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~(~, y) = ~0(~, y) + r
a) + ~v~ (t, a)}.
(11.6.47)
This function satisfies by construction the relation ~'~uniformly in the region 0 < the boundary conditions
X2
0~ z + f ( x , Y, ~)  0 ( ~ )
oy
(11.6.48)
@ y2 < 1 with  1 + 5 < x < 1  6. Further 5 satisfies
~2(x,y)=qo(x,y)forx 2+y21with
1+5_<x_<16.
(11.6.49)
As to the values of ~(x, y) along the line segments x  •  6) within the disk we only know that these are bounded by some constant independent of e. The remainder R(x, y) defined as R(x, y)  u(x, y)  ~z(x, y) (11.6.50) satisfies the boundary value problem
eAR
OR ~y + f(x, y, u )  f(x, y, ~t) 
eAR
OR of ~yy + ~u (X, y, ~z+ O(x, y)R)R = O(e),
uniformly in the region 0 < x 2 + y 2 < 1,  1 + 6 < x < 1  6 , conditions are R0forx2+y 21, 1+6<x<15 and IRI_<M along the segment x = + ( 1  6 ) .
(11.6.51) while the boundary
304
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
The latter assertion follows from the fact that not only ~(x, y) but also the solution u(x, y) is uniformly bounded in the disk x 2 § y2 < 1. This is easily proved with the aid of the generalized maximum principle of section 6.2 and the barrier functions ~ l ( x , y )  =kKl(1 + y) =t=K2, with the constants K1 and K2 sufficiently large. Choosing finally the barrier function (8.4.4)
~2(x,y)  Ce + M{Xl(X) + X2(x)}, with C sufficiently large we obtain the result: R(x, y)  O(e), uniformly in the region {(x, y ) [ x 2 + y~ _< 1,  1 + 26 < x <_ 1  25}. Summarizing we have obtained THEOREM
3
Let u(x, y) be the solution of the boundary value problem r
~U
O<_x 2 + y 2 < 1
oy
with the condition u(x, y) = 9~(x, y) for x 2 + y2 = 1. If f and ~ are smooth functions of their respective arguments and ~(x, y, u) is bounded with
of <: o u ( X ' y ' u ) < 6 < ~ f o r 0 < x 2 + y 2 < 1,  e e < u < + c ~ , and if the reduced boundary value problem has a smooth solution wo(x, y) then the function u(x, y) is approximated as
~(~, y)
= ~0(~, y) + r
 p 0) + o ( ~ ) s
uniformly in any region {(x,y) [ 0 _< x 2 + y 2 < 1 ,  1 + 6 < x _< 1  6 } , where the boundary layer function vo is given by (11.6.43) and where 6 is an arbitrarily small positive number, independent of e. Exercises 1. Give a first order approximation for a solution of the boundary value problem
d2U=u2
~~
(du) 2 
~
,
with
~(0) = ~ > 0,
u(1) = fl > 0,
t3e>a>flora
2. Give a first order approximation for a solution of the boundary value problem
d2u
(du) 2
Exercises
305
with u(0)a>0,
u(1)=/3>0and0<x/~
1
< ~,
0< ~
<
1
Check the restrictions on the boundary values. 3. Give a generalization of Theorem 1 of this chapter by considering the following boundary value problem in an annular domain ft:
02U
10u 1 02u \ Ou + ," ~ + ~d~ )  a(~, ~, ~) ~ + b(~, ~, ~)  0
0 < rl < r < r2, 0 ~ ~) < 2~r, with the boundary conditions
~(~, o)  ~ ( ~ ) ,
~(~, ~) = ~ ( ~ ) .
The assumptions regarding the coefficients a, b and the boundary values T1 (0) and ~2 (~) are similar as stated in Theorem 1.
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Chapter 12 PERTURBATIONS 1.
OF H I G H E R O R D E R
Introduction
In this final chapter we direct our attention to more general scalar perturbation problems. Up till now our perturbations were all of second order. A much more general situation is presented by the operator r
+ L1,
(12.1.1)
where L2 is a differential operator of arbitrary order m and L1 a differential operator of order k with 0 <_ k < m. To treat this kind of perturbation problems a more detailed specification is required. Whenever L2 may be of any order the same applies also to L1. As long as we deal with linear ordinary differential equations, boundary value problems are in general still tractable, but whenever we have to do with partial differential equations there is a plethora of problems. For instance L2 and L1 may be independently of each other elliptic, hyperbolic or parabolic and so there are already nine different cases of degeneration for r = 0 and in six cases we get a change of type of the operator. Apart from linearity or nonlinearity each problem requires its own theory, in particular if one wants to investigate the justification of a formal approximation of the solution. It is obvious that a general theory is not well possible and existing theories are far from complete. It seems sensible that the direction of research in the multitude of perturbation problems is to be determined by significant problems from mathematical physics, e.g. the equation of NavierStokes for the stream function, (in two dimensions). One of the most advanced texts presenting many boundary value problems with different degenerations is by J.L. Lions [100]; this author applies variational methods which are often used in modern textbooks on partial differential equations. Another important contribution is by M.I. Vi~ik and L.A. Lyusternik [138] treating boundary value problems for ordinary and partial differential equations, in particular the degeneration of an elliptic operator of higher order to one of lower order and the degeneration of an elliptic operator to one of first order. In connection with this paper we mention also the work by Besjes [9, 10, 11], who among other things improved the results by Vi~ik and Lyusternik by providing estimates in the maximum norm instead of the L2norm. Further we refer the reader to the papers by D. Huet with L2 elliptic and L1 of order zero, [70, 71, 72] and by D. Huet [73] and W.M. Greenlee [55] with L2 and L1 both elliptic; these papers
308
Chapter 12. Perturbations of Higher Order
have been written in more or less the same spirit as in the book by Lions. Finally it is proper to mention also the wellknown book by W. Wasow [139], where boundary value problems for ordinary differential equations have been treated. A striking difference with perturbations of the second order lies in the circumstance that with second order problems a simple inspection of the differential equation gives a decisive answer to the question where to locate the boundary layer. This is no longer true for perturbations of higher order and it may even happen that a boundary layer construction induces severe complications, see [142]. It is outside the scope of this textbook to give an extensive treatment of perturbations of higher order and therefore we restrict our considerations to the case of ordinary differential equations and to the case where both L1 and L2 are elliptic. In section 2 we deal extensively with boundary value problems for singularly perturbed ordinary differential equations. We investigate the location of the boundary layers and we give only a formal approximation. In section 3 we treat partial differential equations of elliptic type with an elliptic degeneration and we use the variational method to obtain an approximation of the solution of boundary value problems in arbitrary bounded domains in ] ~ . Using some theory of elliptic partial differential equations, it will appear that the construction of an approximation of the solution of the boundary value problem and the proof of its validity is very simple if we assume that the operators L1 and L2 are uniformly strongly elliptic and besides this that L1 is positive. Therefore we give first a short survey of the theory of elliptic equations stating the most important definitions and theorems; for proofs the reader is referred to wellknown textbooks, e . g . A . Friedman [43] or S. Agmon [2]. One of the reasons to discuss the ellipticelliptic degeneration is that it has a direct interesting application in the theory of thin shells. The boundary value for the displacement of a plate Ft, under lateral pressure and clamped along its edge OFt, reads r
(~u  A u = p(x, y), (x, y) C Ft with u[o a = ff~n[Oa = O.
