , E < ) is t h e

p r i n c i p a l symb02 of a. Furthermore, assume t h a t a-a

0

s a t i s f i e s (3.13.69).

Consider t h e i n i t i a l value problem

'

u +A u = f ( x , t ) , (3.13.71)

where ut

= au/at

and

(x,t)

E

IRnxn+,

A~ = o p a .

Then t h e r e e x i s t s a c o n s t a n t c , which does n e t depend on u and such t h a t for t h e s o l u t i o n u o f ( 3 . 1 3 . 7 1 ) holds

E,

:

I n d e e d , f i r s t , c o n s i d e r ( 3 . 1 3 . 7 1 ) w i t h f Z 0 . Then a f t e r t a k i n g t h e i n n e r p r o d u c t w i t h u i n t h e d i f f e r e n t i a l e q u a t i o n ( 3 . 1 3 . 7 1 ) , one f i n d s , u s i n g (3.13.70) :

3.13. Ghding's Inequality

46 1

Hence, G r o n w a l l ' s lemma y i e l d s :

I lu(.tt) I I(0)

2

I l U ( . J O ) I I ( 0 ) ,E

2

,€

e x p ( (-y+cE) t )

.

I n o t h e r words, t h e semigroup o f o p e r a t o r s :

*

IR+ 3 t

e x p ( - t A E ) E Hom(H ( 0 ) , E ( R n ) i H

i s a contructionuniformly with respect t o

E

E

(0,E

0

1,

provided t h a t

E~

is

s u f f i c i e n t l y s m a l l , a n d , moreover, one h a s : (3.13.73)

1 lexp(-tAE) I I H ( 0 ), E

S i n c e t h e s o l u t i o n of

(IRn)

6 e x p ( (-Y+cE) t ) , V t E R + ,

v

&

E (O,Eol.

( 3 . 1 3 . 7 1 ) i s g i v e n by t h e f o r m u l a :

t U(X,t)

=

( e x p ( - t A )'$)

(X)

eXp(-(t-T)AE)f(X,T)dT,

+

0 one g e t s i m m e d i a t e l y ( 3 . 1 3 . 7 2 ) , u s i n g ( 3 . 1 3 . 7 3 ) . I n p a r t i c u l a r , c o n s i d e r (3.13.71) f o r A_ = Op(a) w i t h f ( x , t )

Z

f(x)

and a ( x , E , c ) = <E
1

V x E Wn

I

w i t h some c o n s t a n t

support i n Rn.

y > 0 , b ( m ) e x i s t s and b ( x ) - b ( m ) h a s compact

Of c o u r s e , w i t h t h i s s p e c i f i c c h o i c e o f A E, one can

r e f o r m u l a t e (3.13.71)

i n t h i s case a s a d i f f e r e n t i a l i n i t i a l v a l u e problem

of Sobolev t y p e i n t h e f o l l o w i n g f a s h i o n : (1-E

2

Ax) ( u t - f ) + ( l + E 2

C b k j ( x ) Dk D j ) u = 0 , 1 6 j ,k5n

(3.13.74)

A s a consequence of

(3.13.721,

( x , t ) E IRn

X

W,

t h e s o l u t i o n u m ( x ) of t h e c o r r e s p o n d i n g

s t a t i o n a r y problem (3.13.75) ( l + E with g

=

2

c bkj(x)D D )um(x) = g(x) k j lsk, j
( l - ~ ~ A ) if s, t h e o n l y a t t r a c t o r f o r ( 3 . 1 3 . 7 4 ) . which i s , moreover,

asymptotically s t a b l e f o r (3.13.74), a s t

+

9.

3 . Singular Perturbations on Smooth Manifolds without Boundary

462

Furthermore, (3.13.72) yields the following estimate for the difference v = u-u,:

min pk(x) > 0, pk(x) > 0 being the min{l,p} with p = inf xEIRn l
=

One can use (3.13.74) for solving numerically (by using the iterations) the stationary problem (3.13.75), as it is indicated in Examples 3.13.15, 3.13.16 in the case n = 1. Now a corresponding version of the sharp Ggrding's inequality,the LaxNirenberg theorem, will be proved for classical pseudodifferential operators in Op S'(pin)

with

V

E

IR

. Again

the sLymbols considered are valued in

Horn(CP;CP), i.e. they are p x p matrices. We follow here essentially the mollif ier method introduced in [ Friedr., 2 over in [Vai

r

1

1 with the simplifications brought

I.

We start with the case of pseudodifferential operators of order zero. Theorem 3.13.19.

. Then

L e t p(x,E;) be hermitian non-negative d e f i n i t e m a t r i x , p E

f o r t h e corresponding p s e u d o d i f f e r e n t i a l operator P (3.13.76) Re(Pu,u) 2

where

I / . I l -t

-C\ luj

I",

V u

E C:(IRn)

= op(p)

one has:

,

is t h e cZassicaZ Soboieu-Slobodetski

norm of order - t and

where c is a c o n s t a n t , which does n o t depend on u. Proof. ___ The idea is to produce a symmetric non-negative quadratic form Q ( u , u ) such that with some constant C > 0 holds:

First, we notice that

2 (3.13.78) I/dyd< 2 0, V w E L (BnxIRn), where, as usual, < , > stands for the inner product in Cp. We relate w(y,<) and u by setting

3.13. Girding's Inequality where t h e f a m i l y of symbols: lRn 3 5

-f

463

@ ( 5 , < ) E lR w i l l be chosen l a t e r on

a s a consequence o f n a t u r a l r e q u i r e m e n t s l e a d i n g t o ( 3 . 1 3 . 7 7 ) . One h a s :

p*u

=

(2n)-nj/ei<x-Y1S' P ( Y t 5 )

( Y ) dYd5 r

so t h a t an e a s y c o m p u t a t i o n y i e l d s :

F i r s t , we e s t i m a t e t h e d i f f e r e n c e Q ( u , u ) - ( P u , u ) 0

Rewrite Q ( u , u ) as follows: 0

w e c o m e t o t h e c o n c l u s i o n t h a t 0 ( 5 , < ) s h o u l d b e chosen t o g u a r a n t e e a good a s y m p t o t i c a p p r o x i m a t i o n o f p ( y , < ) by t h e i n t e g r a l

1

de f (3.13.81)

PQ( Y

J0 ( 5 , , < )

=

2

p (y,5 )d < .

For d o i n g t h a t w e i n t r o d u c e 0 ( 5 , 5 ) a s f o l l o w s : (3.13.82) 0 ( E , < ) = 6 n ' 2 $ ( 6 ( < - 5 ) ) , where 6 = 6 ( 5 ) + 0 , a s 5 m n c B~ @ E c o ( ) ~, supp (3.13.83)

/ @ ( e 2) d8

=

-f

a,

i s t o b e d e f i n e d l a t e r on and where

=

~e

E nin

I lei

< 11,

0 2 o and

1,

t h e l a t t e r c o n d i t i o n b e i n g n e c e s s a r y i n o r d e r t o have an a s y m p t o t i c a p p r o x i m a t i o n of p ( y , S ) by p , ( y , < )

as

5

-f

m.

3. Singular Perturbations on Smooth Manifolds without Boundary

464

I n d e e d , w i t h t h e c h o i c e o f O(E,C,) t h e change of v a r i a b l e s 6 ( < - 5 ) (3.13.84) p,(y,S)

=

/$(el

2

a n d , as a consequence o f

a s i n ( 3 . 1 3 . 8 2 ) . one f i n d s u s i n g

= 8:

p(y,5+6-'a)dB

(3.13.83),

( 3 . 1 3 . 8 4 ) and h a v i n g i n mind a n

a s y m p t o t i c a p p r o x i m a t i o n o f p ( y , s ) by p ( y , S ) a s €2 O

m,

-f

one r e a c h e s t h e

c o n c l u s i o n , t h a t 6(5) s h o u l d b e c h o s e n t o s a t i s f y t h e c o n d i t i o n : ( 3 . 1 3 . 8 5 )
+ m,

as

<

-f

m.

F u r t h e r m o r e , u s i n g T a y l o r ' s f o r m u l a f o r p(y,c+6-1'8) w i t h t h e r e m a i n d e r of s e c o n d o r d e r i n t h e i n t e g r a l form, one g e t s f o r p ( y , < ) t h e f o l l o w i n g O

r e p r e s e n t a ti o n :

2 ( 3 . 1 3 . 8 6 ) p O ( y , s ) = P ( Y , S ) + ~ ( C ) - ~$/( e ) < e , v 5 p ( y , t ) > d e +

nn

l

2

+ 6 ( ~ ) - ~ . r ~4 (8) ( i - t ) < p E 5 ( y , ~ + t ~ - l ae ,)e > d t d e . Rn

0

Hence, one g e t s a b e t t e r a s y m p t o t i c a p p r o x i m a t i o n o f p ( y , 5 ) by r e q u i r i n g that

2

( 3 . 1 3 . 8 7 ) / n 8 $ ( 8 ) dB

=

0.

Q7

F o r i n s t a n c e , c h o o s i n g $ ( a ) t o be e v e n : $ ( - € I )

=

( 3 . 1 3 . 8 7 ) i s a u t o m a t i c a l l y s a t i s f i e d , so t h a t i f 6 ( 5 ) w i t h some

CY

$

('a), V 8 E

-

IRn

-a <<> as 5

-f

, a,

E ( 0 , 1 ) , then

( 3 . 1 3 . 8 8 ) ( p -p) E s 2 ( a - 1 ) (IRn)

,

a s a n immediate consequence o f

(3.13.86),

(3.13.87).

N o w w e p r o c e e d t o e s t i m a t e t h e q u a d r a t i c form

(3.13.89) R ( u , u ) =

( 2711 -nJ/J/O ( 5

d ef =

, 5 ) ( Q ( 5 2, C, ) -0 ( 5

,c ) )

Again, w e u s e T a y l o r ' s f o r m u l a :

Furthermore,

(3.13.82) y i e l d s :

ei<'

1-52>

(5 )

, (5

)> d5 d5 2dydL

3.13. Gdrding's Inequality

where we have denoted by

R 1 (u,u) the

465

quadratic form, which corresponds to

the remainder in the integral form on the right-hand side of (3.13.90) Integrating by parts in the first integral on the right hand side of (3.13.93), after having rewritten $$

5

as ( f )

2

($ )<,

one gets:

(3.13.94) R(u,u) =

+ R(u,u) + R1(u,u),

where we have denoted by R(u,u) the quadratic form defined by the second integral on the right-hand side of (3.13.93). As

previously, one gets the conclusion that

(3.13.95) 1 0 (5,5)2 1 P ~ , ~ , ( Y , < E) ~S-l(Rn) < l<jSn J

.

Furthermore, one finds (after the change of variables 6(5-E,)

=

8):

so that (3.13.88), (3.13.97) yield the optimal choice of a by equating 2(a-1)

=

-1, i.e. a

=

f.

Notice that for each pseudodifferential operator a

E Op S-'(Rn)

holds:

3. Singular Perturbations on Smooth Manifolds without Bounda y

466

I Indeed

one gets immediately the last inequality by introducing

-1

v = 2 u and applying Theorem 3.4.1 to -'a-+

E Op

S ( O f O ' O ) (mn)

with s = (O,O,O). Thus, the last inequality, (3.13.88), (3.13.97) with a =

+

yield:

where we have denoted by Po the pseudodifferential operator with symbol pa given by (3.13.84).

Now we proceed to estimate quadratic form R (u,u), 1 (3.13.99) ? ? , ( u , u ) =

3.13. Girding's Inequality

467

one can rewrite (3.13.99) as follows (after the integration by parts with respect to y) :

(3.13.102)

/I

(1-A

Y

)

k r1(y,clrL2) Idy 5 C, V (5,,5,)

E RnxRn.

Hence, (3.13.101), (3.13.102) yield: IR1(utu)I 5

CI

/uI

2 I-+,

so that one finally gets: (3.13.103) IQo(u,u)-P(u,u)1 6

CI

lul I:+,

V u

E C;(Rn).

Now, since (P*u,u) = (Pu,u)* and QQ(u,u) is real-valued, one gets, as a consequence of (3.13.103), the same inequality for the adjoint P*: (3.13.104) lQ (u.U)-(P*U,U) Q

I

5 CI

l2

v

-f'

u E C;(Rn),

so that (3.13.103). (3.13.104) yield: 2

(3.13.105) IQQ(u,u)-Re(Pu,u)I 5 CI

1 -4'

using the non-negativity of 2 ( u , u ) and applying (3.13.105), one Q finds: -Re(Pu,u) 5 Q (u,U)-Re(PU,U) 5 CI 0

so that

Re(Pu,u) t -C(/uI

/-*f '

v

lU1

u E C:(Rn).

2 I-+,

I

Remark 3.13.20. One is tempted to take 6(5)

G

1 (a

= O),

by (3.13.92), vanishes: R ( < , S ) E 0.

since in that case R(S,<), defined

3. Singular Perturbations on Smooth Manifolds without Boundary

468

However, while choosing a = 0, one looses control over the quadratic form -a R (u,u) defined by (3.13.99), which, on the other hand, with 6(5) = <E,> , 1

2

a > 0 , can be estimated by CI /uI1 -

. Thus, actually, the optimal choice of

a is assigned by (3.13.88) and by the estimate of R1(u,u), i.e. by equating 2(a-1) = -2a, which yields a = $ .

#

The following result is a combined version of Theorem 3.13.8 and Theorem 3.13.19 for singular perturbations of order ( O , O , O ) .

It can be

established in the same manner as Theorem 3.13.19 with the choice of

6

= E

f <ES> -l’, so that its proof is left to the reader.

Theorem 3.13.21.

Let P

=

P(x,E,S). p E s ( o ‘ o r o ) ( x n )

. Assume

t h a t t h e p r i n c i p a l symbol

c2(xnxxn)and i s a non-negative hermitian m a t r i x . Furthermore, assume t h a t t h e r e e x i s t s a c o n s t a n t c > 0 such t h a t pO(x,~S)of p(x,c,<) belongs to

(3.13.106) <x> n+l< E S > ~ DB ~ ( P ( X , E , S ) - ~ ~ ( X ,IE S )C)E ,

Then f o r P

=

! B i 5 ncl,

op(p) holds:

where t h e c o n s t a n t k does n o t depend on

E

and u .

The following result is a finite difference version of Theorem 3.13.19 for difference operators of order (O,O), whose full proof for difference operators of any order (with the corresponding necessary modifications in the formulation of Gsrding’s inequality in this case) can be found in [Fr.,9,101. Theorem 3.13.22.

L e t p(x,h,h<) E F ( o ’ o f o ’ 1,o

(an) b e Hermitian non-negative m a t r i x . Then f o r

t h e corresponding d i f f e r e n c e operator Ph = p(x,h,hD) h o l d s :

where t h e c o n s t a n t X does n o t depend on h and u

.

Proof. ~

One has just to repeat the proof of Theorem 3.13.19, a suitable choice of the finite difference substitute for the family of pseudodifferential operators 5 + O(D , 5 ) in (3.13.79) being the family of difference operators

3.13. Girding's Inequality

where

$(e)

i s t h e s a m e f u n c t i o n a s i n t h e p r o o f o f Theorem 3.13.17 and

where, a s p r e v i o u s l y , w ( r l ) Re w(nk )

=

469

(u(rll),.,.,u(n

=

) ) with

u(n

k

)

=

i(l-exp(iqk)),

I

s i n qk.

One s h o u l d stress t h e d i f f e r e n c e between Theorems 3.13.8

and 3.13.22,

0 which c o n s i s t s i n t h e f a c t t h a t t h e c l a s s o f symbols p E F ( I R ~) , s a t i s f y i n g i s more r e s t r i c t i v e t h a n t h e one i n

t h e a s s u m p t i o n s of Theorem 3 . 1 3 . 8 ,

Theorem 3 . 1 3 . 2 2 , t h e l a t t e r g i v i n g r i s e t o a s t r o n g e r r e s u l t ( 3 . 1 3 . 1 4 ) . 2 2 2 ( N o t i c e t h a t one h a s a l w a y s h / O , h 5 (hO+4n) V h E (O,hol . I lul

1111 1

I

See a l s o Remark 3 . 1 3 . 9 . C o r o l l a r y 3.13.23.

Let b ( x , S ) E S 2 m ( l R n ) , m E R , b e such t h a t i t s p r i n c i p a l symbol b (x,S! i s a non-negative hermitian m a t r i x : b o ( x , S )

2 0 , V (x,C)

0 E IRnx(Rn,{O}).

Further, assume t h a t t h e order of b ( x , E ) - b ( x , < ) is 5 2m-1. Then t h e r e 0

e x i s t s a c o n s t a n t k such t h a t f o r t h e p s e u d o d i f f e r e n t i a 2 operator

B = Op(b)

ho Zds (3.13.110) R e ( B u , u )

where

, ) s,

(

(u,vlS

=

s

E R

2 -kl lul

,

2 1 s+m-f ' v

u E C;(Rn

),

stands f o r t h e i n n e r product d e f i n e d by ',

(~U.~V)

with

,

(

)

=

0

.

i.e.

L ~ ( I )R ~

-s-m

s+m

u , P = S-mB and n o t i c i n g t h a t -2m ( 5 1 a (x,C) i s a g a i n a n o n - n e g a t i v e

I n d e e d , i n t r o d u c i n g v = t h e p r i n c i p a l symbol p (x,€,)

t h e i n n e r product in

0

h e r m i t i a n m a t r i x , one g e t s (3.13.110) by a p p l y i n g Theorem 3 . 1 3 . 1 9 t o t h e 0 p s e u d o d i f f e r e n t i a l o p e r a t o r Po w i t h t h e symbol P o ( x , 5 ) x ( 5 ) E S ( R n) , where x E Cm(IRn ) i s a c u t - o f f f u n c t i o n , which v a n i s h e s i d e n t i c a l l y f o r 1 5 ) 5 f , is identically 1 for

(c(2

1 and i s n o n - n e g a t i v e o t h e r w i s e :

The f o l l o w i n g s l i g h t g e n e r a l i z a t i o n o f Theorem 3.13.19

x

2 0, V

5 E Rn

c a n be proved

by t h e s a m e argument.

Let p

=

p ( x , e , ~ ) ,p E

p r i n c i p a l symbol p o ( x , n ) , n

s ( o ~ o( ~ m nu) )w i t h some u E R . Assume t h a t t h e 2 = E E , o ~ P ( x , E , ~ )belongs t o c ( I R ~ X I R ~ ) and is x r l

a non-negative hermitian m a t r i x . Furthemore, assume t h a t

1

~ x > n + l < ~ ~ ~ lB- u ~ ~ X ( P ( X , ~ , ~ ) - p5 Oc E( ,x , 1E8~1 )5) n + l ,

v

(X,E,S)

E ~ n x ( O , E o ]E n .

.

470

3. Singular Perturbations on Smooth Manifolds without Bounda y

Then f o r pE

= op(p)

Re(P u,u) E

and f o r each

(S),E

s

3

E IR holds: 2 (s1,s2,s3+(v-1)/2)

2 -csl/u//

where t h e c o n s t a n t c does n o t depend on

E

.El

v

E

E (O,EO1'

and u.

Corollary 3.13.24. (0.2m) m E R be such t h a t i t s p r i n c i p a l symbol Let b(x,h.W) E F l , o ho(x,n) i s a non-negative h e m i t i a n m a t r i x : h (x,n) 2 0, 0

{O}). Further, assume t h a t t h e order o f b(x,h,hc)i s 5 2m-1. Then t h e r e e x i s t s a c o n s t a n t k such t h a t f o r t h e

V ( x , n ) E IRn x(T:

h-2%o(x,h<)

\

corresponding d i f f e r e n c e operator B~ (3.13.111)

where

(

=

op (h) holds:

Re(B u,u) h s,h 2 -kl IuI 1 2 s+m-! ,h' V h E (O,hol, v u E Hs+m,h(J()

, )s,h d e f i n e d by

(2.7.6)

with

z S ~ , < ~ x , h > S ~ )whi,t h ( = (
s1 = 0 , s2 = s ,

,

,

i.e.

)h t h e i n n e r product (2.7.1)

Indeed, one uses the same argument as in Corollary 3.13.23

and Theorem

to show ( 3 . 1 3 . 1 1 1 ) .

3.13.22,

Corollary 3.13.25.

L e t a(x,<) E sm(Rn), m E i ~ be , e l l i p t i c o f order m , i . e . f o r t h e p r i n c i p a l symboZ a of a h o l d s : lao(x,<)1 0

w i t h some c o n s t c n t y

>

2

m y l < / , V (x,<)EIR~~(R"\IO})

0, t h e b e s t p o s s i b l e value yo o f y being t h e

e l l i p t i c i t y c o n s t a n t . Further, ussume ( h a t t h e order o f a(x,c)-a (x,<) i s 0

5 m-1.

Then t h e r e e x i s t s a c o n s t a n t c such t h a t f o r t h e corresponding

p s e u d o d i f f e r e n t i a l operator A

=

op (a)h o l d s :

Indeed, applying Corollary 3 . 1 3 . 2 1 to B = A*

0

A-y 2m, one gets 0

immediately ( 3 . 1 3 . 1 1 2 ) . With

4,

=

Op (a)an elliptic difference operator of order m (a E

Fm1,o

whose ellipticity constant is y o , the difference analogue of ( 3 . 1 3 . 1 1 2 ) true, too, i.e.

(IRn)), is

3.13. Girding's Inequality

47 1

Example 3-L3-.2. The following matrix differential operator B(D

)

appears in the linear

elasticity theory (see, for instance, [Land.-Lif. , 1 1 ) : (3.13.114) B(D

)

=

2 UD I d + ( V + A )

I IDx .Dx 1 1 lij,kcn' 3

k

'-

C

DX -

l
2 D x ' k

where Id E Hom(Cn;Cn) is the identity and p > 0, X 1 0 are given constants. It is easily szen that the matrix R([) non-negative hermitian. Indeed, for each z

=

1

= =

I ( [ 7.[ k 1 1 l<j,k
5 E I R ~ ,is

E Cn holds:

2

Sjzj( , V 5 E R n ,

Z l'j
where, as usual, <

,>

Hence, B([) -p 15 I 'Id

is the inner product in C". 2 0,

5 E Rn and using the Fourier transform, one

finds easily that ( B ( D x ) ~ , ~ )2s l i / ( 2~ ( l ~v+ s~ E , R , V u E Ci(IRn). If p

=

p(x) 2

u0

>

0, V x E m n , X = X(x) 2 0, V x E m n , X E S ( R n ) and

= u_+u'(x), with p ' E S(Rn), then applying Corollary 3.13.21 to ~(x) 2 B(x,D )-!~(x)It) Id, one finds:

2 2 (3.13.115) R ~ ( B ( X , D ~ ) U , U2 )u~o \ ( u !/s+l-C1/ U (Is++,

'd

*, v

E

S

U

E Ci(Rn),

where the constant C does not depend on u. The following finite difference approximation B(D

x,h

,D:,h)

of B(Dx)

given by (3.13.114) is elliptic of order 2.

u(x)

=

u_+u'(x), p' E S(IRn):

then the difference analogue of (3.13.115)

holds uniformly with respect to h E (0.h 1 with the same uo and a constant 0

C, which does not depend on h E (O,hol and u E Hs+l,h(<

),

b u t may depend

I

on s E 37. Example 3.13. 27. Let a(x,t) a(x,t)

=

=

\lakj(x,t)l/,a(x,t)* = a(x,t) t a0Id, . a

a_+a'(x,t), a(x,t) E C;(mRn

xE+)

> 0, V (x,t)ERnx%+,

and let V(x,t) 2 vo

0,

3 . Singular Perturbations on Smooth Manifolds without Boundary

412

v

(x,t) E lRnxiii+, V(X,t) = V,+V'

(x,t), V' (x,t) E

c p nG+).

Denote At = +V(x,t),

-

IR+3 t

+

At E Op

S

2

n

(IR

being a family of second order elliptic

)

differential operators with the ellipticity constant a

0'

As a consequence of Corollary 3.13.23, for each s E IR there exists a constant C > 0 such that

On the other hand, the integration by parts yields 2

(3.13.117) (Atu,u) 2 yoI lul 11, V u E C;(lRn), where yo = rnin(ao,Vo) > 0. Consider the initial value problem

( at (-A

x

+I)+A )u(x,t) = 0 , t

(x,t) E

mnx=

(3.13.118) U(X,O)

=

u

0

x E IRn

(XI I

a n where uo E C (IR ) and A is the Laplace operator in x E lRn. 0 We are going to show that for each integer m > 0 there exists a constant

C

m

such that

After taking the inner product, (3.13.117), (3.13.118) yield (f)d/dt//u(.,t)([,+y,llu(.,t)ll, 2 2 5 0,

V t 2 0.

so that

(3.13.120)

I lu(.,t) I l 1

5

.-Yoti

luol 11, V t 2 0.

( a t (1-Ax )+At )u = 0 and taking the inner product with one finds using (3.13.116):

Applying ' to CD >'u,

2 0 = (+)d/dt(/u(.,t)[ Is+l+Re(Atu,~)s 2 Z

(?)d/dt/lu(.,t) I

Is+l2

+aoj lu(.,t) I

I s +2l

2 I Is++,

-CI lu(.,t)

3.23. Girding's Inequality

473

so that Gronwall's lemma yields:

Taking s

=

f in (3.13.121) and using (3.13.120) to estimate the

integral on the right hand side of (3.13.121), one finds:

We have used the obvious inequality:

1

e

-2a

t

O( -T)-2yo'dT

v

5te-2yot,

t 2 0, . a

2 yo.

0 Now taking s

=

3/2 in (3.13.121) and using (3.13.121), (3.13.122), one

gets the inequality

By induction argument, one gets for each integer m > 0 the following estimate

+ (ct1~~-~/((2m-2) !)e2Yot//uOl.:1 and this last inequality yields (3.13.119) with C m

max Ck/k! 0< k < 2m- 1 Using the interpolation, one can show(but it will not be done here), =

that, in fact, (3.13.119) holds for V m L f . With x E R , y E R and A = D2+D2 equations similar to (3.13.118) t X Y have been previously considered in [Sob. 3 1. I

The corresponding evolution problems being not of Cauchy-Kowslevski's

3 . Singular Perturbations on Smooth Manifolds without Boundary

414

type, they are known as Sobolev type initial (and initial boundary) value problems (see also [Galp., 1

1,

1).

[Fr., 1

I

Example 3.13.28. The following singular perturbation appears in the theory of gaussian perturbations of dynamical systems (see, for instance, [Freid-Wentz. , 1 (3.13.123) a(x,E,D where b

=

-EAX-,

x E IRn, m

m

: lRn

-f

I):

Rn is a C -vector field, b E C (Rn) , and where, as usual,

Ax is the Laplace operator and V

is the gradient in x E Rn

.

Assume that b(x) admits the representation: (3.13.124) b(x) = -2Vx$(x)+g(x),

= 0, V x E Rn

with $ E Cm(IRn), g € Cm(lRn), 4 : Rn

+

IR, g

:

Rn

Assume, furthermore, that there exists a vector

+

$m

En.

E Rn such that

(b(x)-g(x)+$m) E S(IRn), i.e. $(x) admits the representation (x), $ ' (x) E s(lRn).

$(x) = <$,,X>+$' Let us assume that qo = inf

xE mn

points, and denote yo

=

IVx$(x) 1

2

> 0, i.e. $(x) does not have critical

min{l,qo}, yo > 0.

Then the following inequality holds: (3.13.125)

I

exp(-E-'$ (x)) u (x)*a (x,E ,Dx)(exp(E-l$ (x)) u (x)) dx 2

R

where the constant C does not depend on

E

and u .

Indeed, using the equality b(x) = -2V $(x)+g(x), one easily finds: (3.13.126)b (x,E ,Dx

def =

6

= -E

exp ( -E

-1

2 AX-E+I Vx$ (x)I 2-~AX$ (x)

with g(x) defined by (3.13.124). Notice, that

$ (x)) a (x,E ,Dx)exp ( E-'$ (x)) =

3.13. Girding‘s Inequality with C = ( f ) sup

475

Il

,Emn Hence, applying the second part of Corollary 3.13.21 to P ( x , E , ~ )= E

2

2 1512+/$,(X) 1 2-EA,$(X)-Y~(~+E 151 2) , p E S(0f0‘2)(X7n) with p (x,rl) =

0 2 2 2 111 +Iox(X) -Yo 2 0 , v (x,n) E RnxlRnI and using (3.13.127), one gets

1

(3.13.125). Consider the corresponding parabolic initial value problem, which describes theMarkov process with the infinitesimal generator a(x,E,Dx) given by (3.13.123) :

Using (3.13.125) one gets the following estimate for the solution u(x,t) of (3.13.128) : (3.13.129) Jnexp(-2~-l$(x)) (u(x,t)1 2dx 5 IR

-1 5 exp(-2~ (yo-CE)t) J

n

2 exp(-2~-’$(x)) luo(x) I dx,

R where yo,C and $(x) are as defined here above and where $(x) is still supposed to have no critical points: $” # 0, V x E Rn. -1 Indeed, putting u = v e x p ( $ ~ ) , v = v(x,t) is the solution of the ‘b

Cauchy problem: -1 (at+€ ~ ( X , E , D))v(x,t)

=

0,

(x,t) E I R ~ X R +

(3.13.130) V(X,O) where v

=

=

uo exp(-s

v (x), 0

-1

0) and b ( x , ~ , D) is defined by (3.13.126)

Now, (3.13.125) yields: (3.13.131) Re(b(x,~,D)v,v) 2 (y -CE)1 /v1I 2 0

Thus, taking the inner product in the equation (3.13.130), using (3.13.131) and Gronwall’s lemma, one gets (3.13.129). The condition $ (x) # 0, V x E lRn , $(x) being linear function for 1x1 sufficiently large, means that, up to the orthogonal to $

component

g(x), the vector field b(x) is diffeomorphic to a constant vector field .0, Of course, the most interesting situations in the theory of random

3 . Singular Perturbations on Smooth Manifolds without Boundary

476

p e r t u r b a t i o n s o f dynamical s y s t e m s (see [Freid-Wentz

,

1)

1

and o t h e r

f i e l d s of a p p l i c a t i o n s of s i n g u l a r p e r t u r b a t i o n s ( f o r i n s t a n c e , r e a c t i o n d i f f u s i o n p r o c e s s e s w i t h l a r g e d r i f t v e c t o r f i e l d s ) , are p r e c i s e l y t h e o n e s when b ( x ) ( a n d , t h u s , a l s o $ ( x ) ) h a s c r i t i c a l p o i n t s . I n t h a t case t h e b e h a v i o u r o f t h e t r a j e c t o r i e s o f t h e s y s t e m of o r d i n a r y d i f f e r e n t i a l equations A ( t )

=

b ( x ( t ) ) i n t h e neighbourhood of t h e c r i t i c a l p o i n t s of

b ( x ) ( s t a t i o n a r y s o l u t i o n s ) p l a y s t h e c r u c i a l r o l e i n t h e i n v e s t i g a t i o n of t h e a s y m p t o t i c b e h a v i o u r o f t h e s o l u t i o n s t o (3.13.128)

and t h e c o r r e s p o n d i n g

s t a t i o n a r y e l l i p t i c problem w i t h a s m a l l p a r a m e t e r i n domains i n Wn 2 I n p a r t i c u l a r , f o r n = 1 , uo E L ( W ) , supp uo 5 IR+ , b ( x ) : 1 ,

.

i n e q u a l i t y (3.13.129) y i e l d s f o r t h e s o l u t i o n u of (3.13.128):

1

exp(-2~-'X)/ u ( x , t )

=-

I 2 dx

-1

5 exp(-2~ (l-Ce)t)I

exp(-2e-'x)

2 / u ( x ) ; dx 0

IR+

i . e . u ( x , t ) f o r t > 0 , x E IR-

w i t h r e s p e c t to ( x , t )

E IR-

xW+

i s exponentially s m a l l a s

E

+

0 , uniformly

w i t h x + t 2 y > 0.

0

N o t i c e t h a t i n t h e l a t t e r case t h e ( g e n e r a l i z e d ) s o l u t i o n u ( x , t ) o f 0 t h e reduced problem ( E = 0 ) a s s o c i a t e d w i t h ( 3 . 1 3 . 1 2 8 ) , i s u ( x , t ) = uo ( x + t ) , 0 so t h a t u ( x , t ) Z 0 f o r x + t < 0 . I Example 3 . 1 3 . 2 9 . L e t m > 0 b e i n t e g e r and l e t a E S ( o ' o r 2 m )( R n) ,

be a s t r o n g l y e l l i p t i c d i f f e r e n t i a l s i n g u l a r p e r t u r b a t i o n of o r d e r (0,0,2m), i . e . t h e p r i n c i p a l symbol a ( x , q ) = a ( x , O , q ) , q = € 5 , o f a ,

0

s a t i s f i e s t h e c o n d i t i o n : R e a ( x , n ) 2 y

0

2m

,

V (x,n) E WnxWn

w i t h some

c o n s t a n t y > 0 , t h e b e s t p o s s i b l e value yo of y being c a l l e d t h e s t r o n g e l l i p t i c i t y constant f o r a. F u r t h e r m o r e , assume t h a t t h e c o e f f i c i e n t s a ( x , E ) s i d e of

(3.13.132)

a r e L i p s c h i t z continuous i n

E

E [O,E

on t h e r i g h t hand

0

1

uniformly with

r e s p e c t t o x E R n , so t h a t , i n p a r t i c u l a r , t h e r e e x i s t s a c o n s t a n t C s u c h that (3.13.133)

]a(x,E,n)-a(x,O,q)

1

5 Ce

2m

,

V ( x , q ) E mnxlRn

C o n s i d e r t h e p a r a b o l i c s i n g u l a r l y p e r t u r b e d i n i t i a l v a l u e problem:

3.13. Girding's Inequality

(a/at+a(x,~ED ))u(x,t)

=

f(x,t),

477

(x,t) E IRnxlR+

(3.13.134) U(X,O)

=

u

0

x)

.

Then, there exists a constant k E IR such that the following estimate holds for the solution u(x,t) of (3.13.134):

where y o is the strong ellipticity constant for a(x,E,ED 1 . Indeed, first consider (3.13.134) with f E 0. Let p

=

Re a, q

Im a. One finds for q(x,E,EDX)

=

(3.13.136) (Re(iq(x,E,~D )u,u) 1 =

+I

=

Op(q)using (3.7.16)

(q(x,E,tD )-q(x,E,ED )*)u,u) I 5

since, as a consequence of (3.17.16), one has:

Furthermore, using again the second part of Corollary 3.13.23 and applying it to a (x,ED )-y ( ~ - E )m ~ Awith Ax the Laplace operator, one 0 x o finds:

where k l is some constant which does not depend on

E

and u,

Combining (3.13.133), (3.13.136), (3.13.137), one gets immediately the inequality:

Hence, (3.13.138), (3.13.134) (after taking the inner product in differential equation (3.13.134) with f F 0) and Gronwall's lemma yield (3.13.135) with f F 0. Now, the same argument as in Corollary 3.13.18, leads to (3.13.135) with V f(.,t) E H

(0),E

(lRn), v t 2 0.