(12.1.2)
This boundary value problem degenerates for 6 = 0 to the membrame problem with
  A w = p(x, y), (x, y) e Ft and W[o ~  O.
2.
Perturbations
2.1
Introduction
(12.1.3)
o f H i g h e r O r d e r in Ordinary Differential Equations
We consider the following scalar boundary value problem with a perturbation of higher orer EmkLm[y] + L k [ y ]  f ( x ) , 0 < x < 1, (12.2.1) where
dm "~ Lm  dx~ + Eai(x) j1
dmi dxm_~
(12.2.2)
2.1 Introduction
309
and
dk k dk_ j Lk = bo(x)~xk + E b j ( x ) dxk_ j ,
(12.2.3)
j=l
with bo(x) :/: 0 in [0, 1] and with the coefficients aj, bj and the righthand side f infinitely often differentiable and finally 0 _< k < m; as boundary conditions we take Bi[y](0) = ai,
i
1,2,...r < m
(12.2.4)
and Bi[y](1)=j3i,
i=r+l,r+2,...,r+s=m,
(12.2.5)
where
d~,~ .x, dj B~ = dx~7 + E 7~J dxxJ' j=o  1
(12.2.6)
with 0_
Our main task is to investigate the influence of the perturbation r on the solution of the reduced problem with ~  0. But here we meet immediately the question how to define the reduced problem. In case k  1 and m  2 one has the choice out of only two boundary conditons and a simple inspection of the unperturbed operator LI tells you which boundary conditon should be taken; in case bo(x) > 0 one fulfils the boundary condition at x  1 and in case bo(x) < 0 the condition at x  0. The choice of the proper boundary condition in the reduced problem is determined by the first term in the expansion of the operator r + L1 after the coordinate x has been stretched as x  r or x = 1  r and it is the stability of the resulting boundary layer solution that fixes the proper choice of the boundary condition in the reduced problem. The same considerations apply also in the case m > 2, but it will appear t h a t we get boundary layer equations at both end points of the interval [0, 1]. They read in first approximation
dm V dkV dm U dkU d~ m [ b0(0) d~  = 0 and (  1 ) m~dr/m + (1)kb0(1) ~  0.
(12.2.7)
The number of stable solutions determine the number of boundary layer corrections at each end point and hence also the number of boundary conditions left over for the reduced boundary value problem which, however, may not always be solvable.
2.2 T h e Formal A p p r o x i m a t i o n We return to our boundary value problem (12.2.1)(12.2.6) and we shall give besides the construction of a formal approximation of the solution also the conditions under which this approximation is possible. The outer expansion
W(x; r
Er rt:O
(12.2.8)
310
Chapter 12. Perturbations of Higher Order
satisfies the recursive system
Lk[wo](X) = f (x), Lk[w,](x) = O,
n  l, 2, . . . (m  k  1 ) ,
Lk[w,~](x) =  L , ~ [ w n  m + k ] ( x ) ,
n > m
k,
(12.2.9)
valid for 0 < x < 1. Because we cannot state a priori the boundary conditions for W ( x ; e) we leave these open for the moment and remark only that each wn is determined by (12.2.9) up to any linear combination of k independent solutions of the homogeneous equation Lk[w] = O. Since W ( x ; e) cannot satisfy all m boundary conditions (10.2.4)(10.2.5)we need boundary layer corrections at x = 0 or/and at x = 1. Stretching the x coordinate as x = e~ we get for the boundary layer function V(~; e) the equation
dm V dkV ,,~ din_iV k dk_ Jv d~~ + b o ( e ~ )  ~ + E e J a j ( e ~ ) d~m_ j + E e J b j ( e ~ ) d~k_ j = 0 , j=l j=l valid for ~ > 0. Expanding V(~; e) as oo
(12.2.10) n=O
we get the recursive system
dmvo
dkvo
d~ m + b 0 ( 0 ) ~
= 0,
d~vn dkvn d( m + b0(0) d~   r=((), where
r,(()
n > 1,
is a linear combination of preceding terms
(12.2.11) ve
and their derivatives
(l < n) with coefficients being polynomials in (. The characteristic equation of the hemogeneous equation reads Ak (Amk + b0(O)) = O,
(12.2.12)
and we denote its roots with negative real part by /zj, j = 1, 2 , . . . q. It follows that the functions v~(~) with the right decaying boundary layer character are given by q
vo(~) = ECjoet'J~,
(12.2.13)
j=l
and
q
v,(~) = E ( c j , . , + ~p~)(~))e t'j~, j=l
(12.2.14)
2.1 Introduction
311
where the coefficients Cjo and cjn are to be obtained from boundary conditions still unknown and where Fin _(1)(~) are polynomials in ~ completely determined by rn(~). Similarly stretching the xcoordinate at the other end of the interval [0, 1] by x = 1  er/ we get for the boundary layer function U(r/; e) the equation (_l)m
dmU
dkU
d~ + (1)kb0(1  er/)~@ +
m
k
j=l
j1
dmJu Eejbj(1 1)kJ dkjU = 0 , E ~jaj(1  er/)(1)m/&Tin7 +  er/) ( d@J valid for r / > 0. Expanding U(~}; e) as oo
u(,;
~
(12.2.15)
= n=0
we get the recursive system (1)mk dmu0
d ku0
(__l)m_kdmun
dkun
d~Tm ~b0(1) d  ~ 0,
dr/~ + b0(1)
d@
= s~(r/)
n _> 1,
where sn(r/) is again a linear combination of the preceding terms derivatives with l < n and with coefficients being polynomials in 7}. The characteristic equation reads now + b0(1)) = 0,
and we denote its roots with negative real part by similarly as in (12.2.13)(12.2.14)
(12.2.16) ul
and their
(12.2.17)
uj, j = 1, 2 , . . . p . Then we have
p
Uo(rl) = Edjoe~'Jn
(12.2.18)
j1
and
p
un(rl)  E (djn + rlp~.2)(rl))e ~'~'1
(12.2.19)
j1
_(2) with the stillunknown constants dj0 and din and where the polynomials pin (~I) are completely determined by s,~O?). Because bo(x) ~ 0 in [0, 1] it is clear that the number p equals the number of roots with positive real part of the equation (Amk + b0(0)) = 0 and so p + q = m  k, whenever this equation has no purely imaginary roots; we assume that this is
312
Chapter 12. Perturbations of Higher Order
the case and the other case with p + q  m  k  2 is not considered here, see Refs. [142], [61]. A formal approximation oo
y ~ Eenyn(x)
(12.2.20)
n0
consists of the outer expansion W ( x ; e), (12.2.8), and the boundary layer expansions V(~; e) and U(rl; e), (12.2.10) and (12.2.15). Hence it is spanned by a fundamental system of k solutions of the homogeneous equation Lk[w] = 0 and by the p + q = m  k exponential boundary layer functions { e t ' ~ } jq= l and { e ~ ' } P = l 9 this m e a n s t h a t we have at our disposition m degrees of freedom which corresponds with the number of boundary conditions (12.2.4)(12.2.5). The boundary layer function V(~; e) can account for q boundary layer corrections at x = 0 and the other boundary layer function U(r/; e) for p boundary layer corrections at x = 1. This leaves for the outer expansion W(x; e) and hence for the reduced problem ( r  q) boundary conditions at x = 0 and ( s  p ) conditions at x = l . At x = 0 we have the conditions ~, : B i [ W + V](0) : Bi[W](0) + e ~~'
dX, Q
d~Xi
,~, 1 dj i7 (0; e ) + E 7 i J e ~  J  ~ (0)' i = 1 2 , . . . r .
(12.2.21)
j=O
Expecting that W ( x ; E) should satisfy for e + 0 exactly ( r  q ) boundary conditions, not more and not less, gives a  At_q+1; further we obtain for wo(x) the conditions Bi[wo](O)  c~i, For r  q + l _ < i _ _ r
i = 1, 2 , . . . (r  q).
(12.2.22)
we have from (12.2.21)
)~i1
dj f/
a~,d~, (" (o,, ~) = ~'~~+' { ~,  B,[W](O) }  ~ ~,j~'J  ~ ( 0 ) , j=0
and therefore
dXrq+lvo
d~Xr_q+l (0)

Olr_q+
1 
Brq+l[Wo](O),
(12.2.23)
and d Xi VO
~
(o)  o,
i 
(~  q) + 2 , . . . ~ .
(12.2.24)
Expansion of W and V in their asymptotic series give the boundary conditons for the higher terms w,, and vn at x  0 . In a similar way we have at x  1 the boundary conditions ~ = B i [ W + U](1) = Bi[W](1) + ~ [(r x' dX'[Td@'(0; e)
+ A,1 ~,j(~lJ ~d J U ( 0 ; e ) ] , j=0
i = r + l, . . r + s = m.
(12.2.25)
2.1 Introduction
313
Because W ( x ; e) should satisfy for e + 0 exactly ( s  p) b o u n d a r y conditions, not more and rtot less, we must take ~ = At+sp+1  Amp+1; further we get for w o ( x ) the conditions B i [ w o ] ( 1 ) = /3i, i  r + l , . . . , r + (s  p)  m  p. (12.2.26) For m  p + l < i < m
dAi~f ( 0 ; drlAi
~) =
we get from (12.2.25)
(1)AI~)'i)'mp+l
{ / 3 i 
Bi[W](1)}  ,x,1 E ~iJ(~))kiJ dJ U j=O
(0; e)
and therefore the boundary conditions d),mp+ l Uo d77Am_p+1 (0) = (  1 ) Amv+1 { f l m  p + l
 Bmp+l[Wo](1)
(12.2.27)
}
and d Ai u 0
dr/A' ( 0 ) = 0 ,
i=mp+2,...,m.
(12.2.28)
Expanding W and U in their asymptotic series we obtain the b o u n d a r y conditions for the higher terms w,~ and v~ at x  1 . Summarizing we have constructed in principal an "Ansatz" for a formal approximation (x)
oo
oo
n=0
n=O
1  x),
x where w0(x), v0(~)
].x and u0(/)
E
n=0
(12.2.29)
E
satisfy the following boundary value problems:
1. Lk[Wo](x) = f(x),
0 < x < 1
with Bi[w0](0)  ai, i  1, 2 , . . . (r  q) and B i [ w o ] ( 1 )  ~ i , i = r + 1, r + 2 , . . .
dmvo
d~~
(12.2.30)
(mp).
dkvo
+ b0(0)=.~  0,
~ > 0
with
d Aiv 0 d~Ai (0)  ~r_q+l,i{o~r_q+l Br_q+l[Wo](O)}, i   ( r  q ) + lim v0(~) = 0, where
(~rq+l,i
is the Kronecker symbol.
1,...
r, (11.2.31)
314
Chapter 12. Perturbations of Higher Order
(_l)mk dmuo dkuo dr/~ + b0(1)~@ = O,
r/> 0
with d A' uo d@' (0)  5m_p+l,i (  1 ) amp+' {Zmp+l  Bm_p+l[wo](1)}, i = m p + 1,...m,
(12.2.32)
lim u0(r/) = 0.