3 . Singular Perturbations on Smooth Manifolds without Boundary

478

One can show in the same way that, in fact, for each s E R3 , the solution u of (3.13.134) one has:

Example 3.13.30. Let A(x)

:

Rn

-f

(3.13.139) a(x,D

Hom(R )

n

,R

n

b(x)

),

:

Rn

-f

R n , c(x) : Rn

= ++c(x),

-f

IR and let

x E R n , (with < , > the

real inner product) be an elliptic second order differential operator with m

C

-coefficients and the ellipticity constant y o > 0. As usual, we assume that there exists Am, bm and cm such that

(A(x)-A_) E S ( R n ) , (b(x)-bm) E S(Rn), (c(x)-cm) E S(IRn) which means that each component of the corresponding matrix- or vector-function belongs to Schwartz's space S(Rn). Consider the initial value problem: (2/2t+a(x,Dx))u(x,t) = 0,

(x,t) E RnxWt

u(x.0) = uo(x),

x E Rn

(3.13.140)

and its discretization in time variable t: (3.13.141) {(Id+rTa(x,DX ))OT -(Id-(I-r)Ta(x,D X ))lvT (x,t) = 0 , vT(x,o) = uo(x),

-

(x,t) E lRnxml T,+

where r E (0,1] is a parameter to be determined later on, and where, as -1

= It = kT, k E Z, k 2 0 ) and OT is the shift operator on T,+ -1 the mesh g:,+, (OTv)(t) = v(t+.r), V t E R T , + , Id being the identity

previously, IR

operator. Let For

v

E

E

T = E*,

O,E

0

3

b(x,E,D

with

E~

)

=

2 Id+rE a(x,D

),

b E Op S(o'or2) ( R n )

.

sufficiently small, there exists a singular

perturbation b-l (x,€,DX) E Op S(080r-2) ( R n ) , such that (b 0 b-l-Id) E op s(-l,O,l) (Rn ) . Indeed, one can define b-l(x,c,Dx) in the following fashion: b-'(x,E,Dx)

=

-1 Op b0(x,€<) with b ( x , q ) = l+r. Then

(3.7.17), (3.7.18) will yield:

0

3.13. Gdrding’s Inequality -1 (b(x,E,Dx)

b

(b-l(x,E,Dx) Therefore for

v

419

(x,€,Dx)-Id)E Op S(-lro’l) ( R n1

I

b(x,€,Dx)-Id) E Op S ( - l r o r l ) ( R n ) .

E ( O , F ~ ] with c0 sufficiently small, there exists the

E

-’ which is a pseudodifferential singular

inverse operator b ( x ,,Dx) ~

perturbation in S ( o ’ o r - 2 1 (IRn ) . 2 Introducing QE = (Idirc a(x,Dx))-l

2 (Id-(l-r)s a(x,D

E Op S ( O r O ’ O ) ( ~ n :

))

one can write the solution of (3.13.141) in the form:

One finds easily that the principal symbol q ( X , E ~ E ) S(o’ofo)(IRn) 0

of Q, is given by the formula: qo (x,n) =

(

l+r)

-’ I(

a n ,rp)

( 1 -r)

1

so that /qo(x,n) 6 1, V (x,n) F IRnxIRn, provided that r E [t,l].

Hence, as a consequence of (3.13.26) (with E~ instead of h), one finds: (3.13.142)

1 I (QE)t”l 1

5

L ~ ( I R+~L)~ ( I R ~ )

l I Q E 1 1 t/T

5 ( l+CT ) t/T 5

L2 (nn)+L2 ( R n )

5 exp(Ct),

v

t E IR

1

.

T ,+

It can be easily shown, but will not be done here, that a s a consequence of the stability result (3.13.142), the family of solutions + vT(x,t) of (3.13.141) converges in L2 (IRn ) to the solution u(x,t) of

T

(3.13.140) and, moreover, one has: I (u(.,tl-vT(.,tl 1 1 5 CTI ( U 1 1 0 2’ 1 L2 ( Rn) V t E ]RT , +, where I 1 . 1 is the classical Sobolev norm of order 2 and

I

C > 0 is a constant, which does not depend on u, vT, uo and T . Instead of (3.13.141), one can consider a full discretization of (3.13.140) on IR:,hx$,+,

using any elliptic finite difference

approximation of a(x,D 1 , for instance, the following one: a (x,h,Dx,h,,:D

h) =
.

h>+ ( + )
h>+c (x)

Then, an argument similar to the one used above, leads to the corresponding to (3.13.142) stability result for the family of difference singular perturbations Q T

QT,h

=

,h ( I d + r ~ a ( x , h , D ~ , ~ , D : , ~ )o- ~(id-(l-r)Ta(x,h,D

x,h ,D* x,h)

3 . Singular Perturbations on Smooth Manifolds without Boundary

480 in n

( 0 ) ,h

that r

(q)uniformly with respect to ( ~ , h E)

E [$,I] and ~

( O , ’ T ~ I X ( O ,provided ~~I

~ are, sufficiently h ~ small.

I

Example 3.13 .Xl . Let A(x,t)

:

IR x[-T,T]

+

E Cm(IRx x[-T,T]) and let

Hom(Rp ;lRp), A

A(x,t) = Am(t)+A’(X,t), A’(.&)

E S ( R x ) , v t E [-T,T].

The operator (3.13.143)L(x,Dt,Dx)

def =

i (DtId+A(x,t)D

)

is said to be strictly hyperbolic for (x,t) E Rx[-T,T],

if the eigenvalues

A.(x,t), 1 5 j 5 p, of A(x,t) are real, A . E I R , V (x,t) E IRX[-T,TI and, 7 1 moreover, there exists a non-singular matrix C(x,t) : R x[-T,T] + ISO(IRp;IRp), C E Cm(lRxx[-T,T])

such that

(3.13.144) A(x,t) = C(x,t)-ldiag(A.(x,t)) < , < C(x,t), V (x,t) E IRxX[-T,T]. 3 l=J,p For instance, if there exists a constant 6 > 0 such that /A.(x,t)-Ak(x,t)(2 6 > 0, V (x,t) E IRx[-T,T], V j # k , then (3.13.143) is

I

strictly hyperbolic. Also, if A(x,t) is symmetric, V (x,t) E lRxX[-T,T], then (3.13.143) is hyperbolic. Consider the initial value problem

x

u(x,O) = uo(x),

E

9.

Let Pk(X’t), P (x,t) k

=

-1 (2ni)

J

/X-Ak(x,t)j = p

( AId-A(x,t)) -‘dh

with p > 0 sufficiently small, be the projector on the one-dimensional invariant subspace of A(x,t) in IRp which corresponds to the eigenvalue

X (x,t) of A(x,t). k

It is

readily seen that (3.13.144)

implies that

P (x,t) depends smoothly on (x,t) E Rx[-T,T] and, moreover, k P (x,t) = P (t)+P;(x,t), V (x,t) E RxX[-T,T], with P;(.,t) E S ( I R ) , k krm V t E [-T,T].

Thus, with Pk(x,t)* the adjoint of Pk(x,t) , the matrix M(x,t) M(x,t)

def

Pk(x,t)*Pk(x,t)

=

1(k
3.13. Girding's Inequality

48 1

has the same regularity properties and, moreover, M(x,t) V (x,t) E BxX[-T,T] with some constant m

0

=

M(x,t)* 2 moId,

> 0.

Furthermore, one finds: iM(x,t)A(x,t)D + (iM(x,t)A(x,t)Dx)* =

=

i(M(x,t)A(x,t)-A(x,t)*M(x,t)*)D -(A(x,t)*M(x,t)* ) X'

where ( A * M * ) = ~ a/ax(A*M*) and, as usual, the upper

* stands for the

adjoint of the corresponding operator or matrix. Since Ak(x,t), 1 S k 6 p, are real-valued, one has: M(x,t)A(x,t) -A(x,t)*M(x,t)* = =

C

(Pk (x,t)*Pk (x,t)A(x,t)-A(x,t)*Pk(x,t)*Pk (x,t)) =

l6k5p =

c A k (x,t)(Pk (x,t)*Pk (x,t)-Pk(x,t)*Pk (x,t))

=

0.

l5k5p Hence,

(3.13.146) iM(x,t)A(x,t)Dx+ (iM(x,t)A(x,t)Dx)* = B(x,t), where B(x,t)

dEf -(A(~,t)*M(x,t)*)~: iRx[-T,T]

+

HOm(IRP ;iRP)*

Using (3.13.146), one finds for the solution u(x,t) of (3.13.145):

Therefore, Gronwall's lemma and the strict positivity of M(x,t) yield: 5 exp(Clt/)I / u o / 1 2 + (3.13.147) 1 lu(.,t) 1 1 2 L2 (IR) L2 (IR)

+ C'

It1

0

2 lIf(.,T)l\

eXp(C((t!-T))dT

L (R)

3. Singular Perturbations on Smooth Manifolds without Bounda y

482

where C , C ' and C" are some positive constants. Using the same argument one can easily show that, in fact, for each s

E

IR3

the following a priori estimate holds for the solution u of

(3.13.145) :

V t E [-T,T],

with some constant Ct which may depend on t. Introducing the operators a(x,t,~,D) , a(x,t,E,DX) = A(x,t)Dx<EDx>-l E Op S(orlr-l)

( R X)

r

and considering the initial value problem:

(3.13.149)

i(D +a(x,t,€,Dx))uE(x,t) = f(x,t) t u (X,O) = u (x),

0

one shows in the same way, as previously, that the same a priori estimate, as ( 3 . 1 3 . 1 4 8 ) , with C

holds for u (x,t) uniformly with respect to

which does not depend on u,uo,f and

t Since for each given

E

E

E (O,E

0

1,

i.e.

6.

0

> 0 one has: a(x,t,E,Dx) E Op S ( R x ) , one

shows using the Picard method and the contruction mappinq arqument that the solution u (x,t) to ( 3 . 1 3 . 1 4 9 ) unique. Now, using ( 3 . 1 3 . 1 4 8 )

exists for t E [-T,T] and is

with s = (O,r,O) with appropriate r E R , and

E 0 that for each 2 uo E H ( R x ) , each f E L ([-T,T];H (R ) ) there exists a unique solution

the compactness argument, one shows by letting

u E

o

O'

-f

x

([-T,T];H (IR ))of initial value problem ( 3 . 1 3 . 1 4 5 ) . a x We are going to consider the following finite difference approximation

C

of ( 3 . 1 3 . 1 4 5 )

which can be successfully used for its numerical treatment.

Let, as previously, let I

= imT1

sT,T

5

= hZ C IR be a grid with meshsize h >

0 and

be the grid on [-T,T] with meshsize T .

Consider the two parameter family of solutions u (x,t), (x,t)E Rh XIT h,T of the following discrete initial value problem

and where the difference operators B hrT following symbols:

qnrT have,respectively, the

3.13. Girding's Inequality

483

b(x,t,h,~,h<= ) Id+ir(2h)-lA(x,t) s i n h<, (3.13.151) a ( x , t , h , . r , h < ) = b ( x , t , h , ,~h c ) -iTh

-1

2 A ( x , t ) s i n h < (1-4y s i n ( h < / 2 ))

y E IR b e i n g a p a r a m e t e r , whose a d m i s s i b l e v a l u e s a r e t o b e d e t e r m i n e d t o g u a r a n t e e t h e s t a b i l i t y of

.

(3.13.150)

i s s a i d t o be a s t a b l e

The i n i t i a l v a l u e problem (3.13.150) approximation of (3.13.145) i n

H(o,,h(%)

where t h e c o n s t a n t C d o e s n o t depend on

E

for t

I

,t

> 0,

i f one h a s :

h , u ~ , uo ~ ,and f .

T,

N o t i c e t h a t f o r e a c h T = r h w i t h r > 0 a g i v e n c o n s t a n t , t h e f a m i l y of def d i f f e r e n c e o p e r a t o r s Bh = B H ( 0 ), h ( % ) has an h , r h ' Bh ( 0 ) ,h(iRh) inverse B-l(iR ) whose norm i s u n i f o r m l y bounded, h . H ( 0 ), h ( % ) -t H ( 0 ) , h h w i t h r e s p e c t t o h E ( 0 , h 1, p r o v i d e d t h a t ho i s s u f f i c i e n t l y s m a l l .

'

'

0

I n d e e d , f o r e a c h pxp m a t r i x $ ( x ) f1 f o r t h e commutator [$,Oh I d ] = $ ( x ) O f l

E

C1

-Oil$

(z) , $

.

=

I / @ j k ( x1) 1 ,

one h a s

(x) t h e following estimate:

where $ ' ( x ) i s t h e m a t r i x , whose e n t r i e s are ( d / d x ) $ . ( x ) a n d , a s u s u a l , lk s t a n d s f o r t h e norm i n Hom(CP;CP).

1 1

Hence, one f i n d s , u s i n g ( 3 . 1 3 . 1 5 2 ) :

(1) h '

= Id+TR

and t h e s a m e argument y i e l d s : i Im B

h

=

( f ) ( B -B*) h h

= =

r ( A ( x , t ) 8 +O A ( x , t ) - A ( x , t ) o - ' 4 h h h (2) TR h '

where t h e d i f f e r e n c e o p e r a t o r s R i J )

:

H(o) ,h(%

norm u n i f o r m l y bounded w i t h r e s p e c t t o h Thus, f o r h o ,

To

E

(O,h

0

H(?,

)

-f

1

by C / A ' ( x , t )

S u f f i c i e n t l y s m a l l Bh = B h , r h

]) i n v e r s e Br e s p e c t t o ( h , T ) E ( O , h o l ~ ( O , ~ obounded h'

,h(lF$,

OhL 1 A ( x , t ) ) =

have t h e i r

has a (uniformly with

'

H ( 0 ),h("h)

H ( 0 ), h ( % -1 . N o t i c e , t h a t t h e p r i n c i p a l symbol b i l ( x , t , r , h < ) of Bh i s g i v e n by t h e formula:

,

3 . Singular Perturbations on Smooth Manifolds without Boundary

484

-1 (3.13.154) bo (x,t,r,h<) = (Id+i(r/2)sin h< A(x,t))-l the right hand side on (3.13.154) being well defined, V (x,t,h<),since the eigenvalues of A(x,t) are real. Let (3.13.155) Qh,T

-1 Bh,T%,T.

=

being the matrix The principal symbol q (x,h,~,h<) 0

(3.13.156) q (x,t,h,~,h<) = 0

2

Id-ir sin hS(1-4~sin (hS/n))A(x,t)(Id+i(r/2) sin h5 A(x,t))-',

=

one finds that the eigenvalues u,(x,t,r,y,n) of q (x,t,~,h,q)are 0

3

(3.13.157) u,(x,t,r,y,n) = 3

=

1-ir sin n(1-4y sin2 (n/2))X.(x,t)(I+i(r/?)(sinq)Ai(x,t))-' 3

with A.(x,t) the corresponding eigenvalue of A(x,t), so that an easy 3

computation yields: (3.13.158) 1 p . I

2

7

=

=

2 2 1-4yr2sin2q sin (q/2) (1-4y sin (q/2))X Thus,

1 3~ I . 5

j

2 2 -1 (~,t)~(l+(r'/4)(sin n)A.(x,t)

1

1, V (x,t,r,q), if

(3.13.159) 0 < y 6 1/4. Using (3.13.144). (3.13.150), (3.13.155) and introducing (x,t), one gets for v (x,t) the difference (x,t) = C(x,t)-lu hrT h,T h,T equation: v

-1

(3.13.160) V

(X,t+T) = C(X,t) h.7

c(X,t)V

'h,T

Obviously, (3.13.152) will hold for

h,T

(xrt)+T(<:Tf)

(X,t).

if it is valid for v h.T Qh,TC(x,t) is diag(p.(x,t,r,y,q)

-FcT

Since the principal symbol of C(x,t) and 1 p . l 2 1 if y 3

E

(O,i),

its discrete version with conclusion that

3

using Corollary 3.13.11 (or, rather

H ( o ) ,h(\

)

instead of L2(iR)) one gets the

.

3.13. Gdrding's Inequality

485

satisfies the inequality

T

=

rh, V t E [-T,Tl,

where the constant C does not depend on h. Therefore, using (3.13.160), one finds:

The last inequality now yields:

where the constant C2 may depend on T and r = T/h, but does not depend on h, v ~ , uo ~ ,and f. One shows in exactly the same way that (3.13.161) holds with t replaced by -t.

The difference scheme as in (3.13.150), (3.13.151) turns out to be useful for the numerical solution of the initial value problem for the systems of conservation laws of the form (3.13.163) with A

=

a au + -f(u) ax at

-

= 0,

u(x,O) = u,(x)

f'(u) the Jacobian matrix for the vector-function f(U)

However, in the regions where the (generalized) solution u(x,t) to (3.13.163) has discontinuities, scheme (3.13.150), (3.13.151) being not of monotone type, its use give numerical solutions, which might oscillate near the discontinuity lines (shocks and others) of u(x,t), instead of giving just a numerical jump, which should reflect the singularity of u(x,t).

I

3 . Singular Perturbations on Smooth Manifolds without Boundary

486

3.14. Reduction of Elliptic Singular Perturbations to Regular Perturbations Given an elliptic singular perturbation a(x,E,D) in Rn or on a compact smooth manifold M, an explicit construction is provided here for a sit:qularly perturbed pseudodifferential operator s(x,E,D) which reduces a(x,E,D) to a regular perturbation, i.e. the product (soa)(x,E,D) is a family of elliptic pseudodifferential operators of the form:(soa) (x,E,D)= 0

0

a (x,D)+ EYQ(x,€,D), where a (x,D) is the reduced operator for a(x,E,D), y is

some positive constant and the order of Q(x,E,D) is at most the same 0

as the one of a (x,D). The same kind of a left reducing operator can be constructed for any elliptic difference singular perturbation on the greed Rn x,h’ A left reducing operator turns out to be a right one for both elliptic pseudodifferential and difference singular perturbations. Some applications to singularly perturbed boundary value problems are also discussed. First, the reduction procedure will be explained using a specific singular perturbation comingfrom thelinear theory of thin elastic plates. Consider the singular perturbation (3.14.1)

=

a(s,a/ax) :=

E

7.2 A -

A€,

__

c a a a +1, l
I

where the matrix 1 lakj I I&, jSn is supposed to be symmetric and positive definite, A being the Laplace operator. Obviously, one has uniformly with respect to (3.14.2)

AE : H

(0,2,0)

E

E

(0,ll

:

(d-5.

,E

With ao(E,S) the principal symbol of AE,

define the symbol ~ ( € 6 ) : (3.14.4)

~ ( € 6 =) ao(O,S) (a0(€,C))-’,

( E , S ) E K + x (R”\IO})

= Op s , the latter being a -1 family of convolution operators with the kernels S(E,X) = F

and the corresponding singular perturbation SE

S-tXS1

(3.14.5)

( s E u ) ( x ) = JnS(~,x-y)u(y)dy, V u E CE(Rn). IR

487

3.14. Reduction of Elliptic Singular Perturbations

-2 where ho (E,x) is defined by (3.2.23). Since AE given by (3.14.1) plays an important role in the linear elasticity theory, we shall give here a formula for the kernel S(s,x) of the reducing operator S E = Op s ( ~ E . 1for any symmetric positive definite

I

in the case of the dimension n=3.

-1

where S(x) = FS+x s ( S ) . Using the spherical coordinates 5 = pw, with p = 151, w E fin, Rn being the unit sphere in I R F , one finds easily when n=3: S(x) =

I

I(w;x)dw,

fi3 where we have denoted:

2 with 0 (w) = =

1 akjwkwj, the limit on the right hand side of 15k,j53 3 the last formula being interpreted in the distributional sense in S ' ( R ) .

Obviously, one has: m

-3 2 2 2 -1 I(w;x) = (2n) {iQ ( w ) (<x,w>+i0)-'-O4(w)l(p +Q ( w ) ) exp(ip<x,w>)dp}. 0

Furthermore, since I(w;-x) = I(w;x), one can rewrite the last formula in the form: I(w;x) =

(+)

(2n)-3~2(w)~i( (<x,w>+io)-l-(<x,w>-io)-l

-

m

-Q2(w) /(p2+02(w)

exp(ip<x,w>)dp},

-m

for the distributions -1 (xfiO) and the residuum calculus for computing the integral on the right

so that the Plemelj-Sokhotski formulae

hand side of the last formula, one finds: 2 3 2 -1 ) I20 (w)A(<x,w>)-Q (w)exp(-Q(w)I<x,w>\)}, I(w;x) = (161~

3. Singular Perturbations on Smooth Manifolds without Boundary

488

where 6(y), y E R is the Dirac's 6-function. Further, noticing that 2 -(d/dy)2 exp(-@(o)/yl) = 4 (w) exp(-@(w)/yl) + 2@(w)6(y), and using the fact, that

=

Zwk2

1 , one gets finally the following

=

formula for I (w;x): I(w;x) where A

=

=

-(l6n2)-'A

O(w)

exp(-@(w) I<x.w>/),

Z(a/ax.)L is the Laplace operator.

I

Thus, one finds for S(x) when n=3: S(x) = -(16n2 -1 A For a = Id, one has: @(a)

exp(- f /<x,w>l)dw

'

n3 1 and the last formula yields:

5

the latter coinciding with A02(1,x) defined by (3.2.23) when n=3. The same kind of computation carries over for any odd dimension n, since it allows to evaluate I(w;x) by replacing the integral over R

= {p>O) in its expression by the one over R = {-m
and by using the

residuum calculus.

. some constant co > 0 (which Since c - ~ < E ~ >5 -~~( € 5 4) c <ES> -2 with 0

0

depends only on the smallest and greatest eigenvalues of the matrix

I 1 akj 1 1 ) , one gets (3.14.6)

SE :

H

the conclusion that (Rn)

(0,0,-2), E

uniformly with respect to

E

E

( 0 . ~ ~ 1V,

e0 <

m.

Moreover, actually, the family of linear mappings (3.14.6) is an isomorphism uniformly with respect to Further, the product SE (3.14.7) where Ao

SE

=

0

A'

= A

0

+ E

2 E (I

0

AE

E

E (0.~~1.

can be represented in the form:

,

a(O,a/ax) is the reduced operator for

op s(or2r-2)( R n ) ,

AE

and

3.14. Reduction of Elliptic Singular Perturbations (3.14.8)

QE

:

H

489

( Rn)

(0,2,0),E

is a family of continuous (uniformly with respect to

E

E [O,E~])

convolution operators with the (distributional) kernels q(E,x),

1

q(E,x) = F-1

E*X

~

1

~

(aO(E,E))-l.

(Rn)with H2(Rn) and H(o,O,O),e(Rn)with (0,2,0), E one comes to the conclusion that the multiplication of A from

Identifying H L2(Rn

),

the left by

SE

reduces elliptic singular perturbation (3.14.1), (3.14.2)

to a regular perturbation of A (3.14.9)

Ao

:

H2(Rn)

*

0

,

L2(Rn).

It will be shown in this section that for any elliptic singular perturbation A E in Rn or on a smooth manifold M without border, an explicit algebraic construction can be given for a left reducing operator SE, which reduces A'

to a regular perturbation of the operator A

0

.

Now the construction of a reducing operator for any elliptic singular perturbation will be presented systematically. From now on all the symbols and the corresponding singular perturbations are supposed to possess the reduced symbols and the corresponding reduced operators. Let a E Op

L v ( Rn ) (see Definition 3.12.1) be elliptic of order

v = (vl,V2,V3)and assume that it has a reduced operator

ao E op L ( 0 1 v 2 ' o ) (Rn)(see Definition 3.12.3). Our goal now is to produce a singular perturbation r E O p (to be called factorizing) and a singular perturbation s E O p (to be called reducing) such that (3.14.10) a(x,E,D) = r(x,E,D) 0

(3.14.11) a (x,D) = s(x,E,D)

o

0 a (x,D)+EQ~(x,E,D),

o

a(x,E,D) +EQ~(x,E,D),

where for each s = (sl,s2,s3) E R3 the linear mappings:

are bounded uniformly with respect to

E

E (0.~~1.

L

(vl,0,v31 ( Rn)

L (-v1,O,-v3) ( Rn)

3 . Singular Perturbations on Smooth Manifolds without Boundary

490

Lv ( Rn )

It is readily seen that for each elliptic symbol a E

of order

v there exist constants C > 0, R > 0 such that

V (x,E)

E

Bnx

( O , E ~ ] ,V

5 E

Rn,

Given an elliptic symbol a E LV(Rn)satisfying (3.14.13),

x

E Cm(Rn) be a cut-off function such that x(S) 0

x(S)

5

5

(51

2 R.

let

0 for 151 2 2R,

I for 151 5 R, and denote @ ( S ) = 1-x(5). As a consequence of (3.14.13), the following functions are well5

defined :

{s(x,E,~)

0

r(x,c,S) = Q(E)a(x,c,c) (a (x,<))-'+

(3.14.14)

=

-V E

1 x(~),

0 Vl Q(c)a ( x , ~(a(x,E,C))-l+ ) E x(~),

where a0 (x,S) is the reduced symbol of a(x,e,S). Indeed, since (3.14.13)

is valid uniformly with respect to

E,

it

0

holds for the reduced symbol a (x,S), as well.

The symbol a E LV(mn)being e l l i p t i c of order v

Lemma 3.14.1.

=

(v1,v2,v3)

( 0 , V2,O)

and ao E L ( R ~ ) being i t s reduced symbol, t h e f u n c t i o n s r(x,E,S) and s(x,e,S) defined by (3.14.14)are e l l i p t i c symbols of order (v1,0,v3) and (-vl,0,-v3) belonging t o t h e c l a s s e s ~ ( ~ 1 ' ~ ' (Rn) ~ 3 ) and

L

(-V1 ,O,-V3)

so

(w" ), r e s p e c t i v e l y ; furthermore, t h e i r reduced symbols ro and

are i d e n t i c a l l y

1.

_ Proof. _ The latter statement is immediate, the ellipticity of r(x,E,S) and s(x,e,S) (of order (vl,0,v3) and (-v 1 ,O,-v3), respectively) being a direct consequence of the fact that their principal symbols are 0 -1 0 rO(x,EpS) = aO(x,E.5) (ao(x,5)) and sO(x,~,S)= ao(x,5) (a0 ( X , E , S ) ) - ' , respectively. Furthermore a straightforward computation (using the chain rule and the ellipticity of a of order v) shows that r(x,E,E) and s(x,E,<) satisfy

(3.3.1) with (vl,O,v ) and (-vl,0,-v3), respectively. 3

I

Lemma 3.14.2.

The symbol a E L'(R") ao E

being e l l i p t i c of order v = (vl,v2,v3) and

L(o'v2ro) (xn) being its reduced symbol, one has w i t h any

N > 0:

3.14. Reduction of Elliptic Singular Perturbations

49 1

Proof. It is readily seen that with r(x,E,E) and s(x,E,S) defined by (3.14.14), one has:

Now (3.14.15) is an immediate consequence of Definition 3.12.3 and Proposition 3.3.3, since

x

E ~ ( ~ l - ~ (cE-n )~,)v N

> 0.

Remark 3.14.3. Using the following definition of r and s : -V

0

1 X(S),

r(x,E,S) = a(E)aO(x,E,c)(a0 ( x , ~ ) ) - ' + c 0

s(x,s,~)= a(E)ao(x,t) (a0 (~,E,s))-'+ s where . a

0

and . a

v1

x(<),

are the principal symbols of a and 'a

respectively, it is

readily seen that (3.14.16) (a-ra ) E L

"

( v 1-1,v 2 ' 3 )

(-l,V2,O)

( R ~ ) ,(sa-a ) E L

(Rn).

Sometimes, it is more convenient to use principal symbols for constructing r and s . In that case one can choose a fixed R (for instance, R = 1). while choosing the cut-off function

x(S).

I

The following lemma plays an important role in the reduction procedure and is proved by using a more or less standard argument based on Schur's lemma. 3.14.4. Lemma _ _ _ ~ L e t j(x,~,S)E L'(R")

with v

symbol j0 ( x , ~ ): I . v

(x,<)

= (vl,0,v2) and a s s m e t h a t t h e reduced E mRnx R". L e t a(x,c,<) E L"(R~).

Then one has: def (3.14.17)

QE

=

(Op(j)

o

Op(a) - Op(ja)) E Op L V+u-(l,O,O) (Rn),

3

i . e . f o r each s E R t h e folZowing i n e q u a l i t y holds: (3.14.18)

1 1 QEuI I ( s-v,-F1) ,

wi t h a constant c

> 0,

I (s)

v

E

E

(o,Eol,

which does n o t depend on

E

and u.

5 C E / IuI

,E'

v

u E H

( s ) ,E

(Rn)

3. Singular Perturbations on Smooth Manifolds without Boundary

492 Proof. -

As a consequence of Theorem 3.7.5 (with U = lRn, so that the

condition on Op(j), %(a)

of being properly supported is automatically v+u-(O,l,O) n ( R ) . The specific property

satisfied), one has: QE E Op L

(3.14,17)of Qe is a consequence of the fact that Op(j) = ~(x,E,D) has identity as its reduced operator. As

a consequence of Definition 3.12.1 of the symbol class

L v ( Rn ) ,

one has the following decomposition: j(x,E,S) = j,(E,<)+j'(x,E,S), where the functions x

+

j'(x,e,S), x

+

a(x,e,<) = a,(e,<)+a'(x,c,c), a'(x,E,S) are in S(mn), X

v

(E,S)

E

(O,Eo1

x R;.

Given that Op(j)

o

Op(a,)

- Op(jam)

= 0,

one gets the conclusion that

for proving (3.14.17) it suffices to show that (3.14.19) Op(j')

o

Op(a') - Op(j'a') = E Q ~ ,

(3.14.20) Op(jm)

o

Op(a')

where QE E LV+'(lRn),

-

Op(j,a')

=

EQ~,

i.e. (uniformly with respect to

E

E (O,Eo]) for each

s E R 3 'holds:

(3.14.21) Q? J

:

H ( S ),E

First, (3.14.19) (with QE satisfying (3.14.21)) will be proved. 1 One has the following Fourier representation for QEu:

~ ' ( s - T , E , =~ IF~+~-,~'(X,E,~), ;1(7-n,~,n)

=

~~+~-~a'(x,~,n).

By Schur's lemma on boundedness of integral operators in L2 (see Lemma 3.12.8) for proving (3.14.19), (3.13.21), it suffices to show that s

(3.14.23) and

E

-v

-s

-Ll

3<,1>

-S

2<E11> 31K(S,E,n)/dS5 CE

3.14. Reduction of Elliptic Singular Perturbations v +lJ

(3.14.24)

E

s

with some constant As

-v

I <S> nn C >

s -v -lJ

-s

3

2<ES>

-S

2<~n1>

0, which does not depend of

3

493

I K ( S , E , ~ )/dn 5 CE,

(S,E,~). 0

a consequence of Definition 3.12.3 and the assumption: j (x,S)

1,

one has:

Furthermore, one finds : -V

jDx(I'(x,E,~)-j'(x,~,rl))/ a . 5 IDa(b'(x,€,~)-b'(x.€,~))€ '<ET>'

3 It

Using the representation I

bl(X,E,~)--b'(~,E,n)=

I

(a/ae)bi(X,~,q+e(T-n))de,

0

the inequalities (3.12.1) with v

=

(-l,l,-1) and inequality (3.4.7), one

gets:

V ( x , E , T , ~ ) ,V a E N n ,V k L 0,

with m = 3 + / v 1 , and with some constants Ca,k > 0, which may depend only 3 on their subscripts. Hence, one has for j inequalities: E,O) (3.14.25) I~'(y,s,~)-~'(y

I

1-v

2 C < T - ~ > ~ E'<ET>'~<~>-~, t/ y,E , T

k

,n ,k.

Similarly, one has:

Using (3.14.25) and the estimate

' :1

(y,E,n)1 5 CkE

-lJ

lJ lJ ' 2 < ~ q >3-k,

V y , ~ , ,k, n

(which follows from the fact that a'(.,~,n) E S(lR:)

and from (3.12.1)),

one can estimate the integral on the left hand side of (3.14.23) in the following fashion:

3 . Singular Perturbations on Smooth Manifolds without Boundary

494

where

One chooses k > 0 to be sufficiently large in order to guarantee convergence of the integral on the right hand side of (3.14.27), and that ends the proof of (3.14.23). The same argument can be used to show (3.14.24). Hence, we have proved (3.14.191, (3.14.211. Using (3.14.26) instead of (3.14.25), one gets (3.14.23), (3.14.24) with K replaced by the kernel: K,(~,E,I?) = (j,(E,5)-j,(€,~))at(5-n,E,q).

This yields (3.14.20) with Q, satisfying (3.14.21).

1

Remark 3.14.5. A slight modification of the proof shows that Lemma 3.14.4 is still true if

u2 is not necessarily zero

(j(x,E,<)

=

1+Ea(x)1512 is an example of a 0

symbol, whose order is (O,l,l) and whose reduced symbol j (x,S)

5

1).

How-

ever, in applications the most incountered situation is the one, where the order of j is (vl,0,v3). Assume that j E L ( o r o ’ ” ) ( E n ) and j = j (x,E<). Then the following argument gives a simple explanation to the assertion in Lemma 3.14.4. One has in that case (see Theorem 3.7.6):

where

QE

is the singular perturbation whose symbol has the asymptotic

expansion

3.14. Reduction of Elliptic Singular Perturbations

the symbol of QE being in Lv+' (En )

;

495

here the following notation has been

used:

Notice, that nowhere in the proof of Lemma 3.14.4

the fact that the

symbols j(x,E,S) and a(x.E,S) are smooth in 5 has been used. In fact, the assertion of Lemma 3.14.4

is still true for symbols a which satisfy a kind and still admit the decomposition

of Lipschitz condition of the form (3.1.9) (ii) in Definition 3.12.1

with a' satisfying (3.12.1)

for V@ and cx = 0.