~/4oo
At this point we have to make the important remark that this construction makes only sense whenever q < r and p < s, (12.2.33) and the boundary value problems (12.2.30)(12.2.32) have a unique solution. The condition (12.2.33) is rather severe for the generality of our procedure, because the 1
number of roots (b0(0)) k with positive or negative real part has nothing to do with the distribution of the boundary values over the endpoints of the interval [0, 1]. The condition concerning the unique solvability of the boundary value problems (11.2.31) and (11.2.32) amounts to the unique solvability of the sets of algebraic equations q
ECjo#jAi
=
6i,r_q+l{~r_q+l
i= (rq)+
 B r  q + l [ W o ] ( O ) }
1,...,r.
j=l
and
P E d j o @ ' = 5i,mp+l (  1 ) x=p+I {]~rnp+l
 Bmv+l[w0](1)},
i=(mp)+l,...m.
j=l
The coefficient determinants are related to the Vandermonde determinant and these equations have a unique solution if and only if the integers Arq+l, Ar_q+2,..., A~ are distinct modulo ( m  k) and the same for the integers A,~_p+l, Amp+2,...A,~. Omitting the proof that our formal approximation is also a good asymptotic approximation we arrive at Wasow's theorem THEOREM
I
Assume that the data of the boundary value problem (12.2.1)(12.2.6) satisfy the following conditions i) q<_r, p<_s and p + q = r a  k , ii) the reduced problem defined by (12.2.30) has a unique solution wo(x), iii) None of the integers A~q+l, A~_q+2,... A~ are congruent to each other modulo ( m  k) and the same for the integers A,,~p+l, Amp+2,... Am. Th~n th~ boundary ~lue p~oblem (~2.2.~)(~2.2.~) po~e~e~ ~o~ ~ ~umci,nt~y ~m~ll
unique solution y(x; e) with the property limy(x;e)=wo(x) inS<x< ~$0

15, 
(12.2.34)
315
3.2 Elliptic Partial Differential Equations
with 5 arbitrarily small positive, but independent of e. See also Wasow, Ref. [142, 139] or Vi~ikLyusternik [138]. The relevance of the conditions in this theorem is illustrated by the following examples, which we have taken from [139]
d4y
dy
~ x 4 + dx = O, with y(O) = y'(O) = y" (0)  0 and y(1) = 1,
d2y e dx5 d4y ~ dx
dy dx = O, with y' (0) = 1 and y.(1)  O, dy dx = O, with y(O) = y'"(O) = 0 and y(1) = O, y'(1)  1.
It is not difficult to show that the solution of each of these boundary value problems has the property that y(x) diverges when e + +0. The reader can easily check that in each example one of the conditions of Wasow's theorem is violated. An interesting and relatively simple example is given by the bending of a loaded elastic beam clamped at both ends.
3.
Elliptic Perturbations of Elliptic Equations
3.1
Introduction In this section we study singular perturbation problems involving equations of the
type eL2m[u] + L2k[u] = f(x).
(12.3.1)
These problems are in a certain sense a generalization of the theory of the preceding section in so far as we consider now partial differential equations for functions depending on an arbitrary number of independent variables; however on the other hand the orders of L2k and L2m are restricted because we assume that both operators are uniformly strongly elliptic in their domain of definition. We include this type of problems in our discussion of higher order perturbations because it provides a simple and elegant application of the theory of elliptic partial differential equations. Because we need some functional analysis and some concepts which may not be familiar to the general reader, we give first a concise introduction to the theory of elliptic partial differential equations however without proofs; for these we refer the reader to the excellent treatise by Friedman [43] and also to the book by S. Agmon [2].
3.2 Elliptic Partial Differential Equations 3.2.1 Sobolev Spaces Let gt be an open set in ][~ and 0~t its boundary; cm(Ft) is the set of all functions m times continuously differentiable in ~t and we denote by (~m(~t) the subset of cm(f't) consisting of all functions ~o E Cm(gt) with the finite norm:
{
1
1
9=
f~
I~lj
(12.3.2)
316 where a
Chapter 12. Perturbations of Higher Order is a multiindex a = ( a l , a 2 , . . . ,
0 , Dj = ~Oxj
an);
we denote
D s  D~'D~2...D~",
lal = ~
a j and
j=l
F u r t h e r we write x s = x~ ix s2 2 . . . x sn, and a! . a.l ! a 2. ! .. . a , , w T h e completion of the space (?m(~2) with respect to the n o r m (12.3.2) is called the Sobolev space H m ( f l ) . A oo Cauchy sequence {~/}1 in (~m(fl) satisfies for 0 _ 1hi <_ m the relation
/
IDS~j  DS~kl2dx ~ 0 for
j, k + co,
~2
and so, because that
L2(~)
is a complete space, there exists a function lim
j+oo
flVS~oj  uSl2dx
= O,
u s 9 L2(~)
such
(12.3.3)
and this is valid for all a with 0 _< lal _< m. T h e function u s is called the strong derivative of order a of the function u 9 H m ( f l ) and we write u s = (DSu)8. Also the concept of weak derivative is useful. W h e n e v e r u and v are b o t h locally integrable in then v is called the weak derivative of u iff
/
uDS
dx
~2

(1)J Jfv d ,
e
(12.3.4)
~2
where C ~ is the subset of C~176 consisting of functions with compact s u p p o r t in Q. We write v = (DSu)~. It can be shown t h a t if u E L 2 ( ~ ) with a strong derivative (DSu)8 then also (DSu)~ exists and they are equal. Also the converse is true: If u 9 L2(Q) has a weak derivative (DSu)~o then also (DSu)8 exists and again they are equal. According to the particular situation b o t h concepts m a y be used and we omit henceforth the suffix w or s. The space H m ( Q ) is a Hilbert space with scalar product
(u,v)m= ~
(DSu, DSv)i2= ~
Isl<_m
(12.3.5)
Isl_<ma
where the bar denotes complex conjugation. Similarly the space H ~ ( f l ) is defined as the completion of the space C ~ ( f l ) with respect to the norm (12.3.2) and also H ~ ( f l ) is a Hilbert space with scalar p r o d u c t as defined by (12.3.5). Finally we r e m a r k t h a t we have the obvious inclusions Hm(~) C H k ( ~ ) and H~(~2) C H0k(~), k _< m.
317
3.2 Elliptic Partial Differential Equations
3.2.2 Elliptic O p e r a t o r s , B i l i n e a r F o r m s a n d G h r d i n g ' s I n e q u a l i t y It will appear very useful to write our differential operators in the socalled divergence form L[u] = E (1)IplDP(aP~ ' (12.3.6) 0
E ~PaP~ Ipl=l~l=m
> c~
:= c~
~2 + ' " +
~)m,
(12.3.7)
for all ~C]R~ and all x C ~ . In case co(x) can be chosen independent of x then L is uniformly strongly elliptic and co is called the molulus of ellipticity. The formal adjoint L* of L is defined by L*[v] =
E (1) I~ID~ (aP~(x)DPv) 0
(1)IplDP(a~P(x) D~v)"
(12.3.8)
O<[p[,[a[<m
For u, v C C ~ (gt) we have
with B[v, u] =
E (DPv, aP~D~u) O<_lpl,l~l<~
(12.3.9)
Henceforth we write (v, u)0  (v, u), the usual L2 scalar product. The expression B[v, u] is called the bilinear form associated with L; this form is defined for ap~ bounded in ~ and for v and u belonging to Hm(Ft). This form satisfies the following theorem, important in the theory of elliptic partial differential equations, particularly for the proof of the existence of solutions of boundary value problems. THEOREM A
Suppose that 1. L is strongly elliptic in a bounded domain f~ with modulus of ellipticity co, independent of x. 2. The coefficients a p~ are bounded in f~ for 0 <_ [p[, [a[ <_ m with upper bound r
for I P l  I o l  m, for all x and y in f~, and
3. laP~(x)ap~(y)l <_ c 2 ( l ~  y l ) c ~ ( l ~  yl) ~
0
for
I~  yl ~
0.
318
Chapter 12. Perturbations of Higher Order
Then there exist constants
c and ko d e p e n d e n t on co, cl, c2 and ft such that
ReB[u, u] > c]lul] m2  ko]lu]] 2 ~
Vu 9 H g ( a )
"
(12.3.10)
This inequality is known as Gg~rdings inequality and a bilinear form satisfying (12.3.10) is called coercive. In the following we will suppose that the elliptic operators to be considered satisfy the three conditions stated above. 3.2.3
Generalized
Dirichlet
Prolems
We are interested in Dirichlet problems L[u] = f in 12 C IR'~
(12.3.11)
with OJu 0uJ = gj in 09t,
j = 0, 1 , . . . , m  1,
(12.3.12)
where L is an elliptic operator of order 2m and 12 is a bounded domain with boundary 012 of class C m1. f and 9 are functions continuous in ~ and Oft respectively and o denotes differentiation in the direction of the outward normal of 012. If u E C2m(12)N C m  l ( f i ) and if u satisfies (12.3.11)(12.3.12) then u is a classical solution of the Dirichlet problem. In case 012 E C 2m+1 and 9j E C2m(012) the classical solution u is also determined by the classical solution of the h o m o g e n e o u s Dirichlet problem (12.3.13)
L[v] = f* E 12, OJv =OinOft, OuJ
j=O, 1,...,m1,
(12.3.14)
with v = u  9 and f* = f  L[~], while ~ is an appropriate function belonging to C2m((~) with the property ~O J ~  9j, J = 0, 1 , . . . , m  1. It follows that it is not a too strong restriction to discuss only homogeneous Dirichlet problems. We shall now generalize these problems in such a way that under certain conditions the solution of the generalized problem is also a classical solution; the advantage is that the proofs for existence and regularity of the solution become more transparant and elegant. The generalization of the differential equation L[u] = f is obtained by the associated bilinear form B[~o, u] = (~o, f), V~o G C ~ ( a ) , where the lefthand side is defined for u E H'~(ft)
and derivatives up to only order OJu
m are required. The generalization of the homogeneous boundary conditions ~ Ion = 0, j = 0, 1 , . . . , m  1, is given by the assumption u E H~(gt). This is justified by the following lemma. LEMMA
Suppose
Oft
0 <_ j <_ r n  1 ;
is o f class
C m.
If
u E g ~ ' ( f t ) N cml(~'~),
also conversely, if u E c m ( f i )
oJ~ =0 and 5~]o~
then
0~ ~ = 0 5~[o~
for
for O < j < m  1 ' then
319
3.2 Elliptic Partial Differential Equations
u C H~(gt). These results lead to the following definition of a generalized solution of a homogeneous Dirichlet problem. DEFINITION u is a generalized solution of the Dirichlet problem boundary conditions if u C H ~ (f~) and
B[~, ~] = (V,/),
L[u] = f
with homogeneous
V~ e C3~
(12.3.15)
where B is the bifinear form associated with the differential operator L. The problem to find a function u E H~(f~) that satisfies (12.3.15) is called the generalized Dirichlet problem The relation between a classical and a generalized solution is given by the following theorem THEOREM B Let Oft be of c/ass C m and ap~ E clpl(~t). I f u is a classical solution of a homogeneous Dirichlet problem of order 2m with the property u E cm(~t) then u is also a generalized solution. Also conversely, if u is a generalized solution belonging to Hg'(a) . n d with th~ p ~ o p ~ t y ~ c C 2 ~ ( ~ ) n C ~  ' ( ( ~ ) t h e . ~ i~ ~1~o ~ c l ~ i c ~ l solution of the Dirichlet problem with homogeneous boundary conditions. The consequence of this theorem is that the generalized solution yields also the classical solution whenever the data of the boundary value problem are sufficiently smooth. 3.2.4 E x i s t e n c e o f G e n e r a l i z e d S o l u t i o n s The existence of generalized solutions of Dirichlet problems with homogeneous boundary conditions may be proved with the aid of the LaxMilgram theorem, a generalization of the wellknown representation theorem of Riesz. The latter reads as follows
THEOREM OF RIESZ Let H be a Hilbert space and let F[v] be a linear bounded functional on H, there exists a unique element u C H such that
then
F[v] = (v, u). The LaxMilgram theorem is a generalization in so far as the scalar product replaced by the bilinear form B[v, u].
(v, u) is
LAXMILGRAM THEOREM Let there be defined on a Hilbert space H with norm [1" [[ a bilinear form B[v, u] with the properties. i ) There exists a constant cl independent of v and u such that
IB(v, u)l < clllvll. Ilull, v,,,,., e H. ii) There exists a constant c2 independent of v such that [B(v,v)[ ~ c~ll~ll ~,
Vv e H.