In fact, the same informal argument (based on the use of Theorem 3.7.6)

LV(En ) ,

shows that for each j E @(a)

one has:

OOp(j) =a(x,E,D) 0 j(x,E,D) =

-

1

C pp(a(U)(x,Er5) ) ' ( j

(x,E,~))=

Op(aj) + EQ',

where Q"

E - ~

N

0

Since j ( x , S )

/'I

Z

1, one has, according to Definition 3.12.3

for each

> 0: IE-lj(al

(~,E,s) =

v

-vl

>c

E

<5>

*

1'

-1 a

D~(I(X,E,S)-I) I 5 ,

v3+1

-1 <ES>

i.e. Q" is a continuous linear mapping (uniformly with respect to

n ( S ),E (mn ) into H (s-v-')

A

,E

E)

from

( W n ) for each s E E3 :

rigourous proof of the formula Op(a)

o

Op(j) -Op(aj)

= cQE

with QE

satisfying the above continuity condition, consists of repeating alnost word after word the proof of Lemma 3.14.4. Hence r(x,E,D) and s(x,E,D) defined by their symbols (3.14.14)

are

not only left but also right factorizing and reducing operators, respectively, for a given elliptic singular perturbation a(x,E,D) having 0

the reduced operator a (x,D) (see [Fr-W, 1,3] statement).

for a detailed proof of this

3. Singular Perturbations on Smooth Manifolds without Bounda y

496

Theorem 3.14.6.

L e t a € LV(nn) b e e l l i p t i c of order v = (vl,v2,v3) a admits t h e reduced symbol ao E 1

'

(-v1 ,o,-v.

)

and s E 1

Then f o r each

(0.V2

to)

3 (R )

s

be d e f i n e d by E x3 t h e diagram

€

n3 and assume t h a t

(1~"). L e t r E 1

(Vl'O,V3)

3 (E )

(3.14.14).

(3.14.28)

i s commutative modulo o p e r a t o r s , whose norm i s bounded by

CE

w i t h some

p o s i t i v e constant C.

In o t h e r words, t h e f o l l o w i n g e s t i m a t e s hold (3.14.29)

I /Op(a)-Op(r)

0

0

Op(a )

11

1 IOp(r)

0

Op(s) - Id1 I H(s-(0,v2,0))

(3.14.31)

I /Op(s)

o

lH

Op(r) - Id/

(S-V) , E (3.14.32)

I IOp(s)

0

0 Op(a) - Op(a )

+H

1 IH (s) ,E

(vl ,0,v3)

Proof. Since r E

L

~

C >

0 does not depend on

Finally (3.14.32)

+H

5 CE,

(s-(O,V2,O))

,E

5 CE, (S-V)

+H

,E

5 CE.

(s-(O,V2,O))

(-v1,0,-v (3.14.31)

Furthermore, the Lemmas 3.14.2,

where

,E

,E

3.14.4

,E

)

(lRn), s E 1

one gets immediately (3.14.30),

E + 0:

5 CE,

H ( s ),E+H(s-v) (3.14.30)

as

( X n ) and ro E 1, s o E 1, by applying Lemma 3.14.4. imply:

E.

is an immediate consequence of ( 3 . 1 4 . 2 9 ) - ( 3 . 1 4 . 3 1 ) .

I Remark 3.14.7. Obviously, Lemma 3.14.4

and Theorem 3.14.6 are still valid for matrixv n valued symbols in Hom(Cm;Cm), a E L (R ) being elliptic of order V, i.e.

3.14. Reduction of Elliptic Singular Perturbations satisfying (3.14.13) with la(x,E,S) being defined as in (3.14.14) with

I

497

the norm in Cm and r(x,E,S), s(x,E,S) -V

8

'~(5) replaced by E-'~x(S)I~, where

Id E Hom(Cm;Cm) is the identity. A s a first application of the reduction procedure, we give here a

proof of the statement in Remark 3.13.17, based on the use of Lemma 3.14.4. Theorem 3.14.8.

L e t p E Lo(Rn ) , p being valued in Hom(Cm;Cm). L e t t h e principaZ symboZ pO(x,~S)of P(X.E,~) be

a hermitian non-negative m a t r k : p

0

2

v (x,n) E d'x

(X

'

n)

= p (X, n ) * 2 0 ,

and assume t h a t po E c ( m n x m n ) . Furthermore, assume t h a t w i t h some constant c > 0 t h e fo-dloming i n e q u a l i t i e s hold: (3.14.33) <x>~+'<ES>ID 6 (p(x,~,S)-p~(x,~S)) I 5 CE,

v Then one has for

PE =

161 5 n+l,

(X,E,S) E mnx

(O,Eo1

x

2.

)

stands

P(X,E,D~)= op(p):

where t h e c o n s t a n t c does not depend on

E

and u and vhere

(

,

for t h e i n n e r product i n L2 (nn). Proof. We give here the main steps in the proof of (3.14.34) without going ~

into detail. First, we notice, that on the ground of the same argument, as in the proof of Theorem 3.13.6, one can assume, without restriction of generality, that p(x,~.S) 5 p,(x,~S) and, thus, one has to show (3.14.34) only for P:

=

P~(X,ED)= op(po~Furthermore, since po(x,n) E C2(Rn x R n

)

, the

reduced symbol

0

0

P (x,S) Z Po(x,O), i.e. the reduced symbol P (x) is j u s t the multiplication 0 by the hermitian non-negative matrix P (x,O)OE 1o ( Rn ) . 0

Further, introduce the symbols:

Obviously, one has: r(x,E,S) * = r(x,E,S), s(x,E,S)* and

=

s(x,E,<), rs

=

sr = Id, rtId, s>Id,

3. Singular Perturbations on Smooth Manifolds without Bounday

498 0

0

r (x,S) = s (x,S) = Id, where, as usual, ro and so are the reduced symbols of r and s , respectively. Applying Lemma 3.14.4

(with j

r or j = s ) , or using an argument very

=

similar to the one in its proof, one finds with RE = r(x,E,D) = Op(r), SE = s(x,E,D) = Op(s):

REoSE = Id+EQ:, € = P~+EQ:, 0 (3.14.36) sE"pO

( sE ) * -sE =

SEoRE = Id+EQg,

: P

En:,

= R'~P~+EQ~, O E

(R€)*-R€ = E Q ~ , 0

where Po is the multiplication operator by Po(x), and Id, QS, 16j56, are 0

identity and (uniformly with respect to mappings in the corresponding spaces H

E

E

1) continuous linear

(O,E

0

( W n)

(s) ,E

.

Now, using (3.14.361, one finds:

2

.

Furthermore, let r(x,E,S) = (b(x,E,S)) wlth b = b* 2 Id the positive definite square root from r = r* t Id. Denote BE = b(x,E,D) = Op(b); one has also for BE (as a consequence of Lemma 3.14.4):

where again QE

J'

j = 7,8, are (uniformly with respect to

linear mappings in corresponding spaces H

(s)

,E

(Wn )

E)

continuous

.

Thus, using (3.14.37), (3.14.381, one finds: Re(PEu,u) = Re(BEoBE(P:U)

,U)

1

I

+ O ( E ) j u / 2( o ) , E =

0

0

=

Re(BE(PEu),(BE)*U) +

=

Re(Po(x) (BEu),BEu) +

0

o

since P (x) = (P0(x)) 0

*

O ( E ) \ / U I /2 ( 0 ) , E= O(E)

[

jul

[ 2( o ) , €

b -cEI / u l

I 2( 0 ) ,El

t 0.

In this argument, besides Lemma 3.14.4, also the following statement has been systematically used:

3.14. Reduction of Elliptic Singular Perturbations Let J~ = op(j), j E 0

(JE)*-JE = EQ',

j (x,S) E Id. Then QE

: H

(XIn)

( s ) ,E

respect to

E

E

+

L (O,O,v)(=n), (j(x,E,S))* = j(x,E,S)

H(s

(O,E

0

))

,s

2,S3-V),E

499

2 Id,

where J E is the L -adjoint of JE, and 2 (mn)is a family of (uniformly with

continuous linear mappings.

The proof of this statement is very similar to the one of Lemma 3.14.4

n

and will not be given here. Remark 3.14.9.

Obviously the statements in Lemma 3.14.4 and Theorem 3.14.6 carry over to m

the case, when Rn is replaced by a compact C

manifold M, dim M = n.

Indeed, let {(Ui,ai), lSi5NI be an atlas on M. i.e. {Ui'lZiQJ aimi)

C

and

mn . l5iSNI be a partition of unity subordinate to the covering

Let

{UiIlsisN. One can choose Ui, lSi
with def Qi,k,l

=

-

Op(Oij) oOp($ka)O O P ( Q ~ ) - ~ I ? ( $ ~ $ ~ $ ~ ~ ~ )

If

maps { u E H ( s ) ,E (M), supp u C U1 into (M), supp f C Uil. In the local coordinate system H(sl,S2-"2,S3),E

aikl

:

Each

QE

lrktl

vikl

+

m n , one can view Q?

lrkr1

as a singular perturbation in m n .

Similarly to (3.14.22), one has the following Fourier representation for Q? i,k,l'

where

It is readily seen that (3.14.23), (3.14.24) hold for Ki,k, since the argument in the proof of Lemma 3.14.4 for the kernel K given by (3.14.22) is still applicable when a' is replaced by $ a and j' is replaced by $.j. k

3. Singular Perturbations on Smooth Manifolds without Bounda y

500

Therefore, we conclude that (3.14.18) holds for each Q?

l,k,l

fore, also for

QE =

Op(j).Op(a)-Op(ja)

and, there-

considered as a singular perturbation

on M . Furthermore, defining locally r(x,~,E),s(x,E,E) by the same formulae (3.14.14) and using the local coordinates and a subordinate partition of

unity, one defines globally the corresponding factorizing r(x,s,D) and reducing s(x,E,D) singular perturbations such that (3.14.29)-(3.14.32) in corresponding spaces H uses only Lemma 3.14.4,

( s ) ,E

(M),

hold

given that the proof of Theorem 3.14.6

the latter being true with

mn

replaced by M .

Corollary 3.14.10.

If A€ E op

i s e l l i p t i c of order w

s'(M)

= (v1,v2,v3) and

t h e reduced operator

i s an (algebraic and topological) isomorphism uniformly w i t h respect t o E

E ( 0 . ~ ~ 1then , f o r zO s u f f i c i e n t l y small t h e family of l i n e a r mappings

AE

:

H

( s ),E

*

(')

H(s-V) ,E ( M ) ,

E

E (o,Eol,

i s an (algebraic and topological) isomorphism u n i f o m l y w i t h respect t o E

E ( 0 . ~ ~ 1as , well.

Indeed, the multiplication from the left by a reducing operator reduces A E to a regular perturbation of Ao

:

H

( s ),E(M)

SE

+ H(s1,s2-V2,S3) ,E

(M)

Actually, it suffices to assume that 'A

:

H

(MI

-f

L2(~)

v2

is an isomorphism in order to reach the same conclusion for 3 V s E R (see [Fr.-W, 1,3 1 ) . AE : H (M) + H(s-"), € ( M ) , ( S ) ,E

Example 3.14.11. Let B1 = {x E R

2

, 1x1

< 11, Rl = aB1 = {x

E

R 2 , 1x1 = 11. Consider the

following singularly perturbed boundary value problem:

where N is the inward unit normal at x' E Rl, and a(x'), g(x') are given X' smooth functions on R 1' Using the polar coordinates x

=

(

1x1 ,XI) = (r,B), one can represent

.

3.14. Reduction of Elliptic Singular Perturbations

50 1

any harmonic function u(x) in the disk B1 as the following Fourier series: (3.14.40)

uE(x) = (2Tr)-l

GE(k)r1k1exp(ik8),

Z

kEZ where u (k) are the Fourier coefficients (discrete Fourier transform) of u

(XI.

Substituting ( 3 . 1 4 . 4 0 )

on R l , one

into the boundary condition for u

finds for u (0) the following singularly perturbed equation on f i l :

since in the case considered -a/aN

X‘

a/ar.

=

Using the projection operator Il+ defined by ( 3 . 1 0 . 5 1 ) , one can rewrite ( 3 . 1 4 . 4 1 ) in the following form: (3.14.42)

where II-

(E~,(n+-n-)+a(e))u€(e) = g ( e ) , Id - II+ and, as usual, D

=

e

=

eE

fil,

-ia/ae.

As it has been shown in Example 3 . 1 0 . 1 5 ,

the operator EDe(n+-n-)+Id

is a singular perturbation in Op S ( o r o r l ) (R 1 ) whose principal symbol is 1 +15~1 . Hence, the singular perturbation

is a singular perturbation in Op S ( o f o ’ l ) (fi 1) , as well, whose principal symbol is: a0 (e,Ec) = a ( e ) + E / c I . Hence AE E Op Ia(e)+El5l

v e E

I

S(ofonl)

(ill) is elliptic of order (0,0,1)

iff

2 C < E with ~ some constant C > 0 , i.e. iff Re a ( 0 ) > 0 ,

fi1.

one can use ( 3 . 1 4 . 4 )

Assuming AE to be elliptic of order (0,0,1),

in

order to reduce ( 3 . 1 4 . 4 2 ) to a regular perturbation equation on R 1 , the a reduced symbol here being, evidently, :

=

a (8).

However, it is more convenient in this case to use the construction of a reducing operator indicated in Remark 3.14.3.

Actually, one can take

as a reducing operator here any singular perturbation whose principal symbol is: so(e,Ec)

= a ( e )( a ( e ) + E / c I )

-1

,

SE

E Op

S(oro‘-l)

(a1)

3 . Singular Perturbations on Smooth Manifolds without Boundary

502

and whose reduced symbol

s o F 1.

One of the possible definitions is the following one:

c

(sEv)( 8 ) = ( 2 n I - l

a ( e ) (a(e)+EikI)-' G(kjexp(ik8)

=

kEZ = v(B)-E(2n)

-1

X Ik/(cx(e)+E/kl

G(k)exp(ike)

kEZ where G(k) are the Fourier coefficients of v. Anyhow, since the reduced operator Ao (which is the multiplication by

a ( @ )with Re a ( e ) > 0) is invertible, one gets the conclusion (see Corollary 3 . 1 4 . 1 2 ) v

=

(0,0,1),

that A' given by ( 3 . 1 4 . 4 3 1 ,

AE

:H

(S),E

is an isomorphism uniformly with respect to

(0 ) + H E

1 (S-V),E E (O,EO1,

(a1) I

provided that E~ is sufficiently small. m

If g E Cm(R1), then the solution u ( B ) of ( 3 . 1 4 . 4 2 ) with respect ot

E)

is in C (fl,)

(uniformly

and, moreover, it can be represented by the following

asymptotically convergent series:

so that, for instance,

One can consider for a harmonic function u in the unit disk B

1

another

boundary condition on R1 of the form: (3.14.44)

where II,

B(8)

(-a/ak,+a(e)-EB(e)a =

2

/ae 2 ) u E ( e ) =

g(e),

E

E ( 0 . ~ ~ 1

( E 1 ( 8 ) , I 1 2 ( 8 ) ) is a given smooth vector field on R1,

, g(B)

and a ( B ) ,

are given smooth functions.

Using the same argument, one can easily show that under the conditions:

a ( B ) > 0 , B ( B ) > 0, II ( 8 ) case+ !L2(8) sine c 0 , V 8 E R 1 , there exists a 1 uniquely defined harmonic function u satisfying boundary condition ( 3 . 1 4 . 4 4 ) , V

E

E (O,E

0

1, provided that

E~

is sufficiently small. In the latter case

the corresponding singular perturbation on the boundary turns out to be elliptic of order ( O , l , l ) EB(e)c

2

-(.tl(e)

cos e

+

and to have the following principal symbol:

i2(e)

sine) 151.

Furthermore, the reduced boundary condition

3.14. Reduction of Elliptic Singular Perturbations

503

for a harmonic function u in B1, defines an isomorphism from H (B1) onto s2 (R if a ( 9 ) > 0 (see [Fr.-W.,1,31), which means that reduced equation Hs2-3/2 1 (3.14.45) is uniquely solvable in Hs2-+(R1) for each g E Hs2-3/2 (R1). Hence, under the assumptions above on R e ,

(R

H(sl,s2-3/2,s3) 1 small.

V

),

B ( e ) , a ( B ) , the perturbed equation

(R1) (s1,s2-t,s3+1)

(3.14.44) is solvable in H

for each

E ( 0 . ~ ~ 1provided , that

E

E~

is sufficiently

It is obvious that a convenient choice of the spaces H

( s ) ,E

( B 1 ) for

the harmonic functions u , satisfying the boundary condition in (3.14.39) (respectively,boundary condition (3.14.44)) is the one, where s = (s1,s2,s3) satisfies the conditions s2 > f, s2+s3 > 3/2 (respectively,s2 > 3/2, s2+s3 > 3/2) (see also Theorem 2.2.21, since this choice of s guarantees

existence of traces of u and its first transversal derivatives on R 1 = aB1 uniformly with respect to E). Consider for a harmonic function uE in B1 the following boundary condition on a. = aB 1'. (3.14.46) (-Ea/aNe+i)(exp(iKe)n++n-)uE(9)= g(e), where TI+ and II- = Id-II'

are the same, as above, and

Since the trace operator (-Ea/aN +l)u(r,B)

e

singular perturbation AE E Op (AEv)( 9 )

=

(2rrI-l

S(o'otl)

K

E Z.

I r=l defines an elliptic

(R,),

(I+Elkj);(k)exp(ikB)

C

kEZ (see Example 3.10.15),

equation (3.14.46) can be rewritten in the equivalent

form:

where gE(e) = AElg(9)

def =

C

1

(l+Ejkj)- g(k)exp(ik9).

kEZ The operator on the left hand side of (3.14.47) (well known in the literature as Noether's example of an operator with index), has a nontrivial kernel of dimension dimension

K

iff

K

> 0.

-K

iff

K

< 0 and a non-trivial cokernel of

In both cases its index (difference between the

dimension of its kernel and the one of its cokernel) is

-K.