320
Chapter 12. Perturbations of Higher Order
Then every bounded functional F[v] on H may be represented as
F[v] = B[v, ~}, with u uniquely determined by F. An immediate consequence is the following existence theorem THEOREM C If the bilinear form B[v, u] constant c such that
is bounded on
ReB[v, v] >
r
and if there exists a positive
H~(~t)
w
e Hg(f~)
(12.3.16)
then the generalized Dirichlet problem B[~a, u] = (~, f),
V~ E C~(~t),
u e H~(f~)
(12.3.17)
has for any f E L2(f~) a unique solution u with
I1~11,,,< 111filL=.
(12.3.18)
C
The existence of the generalized solution follows from the LaxMilgram theorem by taking F[v] = (v, f). The estimate (12.3.18) is obtained from (12.3.16) and the existence of a sequence { ~ } C C ~ ( f t ) with lim ~ = u in H ~ ( f t ) ; we have the inequalities n~oo
cllull~ ___ Re B[u, u] < IB[u, ~]1 = n lim IB[~o., ~]1   # or = n lim [(So,,,f ) [   (u, f) < [[UiiL= II/IIL= _< Ilu~ll. II/IIL= '  + O0 and so the estimate (12.3.18) is obvious. A bilinear form that satisfies the inequality (12.3.16) is called strongly coercive in constant with coercive as defined by the Gs inequality (12.3.10). Regarding applications to elliptic boundary value problems it is important to give some examples of strongly coercive bilinear forms. 1. Let 12 be a bounded domain and L =  A + k with A the Laplace operator and k some positive constant. We have B[v, u] = E (DPv' DPu) + k(v, u) Ipl=l 
~5&~_.dz+ k _ _ , _ _ .
Svudx,
[1
and therefore
B[u, u] ~ rnin(1, k)llul[ 2.
3.2 Elliptic Partial Differential Equations
321
It follows t h a t the b o u n d a r y value problem
~(~)
+ k~(~)  f ( ~ ) ,
~ e a,
~1o~  0
has a unique generalized solution in H 1 for all f C L2(fl). This solution is also a classical solution if 0~t is of class C 1, u C C 2 ( ~ ) N C ( ~ ) and f C C(Ft) A L2(~t). 2. Let ~t be b o u n d e d and L =  A . To show t h a t we have also in this case a strongly coercive bilinear form we need the Poinca% inequality which may be applied to domains only b o u n d e d in one direction. This inequality reads n
i1~11~ < d 2 
I 112 ; d 2 ~  ~ '
[lO~
V~ C C~(Ft)
(12.3.19)
i1 where d is the m a x i m a l diameter of ft. Suppose ft C {x C 1[~ []x~I < c}, then we have for ~ C C ~ ( f t ) Xn
f ~:~ O~ (x 1, x2 , . . . . , x,~
~(~)=
~, ()d(.
~C
Using CauchySchwarz we get +c
I~(x)l~ _< 2,
+c
Ib~~(xl,x~.,...,,~_l,e)l'd~=d C
Ib~ 1
,...,x,_l,
C
therefore
+c
+c
_< C
~C
integrating once again with respect to XliX2,...,xn_ 1 we obtain (12.3.19). The bilinear form B[u, u] associated with L   A satisfies according to (12.3.19) Re
B[u,
n/
Ou]2 1 2 1 ~ d~ = II~ > ~11~ + 7~ I"~
u] = E i=0
E
1 1 _> min (~, ~dff)ilull2,
Vu e
and so  A is strongly coercive. We remark t h a t coercive bilinear form.
L2
H~(f~), +A
3. Let f~ be b o u n d e d and L = A 2. For any qo E C~(f~)
i
a
n
"= J = l a n [
i=1 j = l f~
02 ~
2
~
does not yield a strongly we have the relation
~ 2
322
Chapter 12. Perturbations of Higher Order Applying the Poincar~ inequality to ~cOx l we have 2
_< d 2 L2
and hence n
0~O ]2
n < d2 ~"~ ixi L2 i=1
i=1
2
1
n
n
02~
=d2EE OxiOxj i=l j=l
L2
Therefore we get for any q0 9 C ~ (f~) I1 11 =
+
+
< (d~+ x)l~~ ~1+ I1,~,~. 2
_< {(d 2 + l)d 2 + l}l~,l~ = {(d ~ + 1)d 2 + 1}B[~o, ~o] and we obtain by u = lim ~0,~ in Ho2(f~) the inequality n~ r
Re B[u, u] = (Au, Au) > cllul[ 2,
Vu 9 Ho2(f~),
from which the strong coercivity follows. To give an example of a coercive but not a strongly coercive bilinear form we consider for U(Xl,X2) the boundary value problem Au+#u0,
0_r=v/x2+x~
#>0,
withu(ro)=0.
This problem has a nontrivial solution
= g0(v~), if V/fir0 is a zero of the Bessel function J0. It follows that the boundary value problem   A u   #u = f (xl,X2), 0 ~ r < to, u(ro) = 0,
# > 0
does not have a unique solution if v/fir0 is a zero of J0 and therefore the strongly elliptic operator  A  # with J0 (v/fi r0)  0 cannot have a strongly coercive bilinear form. On the other hand it has a coercive bilinear form; this follows from Theorem A or by direct computation. As to existence of generalized solutions of coercive elliptic operators we have the following Fredholm alternative THEOREM
D
I f the differential operator L satisfies the conditions 1, 2, 3 of Theorem A and f E L2(~) then we have the next two possibilities 1. B[~o, u] (~o, f ) with ~o e C~(f~) has a unique solution u 9 H~(12), or
323
4.1 The Boundary Value Problem 2. there exists a finite number of linear independent solutions the homogeneous equation
U[v, ~] = o, In this case B[qo, u]  (qo, f) This solution is not unique.
vj, j  1, 2 , . . . g
of
v~ e c g (a).
has a solution if and only if (f, vj) = O, j  1, 2 , . . . ,g.
4 Elliptic Singular P e r t u r b a t i o n s
of Higher Order
4.1 The Boundary Value Problem We study the following Dirichlet Problem for a real valued function u that satisfies the partial differential equation L~[u(x)] = ~L2m[u(x)] + L2k[U(X)] = h(x),
x e Ft C
(12.4.1)
with the boundary conditions
~gu On t (x) = O,
g = O, l, . . . m  1 ,
x e OFt,
(12.4.2)
where Ft is a bounded domain with boundary of class C ~176The differential operators are given by L2k
E (1)IplDP(aP~(x)D~) ' o
(12.4.3)
L2m :
E (1)IplDP(bP"(x)D~) ' O<_lPl,l~l<m
(12.4.4)
and
with 1 < k < m and where the coefficients a p'~, bp" are assumed real valued and of class C~ Moreover, L2k and L2m are uniformly strongly elliptic in ~ while L2k has a strong coercive bilinear form on the Sobolev space H0k(f~). Finally h(x) is also real valued, h E C ~ ( ~ ) and o denotes differentiation in the direction of the outward normal on 0Ft. Since all functions involved are real valued we consider in H0k(Ft) and H~(Ft) only real valued functions. The strong regularity conditions with respect to the data of the problem may be weakened; they have only been introduced to keep the exposition simple and a generalization is of course possible. Instead of the boundary value problem (12.4.1)(12.4.2) we consider the generalized Dirichlet problem eBm[~, u] + Bk[~, u] = e +
E (DP~~ bP~D~u) o
E (DPcp, aP"D"u) = (~, h), gqo 9 C~(f~), 0
(12.4.5)
324
Chapter 12. Perturbations of Higher Order
and it is our task to approximate the solution u C H~(f~) for e + O. This approximation involves the solution w E H0k(f~) of the reduced problem
Blc[cp, w] =
Z
(D~
a~
 (~a, h), V~a e C ~ ( a ) .
(12.4.6)
0
E x i s t e n c e a n d A Priori E s t i m a t e We start with the following lemma.
LEMMA
Whenever L2k has a strong coercive bilinear form on H~ then cL2m + L2k has also a strong coercive bilinear form on H ~ for e sufficiently small. PROOF Gs
inequality (12.3.10) gives for any v E Hy(f~)
Bm[v, v] =
E
(DPv' bP" (x)D" v)
0
>_ c~llvll~  k~llvllo=, where cm and km are constants with coercivity of Bk follows
Bk[v, v] =
c,,, > O; [[vl[0 = [[Vl[L,(a). From the strong
( Dpv, aP"(x)D"v) > ckllvll~, O_
valid for all v E H0k(Ft). Therefore we obtain for v e H ~ ( ~ ) C H~(fl) the inequality
sB,~[v, v] + Bk[v, v] >_ ~ l l v l l = ~  ~k~ll~llo= + ~kll~ll~ > ~11"11~ + ( ~  ~k~)li~llg
1 _> ~ l l v l l ~ + ~ckllvll~ for e sufficiently small. Hence
~B.,[v, v] + Bk[v, v] > ~ l l v l l ~ , 1 c~ which proves the lemma. for O < e < e o  _ ~km,
(12.4.7)
4.3 The Approximation of the Solution
325
From the inequality (12.4.7) it follows t h a t there exists a constant C = { min(cm, ~ck) 1 } 1 , which is independent of e, such t h a t
Ivll~m+ Ilvll~ ~ C{eBm[v,v] + Bk[v, v]},
(12.4.8)
valid for all v E H~(f~) and for 0 < e < e0. Henceforth we use the symbol C as a generic constant, independent of e, which may change its value without violating earlier results, where C has appeared. The l e m m a has the following consequences 1. According to T h e o r e m C the generalized Dirichlet problems (12.4.5) and (12.4.6) have unique solution in H~(f~) and H0k(f~) respectively and due to the regularity of the d a t a we have u e H ~ (a) N C a (~) and w e H0k (f~) M C a (~). 2. W i t h u =
lim ~
n4~
in H~(f~)
and ~,~ e C~(f~)
we have from (12.4.8)
~ll~ll~ + II~ll~ _< C{~Bm[~,~] + B~[~, ~]} = Cli2~ {~B~[~, u] + B~[~, ~]} 1 2 = C n+cx) lira ( ~ , h) = C(u, h) < wllull 2 + C 2 IihiI L2 L2 and therefore ellull ~m+ lull~ _< C l l h l l ~ ,
(12.4.9)
where C is the generic constant independent of e. This relation yields an a priori estimate for u and it appears t h a t u and its strong derivatives up to and including order k are bounded in L2(f~) uniformly with respect to e. This is not necessarily true for the higher derivatives.
4.3
T h e A p p r o x i m a t i o n of the S o l u t i o n It is natural to put
u(x) = w(x) + z(x)
(12.4.10)
where w(x) is the solution of the reduced problem (12.4.6) corresponding with e  0. Hence z C H0k(~t)N C ~ ( F t ) and z satisfies the equation
L~[z] =
~n~[~],
or in bilinear form eBm[~V, z] + Bk[~, z] =  e ( ~ , L2m[w]),
V~ e C ~ ( a )
(12.4.11)
It is impossible to estimate z(x) with the aid of (12.4.9) because z does not belong to H ~ (f~). This follows from the loss of b o u n d a r y conditions in w; therefore we introduce a correction t e r m v with the p r o p e r t y t h a t w + v belongs to H~(Ft). In case we succeed we put u(x) = w(x) + v(x) + 2(x), (12.4.12)
326
Chapter 12. Perturbations of Higher Order
where the remainder 5 belongs to H~(f~). An estimate of 5 in H~(f~) gives an estimate of the precision of the approximation of u(x) by w(x)+v(x) in H~(Yt) and an estimate of v in H0k(f~) together with that of ~ yields the precision of the approximation of u(x) by w(x) in H0k(f~). The function v(x) is a boundary layer function and therefore we introduce a local coordinate system (p, 01, t92,..., 0 n  l ) ; (v~l, 0 2 , . . . , 0,,1) are the coordinates in 0f~ and p denotes the distance from an interior point in f~ to 0ft. Due to the regularity of the boundary, which is compact, there exists a positive number 5 such that the new coordinate system is well defined in the neighbourhood 0 < p < 5 of the boundary 0f~, see Figure 12.1. We introduce also the inner domain fl~o C fl defined as f~o = {x; x c f~, dist(x, 0f~) > 50 } with 0 < 5 0 < 5
and we fix the value of 50.