The reduction of

3. Singular Perturbations on Smooth Manifolds without Boundary

504

(3.14.46) to (3.14.47) shows that the singularly perturbed operator on

the left-hand side of (3.14.46) has the same index, VE. The same is true, when one considers the boundary condition for the harmonic function uE in B1 of the form:

(3.14.48) (-Ea(e)a/aNe+B(e))(exp(iKe)n++K-)u ( 8 ) = 9(e),

provided that a ( e ) ,

B(e)

are smooth functions satisfying the ellipticity

condition:

where C is some positive constant. Indeed, if (3.14.49) is satisfied, then the singular perturbation in Op S ( o r o r l(Q,) ) associated with the trace operator defined by (3.14.48) on harmonic functions uE, is elliptic of order (O,O,l), its principal symbol being

with H ( C ) the Heaviside function. Since the singular perturbation A

(-Ea(e)a/aNe+B(e))uE(r,e) of order (0,0,1)

jr=l

associated with the trace operator

on harmonic functions u in B ~ is , elliptic

and has an invertible reduced operator (which is just the

multiplication by B ( 8 ) # 0, V 0 E R1), the inverse A-’ exists and is an elliptic singular perturbation in Op S ( o c o r - l(al), ) V that

E~

E

E

( 0 . ~ ~ 1provided ,

is sufficiently small (see Corollary 3.14.12).

Thus, the index of the singular perturbation defined by (3.14.48) is the same as the one for the reduced operator, the latter being again that of Noether on the left hand side of (3.14.47). It is readily seen that (3.14.17) can be reformulated in the following equivalent way: find a function $ + ( z ) analytic in B1 = { z E C, I z I < 1 1 and a function $ - ( z ) analytic in CB1 = { z E C, / z / > 1 ) such that exp(i~e)~+(exp(iB))+$-(exp(ie))= g ( € J ) ,

which is the Riemann-Hilbert problem for the circle Q l [:4uskh.,

1

C

C (seefor instance

I,.

Example 3.14.12. We come back to the singular perturbation from Example 3.13.26:

3.14. Reduction of Elliptic Singular Perturbations (3.14.50) a(x,s,D

)

505

-~A~-, x E R n .

=

With +(x) defined by (3.13.124) and b(x,E,DX) defined by (3.13.126), one reduces the equation (3.14.51) a(x,E,D )uE(x) = fE(x) to the following one: (3.14.52) b(x,E,D )vE(x) = gE(X),

X

E Rn,

where

-1 -1 (3.14.53) v (x) = u (x)exp(E +(x)), gE(x) = Ef€(x)exp(~ Q(x)). Assuming that g v

E H E

(s1,s2,s3+2) ,E

E

E H

( s ) ,E

( R n)

,

one gets the conclusion that

(R"), since b(x,E,S) E !-(o'of2)(Rn) is elliptic of

order ( 0 . 0 . 2 ) . 0

Furthermore, the reduced operator b (x,Dx) is just the multiplication =f by q ( x ) de jVX+(x) 2 qo > 0 , V x E R n , q(x) = l$m\2+qo(x) with

l2

q' E S(Rn). Thus, using (3.14.14), one defines a reducing operator

E s(x,E,D~)

L (o'of-2)(Rn) and gets the conclusion that (3.14.52) has a well--

( 1 ~ ~for ) any g E H (En). (s1,s2,s3+2), E E ( S ) ,E One can also use as a reduclng operator the singular perturbation

defined solution v E H E

s (x,ED~) with the symbol: 0

. (3.14.54) so(x,~S)= q(x) ( 1 ~ 521+l
E

+ so

(x,€DX) is a family of integral

3 . Singular Perturbations on Smooth Manifolds without Bounda y

506 where

P(X) and A.

-2

=

2' (IVXtJ(X) I2+(i)/g(x)I 1 2 ,

(E,x) is given by (3.2.23) (it can be also expressed in terms of

Hankel's functions). For n

=

3, one finds using (3.2.24):

2 2 (3.14.56) S0(x,y) = (4dyl)-l(p(x)) /VxO(x)1 exp(-(t)(P(x)lyl+)) 0

Since the reduced operator b (x,D

1 Vx$ (x)1 2, the

)

reduction of the singul:r

is just the multiplication by perturbation given by (3.14.50)

to a regular one, allows one to solve (3.14.51) asymptotically, i.e. to represent u (x) as an asymptotically convergent series:

with some constant C > 0. The following matrix singular perturbation a(x,E,D

),

plays a special role in fluid dynamics with hydromagnetic effects; here H(x) : R3

+ It3

is the magnetic field,

E

is the magnetic viscosity which

is supposed to be a small parameter and v(x)

:

R3

+

R3 is the velocity

field of the fluid. Besides, for both v(x) and u(x) holds: div v(x) = X 3 = divxu(x) = 0. The cross x stands for the vectorial product in R

, VxX

being the usual curl operator on functions w [Lan.-Lif. , 2

R3

-+

R3 (see for instance,

I).

Since div H(x) X

:

=

div v(x)

=

0, one finds easily:

X(v(x)XH(x)) =

vX

= ~(x)-H(~), so that the principal symbol a (x,E,D ) of 0 ,S>)Id with a ( x , ~ , D ~ ) ~ g i vby e n (3.14.56) is: ao(x,~,c)= -(~1<1~+i
Id the identity in Hom(C3;C3).

In some situations (for instance, in some problems of cosmic fluid dynamics with hydromagnetical effects) the influence of the magnetic field on the hydrodynamical characteristics of the stream are very weak,

3.14. Reduction of Elliptic Singular Perturbations

507

so that one can determine the velocity field v(x) independently of ff(x). Besides, v(x) turns out to be practically constant when 1x1 is sufficiently large. Assuming that v(x) satisfies the conditions imposed on b(x) in Example 3.13.26, one can use the singular perturbation s 0 (x,E,D ) defined by (3.14.54). (3.14.56), in order to reduce the singular perturbation given by (3.14.57) to a regular one, i.e. to solve the corresponding equation a(x,E,D )H(x) = f(x) asymptotically when X

E +

0 (see [Br.-Fr., 1

11,

where an asymptotic analysis of this hydromagnetical problem has been done and used for the interpretation of the solar wind phenomenon around the earth). Example 3.14.13. The following singular perturbation appears as a suitable mathematical description of a semiconductor device in the situation, when it can be considered as a one-dimensional stationary pn-junction (see

[Mo, 1

1):

(3.14.58)

-E

2 $"+n-p

f(x), -n"+(n$')' = g(n,p),-p"-(p$')' = g(n,p),

=

where $,n,p are unknown functions (normalized potential and electron and hole densities), upper prime stands for the derivative with respect to x E U 5 R and f(x) , g(n,p) are given functions, g(n,p) =

-1

2

(4)(co-np)(n+p+2co) ,

with co > 0 a given constant, E being a small dimensionless parameter. Introducing new unknown functions u

c0n exp(-ji), v = cop exp($), one can rewrite (3.14.58) in the following form: -E

(3.14.59)

2

$"+c (u exp($) - v exp(-$)) 0

=

=

f(x)

-(e*u')' = G($,u,v) -(e-*vI)*

=

Notice that u E v

~($,u,v)

5

1 is a solution of (3.14.60) if $ 1s a solution

of the following singularly perturbed equation with a given second member: -E

2

*" +

2c

sh $ = f(x),

3. Singular Perturbations on Smooth Manifolds without Boundary

508

whose reduced equation has the following solution ji0 (x) = sh-l(f (x)/2c0). Consider the linearization of (3.14.59) at the point (IJJo(x),1,l), whose principal symbol is the following matrix:

(3.14.60) aO(x,E,6)=

0

where q(x) = 2c0 ch(+O(x)), the reduced principal symbol a0 (x,S) being just a0(x,0,5). It is readily seen that (3.14.60) is elliptic and the singular perturbation with the symbol s (x,E,~), 0

0

0 can be used for reducing the linearization of (3.14.59) at the point ji

=

I$~(X),u = 1, v

=

1, to a regular perturbation when x E R and w

f E C i ( R ) or when x E TI, f E C ( T

1

)

(periodicity conditions).

We shall come back later on to (3.14.58) while we will be considering boundary value problems for elliptic singular perturbations. Let M be a smooth manifold withoutboundary and let a(x,~,D)EOp L V ( M ) be elliptic of order u. Assume that it has a

( M ) for some u 2 E R . Then

(M) onto Ha -v

Hu

2

2

0

reduced operator a (x,D) which maps isomorphically

2

0

a (x,D) : H

(M)

s2

-f

H

s

-v

2

(M)

2

is an isomorphism for V s2 E R . Indeed, one gets this conclusion by using the classical elliptic regularity results for s 2 >

f12,

the duality argument for s2 =< - 0 +v

and

.

interpolation between -u2+v2 and a2 , if necessary (see [Fr -W. ,1,31 for more detail). Let s(x,E,D) be a reducing singular perturbation for a(x,~,D) (as defined above). The singular perturbation q(x,E,D), def 0 q(x,~,D) = 1-(a (x,D))-'s(x,~,D)a(x,E,D),

3.14. Reduction of Elliptic Singular Perturbations

is well defined, provided that

E~

is sufficiently small, and, moreover, as

a consequence of Theorem 3.14.6 and Remark 3.14.9, one has for the norm of the singular perturbation (3.14.61):

where the constant C does not depend on

E (O,E~].

E

Therefore, one can represent the inverse singular perturbation -1

(a(x,E,D))

by the following Neumann’s series: k

O

-1

Z (q(x,E,D)) (a (x.D))

(3.14.62) (a(x,E,D))-l =

s(x,E,D),

k>O C ( q ( x , ~ , D ) being )~ convergent in the Banach space k>O (M)) of all continuous linear mappings in H (M) L(H(s) ,E(M);H (s), E ( s ) .E uniformly with respect to E E (O,E 1.

the series

0

Hence, for each f E H

(s-V)

formula for the solution u E H (3.14.63) u =

,E

(M), one has the following asymptotic

( s ) ,E

(M) of the equation a(x,s,D)u = f:

k O N (q(x,C,D)) (a (x,D))-lS(X,E,D)f+ E g,

Z

O5k
E

E (O,E

If f E Cm(M) uniformly with respect to

0

1

E

with some constant C > 0.

E ( O , E ~ ] and symbols

a(x,~,c)admits an asymptotic expansion: a(x,E,5)

-

z

E

k k a (x,S)

k>O

then (3.14.63) can be simplified. We give here an example to illustrate this situation. Example 3.14.14. On the unit circle R 1 with 8 E [ - i ~ , n ) as a coordinate consider the following singular perturbation a(e,E,D,),

m

where q E C (R1) and satisfies the ellipticity Re(q(0))L > 0, tl 0 E R1. We choose the branch q(0)

=

509

((q(0))2)1’2 such that Re q(0) > O , V 8 E R l .

3 . Singular Perturbations on Smooth Manifolds without Bounda y

510

The reduced operator ao being identity, the singular perturbation defined by (3.14.64) is a family of isomorphic linear mappings uniformly with respect to that E~ given

6

E

( O , E ~ ] of H

(R

(S),E

1

onto itself, V s E R 3 , provided

)

is sufficiently small (if q(8) > 0, V 8 E SZ

E~

> 0 uniformly with respect to

E

1

then it holds for any

E (O,E~]).

One can use a partition of unity and the definitlon of singular perturbations on a smooth manifold without boundary for constructing a reducing operator for a(e,~,D) given by (3.14.64) (see Remark 3.14.9).

e

However, given that 8 E [ - r , n I isa coordinateonR1, one can give a different definition of a left reducing operator for a(B,E,DA) without using a partition of unity. Indeed, define the kernel s(@,E,e-e')

=

as follows:

S(e,E,e-e')

(2n)-1 c (1+E2(q(e))2k2)-1exp(ik('t3-8')). kEZ

Using the Poisson formula (1.2.180), one can represent S ( O , E , f 3 - 8 ' )

in

the form: (3.14.65)

S(e,E,e-e')

=

(2€q(e))-l I: exp(-1e-e'-2nk(/(Eq(e))), kEZ

given that with Re q ( 8 ) > 0 one has:

Define the singular perturbation

SE =

s ( ~ , ED

e)

as an integral

operator with the kernel S: def (3.14.66) (S(8,E,~e)g) (8) = J S(0,E,e-ei)g(e')de, v g E

Crn(Ql).

12 1

Since the principal symbol s

0

0

E s ( o , o , - 2 ) (R 1 ) , one has:

( 8 , 6 , < ) of

s(@,c,Da

is: s,=

"

2 2 2 -1 ( l + ~ (q(8)) ) ,

s(S,E,D ) E Op S(ogo*-2) (nl). Furthermore,

e

SE = s(B,E,De)has identity as its reduced operator. Therefore, as a

consequence of Theorem 3.14.6 and Remark 3.14.9, SE = s(B,E,De ) defined by (3.14.651, (3.14.66), is a reducing operator for a(e,6,De) given by (3.14.64) (in fact, it is a quasiinverse operator for a(B,E,Dg)),i.e. Id+ q (~B ) ,E,D S ( ~ , E , D ~ ) ~ ~ ( ~=~ , E,D where

e) ,

3.14. Reduction of Elliptic Singular Perturbations

51 1

is a family of continuous linear mappings uniformly with respect to E

E

(O,E~], V s

E

R 3 , provided that

E~

is sufficiently small.

Besides, (a(e,E,D,)) -1 =

X (Id-s(e,E,De)oa(B,E,De))k .,s(8,E,De), ktO

the series on the right-hand side of the last formula being convergent ( Q ) ) of all continuous linear 1 mappings in H ( Q ) uniformly with respect to E E (O,E 1 , V s E R3 (s),E 1 0 Furthermore, if fE E C m ( Q ) (uniformly with respect to E E (O,E 1 ) 1 0

in the Banach algebra L ( H

( s ) , E ( Q l ) ' H ( s ) ,E

.

then the solution u (8) of the equation ~(@,E, D (8) = fE(e), e~)U

E

E

(O,E

01, m

is well defined and belongs to a bounded set in C ( i l l ) , V provided that

E~

E

E

(0.~~1,

is sufficiently small (see Remark 3 . 1 2 . 2 3 ) .

Moreover, if f (8) admits an asymptotic expansion,

then so it is also for uE(8), (3.14.67)

(8)

-

c

E

k

uk(e), uk E

m

c (a1),

kZO

where the coefficients uk can be defined by recursion. For instance with f (8)

5

f (8), one finds easily: 0

Besides, the asymptotic convergence in ( 3 . 1 4 . 6 7 ) , space H

( 3 ) ,E

(R

1

(3.14.68)

in each

is an immediate consequence of the a priori estimate

with a constant C which might depend only on s and q, the latter resulting from Theorem 3.14.6 and Remark 3.14.9. A

special attention deserves the case when f(8) = fs(8)+g(8) where

3 . Singular Perturbations on Smooth Manifolds without Boundary

512 m

g E C (Q1) and f (8) is the singular part of f, having a finite number of points in R1 as its singular support. For instance, consider the case, when f (8) = sgn 8, 8 E Ql. Since sgn 6 E H

(0),E = sgn 8:

au

(Q ) ,

1

one has for the solution u (6) of the equation

Using the reducing operator s ( € l , ~ , D ) defined by (3.14.65), (3.14.66), 6 one can represent u (6) in the form:

u where on

E

(el

= s ( e , E , ~ )f (e)+EVE(e), 6 s

I

/v C E 1 1 (0,0,2) ,E E (O,eo1.

C

with some constant

C >

0, which does not depend

Using (3.14.65), (3.14.66), one finds:

, def (3.14.69) ~ ( ~ , e= l

s ( e , E , ~ )f

6

= (2Eq(e))-l c

s

sgn

(6) =

el

exp(-/e-e'-2nk//(Eq(e)))de',

e E nl.

kEZ Ql

Computing integrals on the right-hand side of (3.14.69), one finds explicitly the first term u ( E , ~ ) in the asymptotic expansion u (6) the following formula:

the point

e

= IT

being identified with 8 =

-IT.

Hence, also for u(E,e) the singular support consists of the same two points as a consequence of the pseudolocality property of singular perturbations (see Theorem 3.5.4). Now the construction of a right reducing operator for a given elliptic singular perturbation will be given and some applications will be considered.

3.14. Reduction of Elliptic Singular Perturbations

513

We are going to show that for a given elliptic singular perturbation a(x,E,D 1 E Op Sv(M) the left factorizing and reducing operators r(x,E,D and s(x,E,D

)

defined by their symbols ( 3 . 1 4 . 1 4 ) ,

)

are also right factorizing

and reducing operators, respectively. The essential part here is the following analogue of Lemma 3 . 1 4 . 4 : Lemma 3 . 1 4 . 1 5 .

Let j(x,~,S)E Lv(Rn) w i t h w 0

j (x,~) E I,

v (x,~) E

L e t a(x,c,S) E L ! ' ( R ~ ) .

R~

x

=

( 0 , 0 , v 3 ) and assume t h a t t h e reduced symbol

mn. Then one has:

i . e . f o r each s E m3 t h e folZowing i n e q u a l i t y h o l d s :

w i t h a c o n s t a n t c > 0 , which does n o t depend on

E

and u.

Proof. First a transparent heuristic argument will be presented, which explains why ( 3 . 1 4 . 7 0 1 ,

(3.14.71)

should be true.

Using Theorem 3 . 7 . 6 , one finds:

5

as a consequence of Definition 3 . 1 2 . 3 . Thus,

1, one has:

3. Singular Perturbations on Smooth Manifolds without Bounda y

514

given t h a t t h e o r d e r of t h e symbol on t h e l e f t hand s i d e of t h e l a s t

< v+p, v

inclusion i s v+p-(o,/crj , o ) + ( o , I , I )

IcrI > 0.

QF,

Besides, t h e amplitude of t h e o p e r a t o r which i s t h e k e r n e l of t h e o p e r a t o r

QE,

(see Definition 3.7.1),

coincides with t h e k e r n e l of t h e

operator

t h e o r d e r of t h i s amplitude being a t most v+p. Now a rigorous proof of t h e lemma w i l l be given, which i s very s i m i l a r t o t h e one of Lemma 3.14.4.

Without r e s t r i c t i o n of g e n e r a l i t y , one can

assume t h a t v1 = p l = 0. L e t a'(X,E,S)

def = a(x,E,<)-a,(~,S), j'(x,E,S) = j ( x , c , t ) - j m ( ~ , S ) ,

where a ' , a m and j ' , j m a r e as required i n D e f i n i t i o n 3.12.1 of c l a s s e s Lv(iRn).

Since j m does not depend on x , one has:

Therefore, it s u f f i c e s t o show t h a t (3.14.72) O p ( a ' ) . O p ( j ' ) - O p ( a ' j ' )

= EQ€. 2

( 3.14.73 ) Op (arn) .Op ( j ' ) -0p (amj

= €QE 3'

where

QE

I

'1

E Op LV+'(Xn), j = 2,3.

First,

(3.14.72) w i l l be proved. One has:

where

and t h e h a t

s t a n d s f o r t h e Fourier transform with r e s p e c t t o x

E Rn.

By Schur's lemma on boundedness of i n t e g r a l o p e r a t o r s i n L 2 ( s e e Lemma 3 . 1 2 . 8 ) , i n o r d e r t o p r o v e t h e lemma, it s u f f i c e s t o show t h a t f o r each s = (0,s2,s3)

E

n3 s -p

(3.14.75) f n < q > iR

onehas: -v

s -p

2<En2

-v

-s 3

-s 2<ET>

v

' IE

3/K(ll,E,T)

iRn,

v

E

Idr? 6 CE,

E

(o,Eolp

3.14. Reduction of Elliptic Singular Perturbations

-v

s -p (3.14.76) In<'l> R

s -p

-v

-s

3

2<Erl>

515

-S

31K(Q,E,T) /dT 5 CE.

2<ET>

v

rl

E mn, v

E

E

(O,Eo1,

where the constant C > 0 might depend only on s2,s3,n,v2,v3 and the symbols a and j, but not on As

0 j (x,S)

E,T

and rl.

a consequence of Definition 3.12.3

and the assumption:

1, one finds:

Let b(x,E,S) = b,(E,<)+b'(x,E,S)

with bm and b' as in Definition 3.12.1,

where Ck > 0 does not depend on <,E,x. The same argument shows that D a j ' (x,E,S) satisfy the inequalities (3.14.77) with different constants which may depend on the multi-index a.

Hence, one gets:

with perhaps different constants Ck > 0. Now the mean value theorem and the inequalities in Definition 3.12.1

of classes Lp(Rn ) yield:

where we have used Peetre's inequality: <S>p-p

5

2'pl<<-T>lp'

,

V < , T

E lRn,

v p E R.

The same inequalities being true for Daa' (x,€,S) with some constants 'k,a'

one gets the conclusion

516

3 . Singular Perturbations on Smooth Manifolds without Bounda y Obviously, the inequality

is satisfied as well. Now we have all the elements necessary to prove (3.14.75). Using (3.14.74), (3.14.78), (3.14.79) and the inequality

< n - O 2 2<S-r>, one can estimate the integral on the left hand side of (3.14.75) in the following fashion:

where ml = 1+1u2-11+/u31+js21+1s31+1u 2 +v2 1 and m2 =

l u 2+v 2 I + / s 2 ] + / s 3 / .

We choose k to be sufficiently large in order to guarantee the convergence of the integral on the right hand side of (3.14.81), and that

.

yields (3.14.75)

The same argument can be used in order to prove (3.14.76). Hence, we have proved (3.14.72). In order to show (3.14.73), one will have to estimate the kernel

-

def K _ ( ~ , E , T ) = j'(T-n,&,n) (arn(~,T)-arn(&,n)). Using (3.14.80) instead of (3.14.79), one gets (3.14.75), (3.14.76) with

K replaced by Km. This yields (3.14.73). We have shown that the singular perturbations Q ? , j = 2,3, in (3.14.72),(3.14.73) have atmostorder I that QE = Q:+Q~ E Op LU+' (iRn1, as well.

v+p,

so

Now, using exactly the same argument, as in the proof of Theorem 3.14.6, one shows its analogue with r(x,&,D), s(x,E,D) as right factorizing and reducing operators:

the same kind of inequalities being also true for elliptic singular perturbations in S'(M)

with M a smooth manifold without boundary.

3.14. Reduction of Elliptic Singular Perturbations

517

Corollary 3.14.16.

L e t M be a smooth manifold w i t h o u t boundary and l e t a E o p s'(M) of order v

=

f a c t , a0

H

be e l l i p t i c

(v1,v2,v3)and a s s m e t h a t t h e reduced operator ( 0 '"2 , O ) (M! maps isomorphicaZZy M (M) o n t o L~(M), so t h a t , i n ao E op s V-

:

-v

( M ) + HS

s2

2

(M)

is an isomo$icm

f o r any giuen

s2 E

w.

2

Consider t h e equation a(x,~,D)u(x) = f (x) w i t h f E H

(s-v), E

(M)

giuen.

Then U(X) can be represented i n t h e form: 0

(3.14.84) u(x) = S(x,E,D) (a (x,D))-'f(x)

1 IvI 1 ( s ) , E

where

depend on

E;X

+ EV(X),

w i t h some c o n s t a n t c > 0 , which does n o t

and f.

Indeed, using (3.14.82) and the corresponding version of (3.14.30),

(3.14.31) on a smooth manifold (see Remark 3.14.9), one gets immediately (3.14.84). Example 3.14.17. Let Tn be the n-dimensional torus, Tn = Rn/Zn, with the global coordinates m n x . E [0,1], 1 5 j 6 n. Let q(x) > 0, q E C ( T ) be given, and consider the 7 equation on T ~ :

(3.14.85)

(E

2 2 A -A+q(x))u'(x)

=

f(x),

where f E L2(Tn) is given and A is the Laplace operator on Tn. Using (3.14.84), one can represent the solution u(x) of (3.14.85) in the form:

where s(x,E,D) is a right reducing operator for the singular perturbation on the left hand side of (3.14.15) and

v

=

(0,2,2)

I / v s I I (v),E

I

, with

5 CI/fl

and a constant C > 0 which does not depend on

L (T

E,U

and f.

Moreover, as a right reducing operator, one can choose the one, which is globally defined as follows:

where S(E,x) =

Z kEZn

2 2

(1+4rr E [kI2)-l exp(2rri).

3. Singular Perturbations on Smooth Manifolds without Bounda y

518

Using again the Poisson formula ( 1 . 2 . 1 W ) , S(E,X) =

A.

Z

-2

one finds:

(E,x-k),

kEZn where A.

-2

-1

(€,XI = FS,x(1+c252)-1

is given by ( 3 . 2 . 2 3 )

F o r n = 3 , one finds using ( 3 . 2 . 2 4 ) :

and for the solution u of ( 3 . 1 4 . 8 5 )

one gets the asymptotic formula:

0

where (lu ) (x) is the periodic extension of uo with a constant 5ClIfll 2 L (T )

6,U.f.

c

:

Tn

-f

C onto Rn and

> 0, which does not depend on

Example 3 . 1 4 . 1 8 . Let U aU

C

R2 be a bounded convex domain, whose boundary aU is a Cm-curve,

{x=y(s), 0 5

=

s 5

L}, the parameter s being the arc length on aU

between a fixedpint xo = y(0) and the current point y(s). Let u E ( x ) be a harmonic function in U, which satisfies the following boundary condition on

au

(3.14.87)-Eau~(x)/aNs+a(s)UE(X) = g'(s),

where N

x=y(s), 0 5 s 5 L,

is the unit inward normal to aU at x = x(s), a ( s ) > 0 ,

a ( 0 ) = a(L) and the periodic extension of a

:

v

S

E [o,L],

[O,L] -+ R+ Onto R iS

m

supposed to be in

C (R).

We shall also assume that gE E H ( o ,s2-f ,s3-l) s2+s3> 3/Z

sought in H

caul with

s2 > f ,

(3), E

will be

(U), so that the trace on the left hand side of ( 3 . 1 4 . 8 7 )

is well-defined, V

E

>

0 (see Theorem 2 . 2 . 3 ) .

Denote by P the following integral operator: (Pa)(x) where (3.14.88)

,€

and the harmonic functions uE(x) satisfying ( 3 . 1 4 . 8 7 )

P(s,x-Y

-2

3.14. Reduction of Elliptic Singular Perturbations

519

Introducing

with

T

=

dy/ds the tangential unit vector to aU at y(s), one can rewrite

P(s,x-y(s)) in the form: P(s,x-y(s)) = F-lI~x

(3.14.89)

I I)

exp ( -P 5

1

the right hand side of (3.14.89)

#

being well-defined, since p > 0 , V x E

i,

x#y(s), as a consequence of the convexity of U. Thus, one finds for x

p(s',s)

> 0,

for

s'

+

y(s'),

# s.

Notice that 2

x'(s',s) =s'-s+(s'-s) a(s',s), p ( s ' , s ) =tk(s',s)( s ' - s ) ~k(s',s) , > 0, s'

where k(s,s) = k ( s ) b 0 is the curvature of m

k(s',s) being C

au

#

s,

at y(s), a(s',s) and

in ( s ' , s ) .

Further, since y(0) = y(L), one can consider the operator Q,

with the distributional kernel Q(s,s'-s) defined by (3.14.901, operator acting on functions $ defined on the circle R R , R

=

as an L/2r

parametrized by s E [O,LI. Let {6k}15ksm be a covering of RR by intervals Ak of length 6 sufficiently small and let {

x

~ be} a subordinate ~ ~ ~ partition ~ ~ of unity.

Then one has:

where (Q$

k

) (s')

is the operator of the form:

3 . Singular Perturbations on Smooth Manifolds without Boundary

520

f f qk(s ' , s , S ')exp(i ( s ' - s ) 5 ' ) $k ( s ')dE'ds

(Qa,) ( s ' ) = (2r)-'

R3R

with the amplitude q given by the formula: k (-.14.91)qk(s',s,€,') =

as, ,NS> 1 5

1 exp (i( s ' - s 1 2 a ( s ' ,s ) 5

I-

(f )k( s

I ,

s ) ( s '- s )

It is readily seen that with any cut-off function $ ( S ' ) ,

jc'l

2

15

$(<')

]). 5

0 for

1 for 15'1 2 26, $ E C m ( R 5 , 1 , the amplitude JI(c')qk(s',s,C')

2 6, $ ( 6 ' )

satisfies the conditions of Definition 3.7.1 with v = (O,l,O), i.e. the operator Q with the distributional kernel Q(s,s'-s) defined by (3.14.90) is a singular perturbation in Op qO(t',S')

=

(Or'

S

1r

(62,)

whose principal symbol

o

1511.

Therefore, the operator (3.14.92) (a(S',E,Ds,)@) (S')

def =

is a singular perturbation in Op

(-Ea/aNs,+a(S'))( P o ) (S') S(orofl)

1 ro

(62

R

whose principal symbol

ao(s',E,5') is: (3.14.93) a0(s',€,5') = ~ ( S ' / + a ( s ' ) . Since a(s') > 0 , V s ' E [ O , L J ,

singular perturbation (3.14.92) is

elliptic of order (O,O,l), its reduced operator being just the multiplication by a ( s ' ) . Define a reducing operator as a singular perturbation b in (O,O,-l)

OP

s1,o

(62,)

given by the formula

(3.14.94) (b(s',s,Ds,)$)( s ' ) =

(2w)-'

Z

1
where

Ok

=

I

def =

( l+E

(a(s'))-l<<'>)-lexp(i ( s ' - s ) 5 ' ) $,(s)dg'ds

R R

xk$.

The singular perturbation (3.14.94) is at the same time a left and a right reducing operator for a(s',&.D

,)

given by (3.14.92).

One can also use as a reducing (left and right) operator in the case considered.the integral operator with the kernel (3.14.95)

S (E ,S

' , S ) de=f a ( S ' ) (TE)-1

m

(1+<

) -'Cos

(a(s

) E - l ( S' - 5 )

0

which is the inverse Fourier transform of

( 1+

(E/a( s ' ) ) I ,€ '

1)

:

5 I ) d5

3.14. Reduction of EUiptic Singular Perturbations

Seeking a harmonic function u'(x) the form uE

in U satisfying (3.14.87) on aU, in

with P defined as above, one gets for

= PQE

52 1

QE

the following

singularly perturbed equation on aU: (3.14.96) (a(s',E,Ds,)QE)( s ' ) = gE(s') with gE E H

(0,s2-f,s3-l),E

(au) given.

A s a consequence of Theorem 3.14.6 and Remark 3.14.9, equation

(3.14.95) has a well-defined solution

QE

consequence of Theorem 2.2.3, u E - PQE

H(0,s2+f,s3),E H(O,S2,S3),E

(au).

AS a

(u) for s 2 > f ,

s2+s3 > f . If g E Cm(aU) then to E

(O,E~], V

E~

<

+'

-,

E cm(au) and uE E

uE being

Cm(fi)

uniformly with respect

well-defined, as a consequence of

Theorem 3.12.12 and Remark 3.12.23. 0

Let u (x) be the harmonic function in U which takes the value g ( y ( s )) y ( s ) E au, on

au.

,

Rewriting (3.14.87) in the form:

(-Ea/aNs+a(s))uE(x)-u0 (x)

=

0, for x = Y(s)

E au

and writing the formal asymptotic series

one gets the following formal asymptotic expansion for uE(x),x uE(x)

-

(a(S))-l

c

=

y(S) E aU:

Ek((a(S))-la/aNS) k O (x), x=y(s).

k>O Denote by PD the operator which associates with a given function f on aU the harmonic function in U which takes the value f on aU, so that

The asymptotic series on the right-hand side of the last formula is asymptotically convergent ot uE (x). Indeed, introducing

3. Singular Perturbations on Smooth Manifolds without Bounda y

522

one f i n d s e a s i l y t h a t t h e d i f f e r e n c e 6

v P

def 6 = u

-

z

k k

E U ( X )

O
i s t h e harmonic f u n c t i o n i n U which s a t i s f i e s on aU t h e following boundary condition:

Thus, a s it has been y e t shown, one has f o r vE t h e following e s t i m a t e :

Since up E

m -

c (u),

p = 0.1,

...,

for g

E cm(au)

(classical result for

e l l i p t i c boundary value problems without small p a r a m e t e r ) , one g e t s t h e conclusion t h a t vE

E

E ( 0 , s 0 1 and, 5 CsP+l, V p = 1,2 where

C m ( f i ) uniformly with r e s p e c t t o E

moreover, f o r each m t 0 holds:

1 I vE1 I

t h e constant C may depend on m and p.

,...,

Cm(3

3ne can consider t h e problem of d e f i n i n g a harmonic f u n c t i o n uE i n U, which s a t i s f i e s t h e boundary condition: (3.14.93

- a u E / a i s + a ( s ) u 6 ( x ) - E B o2au/as 2

=

E

( s ) , x = y ( ~ E)

au

where Is i s a smooth u n i t v e c t o r f i e l d on a U , > 0 , V s E

au,

and

S

cx(s),B(s)

v s E

a r e given smooth f u n c t i o n s with r e a l v a l u e s , a ( s ) > 0 , 5 ( s ) > 0 ,

10,Ll. Using t h e same o p e r a t o r P a s above and t h e same argument, one g e t s on

t h e boundary

au

perturbation a

t h e equation of t h e form (3.14.96) with a s i n g u l a r

E Op

S("'")

1 PO and whose p r i n c i p a l symbol a. ao(S',E;')

(0,)

which i s e l l i p t i c of o r d e r v

=

(0,1,1)

has t h e form:

= lC'l+~B(s')15'1 2 0

Besides, t h e reduced o p e r a t o r a ( s ' , D

,)

. being an e l l i p t i c pseudo-

d i f f e r e n t i a l o p e r a t o r i n S ( 0 ~ 1 ~(R,0) ) of ozder ( 0 , 1 , 0 ) , 1,o 0 (a ( s ' , D s , ) b ) ( s ' ) = (-+a(s')) ( P $ ) (x), and having a well-defined i n v e r s e o p e r a t o r ( a o ) - ' ,

x=y(S')

t h e r e e x i s t s a well-

defined harmonic i n U f u n c t i o n , which s a t i s f i e s (3.14.97) on a U .

3.14. Reduction of Elliptic Singular Perturbations Moreover, for each 9 ' E H (o,s2-3,2,s3-1),E(U) $'

of the corresponding equation (3.14.96) :

uE = P$'

$€

523

one has for the solution

E H

(O,s,-f,s,) L J

(U), so that .E

E H(0,s2,S3),E(U) provided that s2 > f, s2+s3 > f .

As a reducing operator one can take the one with the symbol (1+c(8(s')/)15'/)-' which can be represented as an integral

operator with the kernel (3.14.95) with /B(s') substituted to a(s'). The assumption that U is convex is, of course, not essential. Indeed, if U is no longer convex, then the singular perturbation Q above is defined locally by an amplitude qk, which is asymptotically represented as follows:

with a(s,s'), k(s,s') defined as above. Notice, that each term $ ( < ' ) (s'-s)2P(ia(s',s)S'-(f)k(s',s) lS'I)p with

$ ( E l ) the cut-off function, as defined above, is an amplitude which defines

(nR 1 , so that the operator a singular perturbation in S(or-pro) 1 ,o $ + Q$ = -a/aN (P$)(x), x=y(s'), is locally defined by an amplitude q S' k' which is an asymptotic sum of amplitudes, whose orders -p decrease to -m as p

+

m.

Obviously, the argument above extends to any dimension n > 2, the kernel of the operator P being in that case defined in an obvious way as -1 -n P(X,Y') = 2n <x-Y',N ,>Ix-Y*/ , x E Y

being the unit inward normal at y' E aU and N Y' unit sphere in Rn

u,

nn

au,

y' E

being the area of the

.

In the multidimensional case the operator -2 2/as2 in (3.14.97) is replaced by any elliptic second order differential operator in tangential variables y' E aU, whose principal symbol has the form: with a (n-l)X(n-l) positive definite matrix A(y'), V y' E aU

,

the ellipticity

condition being fulfilled, provided that > 0, V y' E aU with

1 the given smooth vector-field on aU. Y' We give here two examples in order to illustrate the reduction procedure and its applications to elliptic difference operators. Example 3.14.19. m

Let q1 (x) 5 0, q 1 E CO(R) and let q(x) = l+q (x). Consider the family of 1

524

3. Singular Perturbations on Smooth Manifolds without Boundary

difference operators a(x,h,hD) on

%1

with the symbols a(x,h,hS),

With iDh and iDE the forward and backward finite difference derivatives on 1 2 the greed % , one has: a(x,h,hD) = h (D D*l2+q(x)DhD; (see also Examples h h 3.11.10). Obviously, a(x,h,hS) is a finite difference approximation of the 2 2 differential operator p(x)D2 = -p(x) d / d x on W , so that (3.14.98) can be considered as a singular perturbation (with the meshsize h as the small 0

parameter) of the difference operator a (x,h,hD) = p(x)D D*, the latter h h being the usual elliptic three point finite difference approximation of the differential operator -p(x) dL/dxL on W

.

One has: 0

a(x,h,hD) = r(x,h,hD) a (x,h,hD) with r(x,h,h3) the following factorizing operator: 2 (3.14.99) r(x,h,hD) = q(x) + h DhD:

=

(2+q(x))-(8 h

h

h being the shift operator on E( : (Bhu)(x) = u(x+h) . Introduce the following operator s(x,h,hD):

f3

s (x,h,hD) =

Op(r (x,h,hS))

Obviously, s(x,h,hD) is a discrete convolution operator on

4 with the

kernel S (x,x-y) given by the formula: h

It is readily seen that s(x,h,hD), acting on the mesh functions u(x) with a compact support in

4

according to the formula:

3.14. Reduction of Elliptic Singular Perturbations with

525

(x,x-y) given by (3.14.100), is a left and a right reducing operator h for a(x,h,hD) S

.

Indeed, one gets this conclusion by applying Theorem 3.11.17 to s(x,h,hD)

a(x,h,hD) and a(x,h,hD)

o

,.

s(x,h,hD), respectively. In both

cases a(x,h,hD) after the multiplication from the left or from the right 0

by s(x,h,hD) is reduced to a regular perturbation of a (x,h,hD) by terms of the form hb(x,h,hD) with b(x,h,hS) E F ( O r 2 ) (IR) of order (0,2) at most. 1P O Example 3.14.20. m

Let q E C (IR) be as in the previous example and let -Ah be the usual five point elliptic approximation by finite differences of -A on the greed in B2 (see Examples 3.11.10). Consider the following discrete boundary value problem in 2 E \ , x2 2 01:

2 %,+= tx

(3.14.101)

{

(l-Ah)u(x) = f(x),

'(,DX

hD:

1'

x E

%2 ? + x2 ,

,h-iq(xl)Dx ,,)u(xl,x2)

2

1

> 0,

Ix2=o

=

$(x,),

x1 E

4,

(3.14.101) being a finite difference approximation of the following differential boundary value problem:

We are going to show that problem (3.14.101) with 2 has a unique ' ( s 1, s 2-2,s3),~,h(%,+) and '(S~,S~-~/~,S~-~),E,~

'

solution u E

H ( s ) ,E,h(%,+

)

, provided

that s2 2 2, s3 b 0

and s

3

being integer. AS a consequence of Lemma 2.8.18 (applied in the variable x2

there exists an extension if E

ff ( s

f in (3.14.101).

,S

1

-2,s3),s,h(<) 2

Introducing v = op(1+/<j2)-lif,v E H ( s ) ,c,h(%

1

E (X$,+

),

of the second member

)'

it is readily seen

that

Thus, introducing w = u-v, problem (3.14.101) is reduced to the case, when f E 0, so that from the very beginning one can consider this situation, without restriction of generality.

3. Singular Perturbations on Smooth Manifolds without Boundary

526

Each solution of the equation (l-Ah)u(x) = 0 in for x2

+

+-, has the

with some $(x

1

)

g,+, which decreases

form:

and 6+ given by the formula:

where, as usual,

2

=

-1 1+1S1I2, C1 = (ih) (exp(ihS1)-l).

Thus, $(x ) is a solution of the following singularly perturbed 1 finite difference equation on

4:

1 (3.14.102) (a(xl,E,h,hol)Q)(x,) = $(xl), x1 E Rh ,

where a(xl,E,h,hD1)has the following symbol: 2 (3.14.103) a(xl,E,h.hC1) = ~ 1 5 ~ 1 Obviously, a E

F;Yb1")

(R

and is elliptic of order ( O , l , l ) ,

since its

principal symbol a

0'

def -1 (3.14.104) a0(x1,p,nl) = p Iwl12+q(xl)/wll( ( I + iwl = exp satisfies the inequality: -1 (3.14.105) ao(xl,p,n1 ) 2 coIwl/

,

1

V (xl,p,nl)E R 1 x R + x Tl,nl

(see Definition 3.12.42). 0

Besides the reduced operator a (xl,h,hD1), (3.14.106) a0 (xl,h,hD1)=q(xl)((l+WDx

>/2)2)f -haxl,h>/2)

1' lfh 0 -1 having ao(xl,h,nl) = h q(xl)lwll ((1+( lw11/2)2)t-(w11/2) asitsprincipal symbol and being elliptic of order ( 0 , l ) in the class Op F ( o ' l ) ( B ) ,

is

also invertible, its inverse being defined by the formula:

Therefore, the perturbed operator has an inverse as well, provided that E

E ( O , E ~ ] with

E~

sufficiently small.

3.14. Reduction of Elliptic Singular Perturbations

527

Furthermore, a(xl,h,hD1) being elliptic of order (O,l,l), one gets the conclusion that the solution I) of (3.14.102) satisfies the condition:

Hence, an easy computation (similar to the proof of the second part of Theorem 2.7.28 concerning the lifting operator Lh) shows that uniformly with respect to (E,h)E

(0,ll

x

(0,1] holds:

As a left and a right reducing operator for a(xl,h,hD1 ) one can take the one defined by the symbol:

An easy computation shows that s(xl,E,h,hD1 ) is a discrete convolution operator with the kernel S(p,xl,xl-yl)given by the formula:

Since k = (x -y )h-' E Z for (xl,yl)E

1 %X

%1 ,

the integrand on the right

hand side of (3.14.11), contains Tschebyshev's polynomials cos(2k arcsint) of degree 2k.

,l, Of course, one has for each $(x 1) with a compact support in m

and one finds easily that S(-,xl,xl-y1 ) 0 otherwise.

I

=

:

6 , i.e. 1 for x1 = y1 and x1 'Y1

3 . Singular Perturbations on Smooth Manifolds without Boundary

528 Notes ___

The concept of vectorial order v E R3 and classes Op P

of singular

perturbations with smooth coefficients were introduced in [Frank, 15,19,22 1 but were implicitly present also in [Vishik - Luysternik, 1

1.

The classes

op Lv (also in the case of variable symbols) were considered in [Frank, 221, [Frank - Wendt, 1,5,101. (U), Op S v ( U ) are natural extensions of the corres1 ,o ponding classes of pseudodifferential operators to singular perturbations,

Classes O p

Sv

as well as the concept of uniform pseudolocality and the results concerning

the C*-algebra structure (adjoints and products of singular perturbations are still in the union with respect to v E iR3 of the classes mentioned above 1 . While presenting the stationary phase, Laplace and saddle point methods, we follow essentially [Fedorjuk, 1 in [Dieudonns, 1 1, [Hbrmander, 4 Examples 2.8.15

-

1. Other presentations are to be found 1 , [Melin - Sjdstrand, 1,2 1 and others.

3.8.18 (also in a slightly more general situation) were

considered in [Frank, 14

1

(see also [Guillermin - Sternberg, 1

1

as far

as Example 3.8.15 is concerned). Examples 3.8.19, 3.8.20 are useful for the theory of coercive singular perturbations (see [Frank, 22 1, [Frank - Wendt, 1,3,10 1) which are the subject of vol. I1 of this book. Example 3.8.22 is taken from [Frank, 14

1.

Example 3.8.23 and general classes of hyper-

bolic difference operators were considered in [Frank,5,7] (see also [Strang,

1

1

for first order systems of hyperbolic difference operators). Fourier integral singular perturbations, as presented in 2.9, are a

natural extension of the local theory of such operators without a small

, 1 1 , [Ludwig , 1 1, [Hbrmander, 3 1 , [Duistermaat , 1 1, [Duistermaat , 1 1 , [Trgves, 1 1 , [Taylor , 1 1 ) . On

parameter (see [Lax Hormander

the other hand they are tightly connected with the local theory of Maslov's canonical operators (see [Maslov

,

1 I).

The idea to use the stationary phase method in order to establish the transformation formulae for the symbols of the classical pseudodifferential operators after smooth diffeomorphisms belongs to M. Fedorjuk (see [Fedorjuk,l,3]). However, earlier this kind of formulae was established in [Hbrmander, 2 ] without using the stationary phase method. In 3.10 the procedure indicated in [Fedorjuk, 1 1 is extended to the classes of singular (U) and Op S v ( U ) , as defined in 3.3. The idea to use Sv 1to an equivalent global definition of singular perturbations on a smooth manifold

perturbations Op

529

Notes without boundary (Definition 3.10.14) comes from [Hdrmander, 2

1

where such

a definition was introduced for classical pseudodifferential operators. Essentially, the difference between both cases consists in the presence of two large

independent parameters 151 and

E

-1

when one considers singular

perturbations, while for the classical operators only 151 ( 5 E Rn

being

the cotangent variables) is relevant. Example 3.10.15 is of interest for the diffraction theory, where the small parameter characterizes the loss of energy near the boundary (see vol. I1 of this book where the eigenvalue problems for such and more general coercive singular perturbations are considered). The C*-algebra of difference operators considered in 3.11 was introduced in [Frank, 4,9,10

1.

A

different kind of one parameter families of difference

operators, adapted to the approximation of first order hyperbolic systems, where considered in [Yamaguti - Nogi

, 1 1.

operators introduced in [Lax - Nirenberg, 1

The algebra of difference

1

was the main tool for proving

the Von N e m n n conjecture concerning the stability of difference evolution equations (Von Neumann's conjecture had been an open problem for many year before the appearance of [Lax-Nirenberg, 1

I).

The difference version of the

local theory of Fourier integral operators, as considered in 3.11, is

1,

useful for hyperbolic finite difference approximations (see [Strang, 1

1)

[Frank, 5

of hyperbolic differential and pseudodifferential operators.

In terms of this kind of Fourier integral operators, one can reformulate the Courant-Friedrichs-Levy stability condition also in the case of general hyperbolic operators. Definition 3.11.31 was introduced in [Frank, 5

1.

The

class of hyperbolic difference operators and asymptotic formulae for their solutions, as the mesh-size vanishes, are to be found in [Frank, 7

1.

The

difference version of Fourier integral operators with a small parameter 0 (thus, they are affected by the presence of two small parameters,

E

>

E

> 0 and the mesh-size h > 0 ) is useful for the numerical treatment of

singularly pertrubed hyperbolic operators. There is a hope to be able to elaborate on this topic in forthcoming volumes of this book. The ellipticity concept for singular perturbations was introduced in [Fife, 1

1

and, independently in the form as it is formulated in 3.12

(Definition 3.12.2) in [Frank,15,19,221. C h a r a c t e r i s t i c t w o - s i d e d a priori estimates for elliptic and coercive singular perturbations (uniform with respect to the small parameter) were stated in [Frank, 15,19,22] and proved in [Frank, 22

1.

The strong ellipticity concept for finite difference

3. Singular Perturbations on Smooth Manifolds without Boundary

530

1.

operators is already present in [Lax - Nirenberg, 1

The concept of

ellipticity for one parameter families of difference operators was introduced in [Frank, 4

1

and independently (for a subclass of finite difference schemes,

which approximate elliptic differential operators) in [Thomee - Westergren, 1

1.

Two-sided a priori estimates for elliptic finite difference operators

were stated in [Frank, 41 and proved in [Frank, 13

1.

For difference

singular perturbations analoguous concepts were introduced in [Frank, 16, 17,21

1

and corresponding two-sided estimates established in [Frank, 23

1.

A priori estimates (uniform with respect to the small parameter) in L P norms for elliptic singular perturbations were established in [Sweers, 1.21. Theorem 3.12.62 is due to Hdrmander and we follow esstentially the scheme

1

in [Hdrmander, 1

for proving it. In order to establish uniform a priori

estimates in L -norms for elliptic singular perturbations, the classical

P

result in [Michlin,l,2 1 (Corollary 3.12.63) suffices. The guide lines for proving results similar to Theorem 3.12.62 are to be traced back, for instance, in [Zygmund, 1

1.

As far as L -estimates for classical pseudo-

P

differential operators and more general classes of operators are concerned, the reader is refered to [Beals, 2

1

1 ,

[Triebel, 1

1

1,

[Fefferman - Stein, 1

1,

[Strichartz,

and others.

For the strongly elliptic differential systems introduced in [Vishik,

11 M.I. Vishik proved that the symmetric quadratic forms associated with the real part of such a system of order 2m (in a bounded domain R

C

lRn

with sufficiently small diameter) defines an equivalent norm in the Sobolev space Hm(fi). A proper extension of this result to arbitrary bounded domains was given in [Gsrding, 1.2

1

and came into use in the mathematical

literature under the name of Ggrding's inequality. This kind of inequality

1.

for singular integral operators was established in [Calderon - Zygmund, 1 Several important results toward the sharp form of G&ding's are to be found in [Seely, 1

1.

inequality

The sharp form of Girding's inequality for

the classical pseudodifferential operators, as introduced in [Untergerger Bokobza, 1 1, [Kohn-Nirenberg, 1

1

-

and for one parameter families of

difference operators (appearing in the stable approximations of well posed evolution problems), was first established in [Lax - Nirenberg, 1

1.

A

simplified proof of the sharp form of Ggrding's inequality was given in [ Friedrichs, 2

1

and [Vaillancourt, 1

3.

For classes F"(Zn

)

(see Definition

3.12.32) of one parameter families of difference operators the sharp form

of Gsrding's inequality was proved in [Frank, 9,lO

1.

For singular

Notes

53 1

perturbations whose reduced symbol is identity it was stated in [Frank, 241 (see also [Helffer

-

SjGstrand, 1

1

for the case of semi-classical pseudo-

differential operators). As far as Gsrding's inequality is concerned in the case of pseudodifferential (and more general) operators without small or large parameters, the reader is also refered to [Beals - Fefferman, 1 [Fefferman - Phong, 1

1

1,

and others. Example 3.13.14 is taken from [Frank, 3

and the last part of Example 3.13.15 from [Frank, 24

1

1. For proving the

Lax-Nirenberg theorem (Theorem 3.13.19), we follow essentially [Friedrichs, 2

1,

[Vaillancourt, 1

relevant

1

(with some minor modifications).Example 3.13.28 is

for the probability theory (see [Friedlin - Wentzel, 1

1

and for

quantum mechanics (see [Helffer - Robert, 1 I). A s far as the difference methods for the conservation law systems are concerned, the reader is refered to [Lax, 2,3 1, [Godunov, 1 1, [Harten - Hyman - Lax, 1 1, [Osher,

1

1,

[Van Leer, 1

1

and others.

The idea of a constructive reduction of coercive singular perturbations to regular perturbations was put forward in [Frank, 19

out in [Frank - Wendt, 1-5,6,10

1

and fully worked

1 and [Wendt, 1,2 1. The Wiener-Hopf

factorization was used in [Eskin, 1

1

in order to reduce an elliptic pseudo-

differential singular perturbation with homogeneous Dirichlet boundary conditions to a regular perturbation. Example 3.14.11 is taken from [Frank - Wendt, 1

1.

devices (see [Mock, 1

Example 3.14.13 comes from the theory of semiconductor

1,

[Markowich, 1 1, [Smith, 1

1,

[Sze, 1 1). Example

3.14.17 is of interest for the linear elasticity theory and Example 3.14.18 is relevant for the diffraction theory. The idea of reducing elliptic finite difference singular perturbations to regular perturbations, put forward in Examples 3.14.19 and 3.14.20, can be consistently worked out for the general elliptic finite difference operators in the same way and spirit as it has been done for the elliptic singular perturbations in Op L v ( M ) .

This Page Intentionally Left Blank

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