X~
X~
Fig. 12.1 The neighbourhood of the boundary Using our experience with the exponential behaviour of boundary layer functions we try the correction
v(x)
ajfi,
= r
e x p [   ~~],
(12.4.13)
j=0 where u is a positive number and the r  r with r
aj
constants to be determined later on. Further
1
 1 for 0 < p < ~ 0 ,
r
3
 0 for p > ~ 0
and r e C~ c~). Since w e C~ (all data are regular), the function w + v belongs to C ~ (~) and by the lemma of subsection 3.2.3 we obtain as a sufficient
327
4.3 The Approximation of the Solution condition for w + v c H ~ ( g t )
the following set of equations
Oe(w +v)[o=o = 0 for 0 < i < m  1. Ope  _
(12.4.14)
The function w belongs to H0k N C ~ (l)) and by the same l e m m a we get
O~w Ope ,Io=o = 0 for 0 _< s _< k  1. Substitution of (12.4.13) into this equation gives immediately ak1  0 and therefore v(x)=
r
(12.4.15) a0 = al = a2 . . . . .
( ~j:k l aj 7.1 pj ) exp [  ~~]. P
(12.4.16)
Inserting this result into (12.4.14) yields a simple recursive set of ( r n  k) algebraic equations for the remaining coefficients ak, ak+l'"am1. These equations read l
j~kaj(~) (1)eJ ev(ej) = _.
Oew ~. = k~ k + l m1. Ope Io=o' "'"
After solving we get the following estimates for
ak =
(12.4.17)
aj
Okw Opk [p=o = O(1) Ok+lw
ak Ip=o + (k + 1)~V  O(e  v )
ak+l =
0~i
a~+~ =
Ok+2w 0p~+~ I,=0 + (k + 2)a~+~ ~ ......................
(k + 1)(k 2 + 2)
~a~
= O(~~)
, etc.
W i t h the choice (12.4.17) we have (w + v) e H~(f~)
aj  O(e(Jk)v),
and
j  k, k + 1 , . . . , m 
1.
(12.4.18)
It will appear useful to have an estimate of the norm ]]vl]p in HP(f~). Because derivatives of r are only different from zero for 1/250 < p < 3//450, with 50 > 0 and independent of e, we have for e sufficiently small
Ilvjl2 <_c E where C is independent of e.
~
dl2
328
Chapter 12. Perturbations of Higher Order
Putting p : e~T we get
OPeOev= r
Og"
~
rn 1
[~(x) Z
Tj
eJVaJ j(er] = O(e(ke)u) ;
j=k
since the volume element df~ has a factor d p  r
we obtain (12.4.19)
Because w + v E H ~ (ft) and w e H0k (it) the boundary layer term v belongs also to H0k and it follows that lim v = 0 in H0k; (12.4.20) e~0
therefore this term is not "observable" for e small in the H0k norm, but it becomes infinitely large in the H v norm for p > k. 4.4
The Estimate of the Remainder and Final Results We have defined two remainder terms z(x) and 5(x) by
u(x) = w(x) + z(x) and u(x) = w(x) + v(x) + 5(x), where w(x) is the solution of the reduced equation and v(x) the boundary layer term given by (12.4.16)(12.4.17). In the following we give an a priori estimate of both remainders. From Le[z] = eL2m[w] it follows that
eBm[~, z] + Bk[~o, z] = e(~a, L2m[w]), Because C ~ ( a )
is dense in H ~ ' ( a )
and 5 e H~'(fl)
V~ E C~(12). we get
eBm[5, z] + Bk[5, z] =  e ( 5 , Lzm[w]) or
eBm[5, 5] + Bk[5, 5] =  e(5, L2m[w])  eBm[5, z  5 ]  Bk[5, z  5] =  e(5, n2m[w])  eBm[5, v ]  Bk[5, v]
< ~11~110IIn~[~]ll0 + ~Cl]~ll~ I].11~ + CIl~llk I1,1]~With the aid of the estimate (12.4.19) there results the inequality
where C1 is again a generic constant independent of e. Because 5 e H g ( a ) we may apply the inequality (12.4.8)
zll~_
[s,
+
[s,s],
Remarks
329
and therefore
~11~11~ + I1~11~< c3{~11~110 + ~(km+ 89
+ ~ 89
1 2 < { 4e2C2 + ~11~11~ + 2el+(k_m+~)2vC ~
2 + 4e~,C2+ ~11~11~}, 1 + 111~ lira
or after some reshuffling ell~ll~ + II~ll~ < c 4 { F + el+2r,(km+ 89 4 er'}.
(12.4.21)
The exponent u > 0 is still free and we make now an optimal choice such that the lefthand side becomes as small as possible; this is the case when u = 1 + 2/2(k  m + 1), L
or
(12.4.22) 1
/]
2 ( m  k)" So we have finally obtained the a priori estimate eli5]] 2 + 1]2,]]~  O(e2(:k) ).
(12.4.23)
The estimate for z follows from
~llzll~m + Ilzll~ __<~11~11~+ I1~112+ ~llvll 2 + I1~112 and with the aid of (12.4.19) we get again e]]z[] 2 + [[z[]2 = O(e=(~') ).
(12.4.24)
Summarizing our results we have proved THEOREM
2
The boundary value problem (12.4.1)(12.4.2) has for e sumciently small a unique solution u(x) and this solution is approximated by the solution w(x) of the reduced equation with e = 0. This approximation satisfies the relations
Ilu wllk  o(~<&~>),
l u wllm  o(~
~).
Remarks 1. This result is rather weak, because the discrepancies between u and w are only in the L2 norm; s h a r p e r results are given in [73]. Pointwise approximations may be obtained by applying Sobolev's inequalities, see [43], [2]; further we refer the reader to [10]. 2. Applications to thin plate bending problems are presented by Jiang Furu in [86]. 3. A generalization to certain quasilinear elliptic equations with ellipticelliptic degeneration is possible, see [82].
330
Chapter 12. Perturbations of Higher Order
Exercises 1. Give a formal approximation of the solution of the following boundary value problems d3u
d2u
du
e~Sx3§
0,
0<x
with the boundary conditions a. u ( 0 ) = l , u ( 1 ) = 0 , ~d~(1) = 1 , b. u ( 0 ) = 0 , adu ~ ( 0 ) = l , u ( 1 ) = l . 2. Give a formal approximation of the solution of the boundary value problem dau e dx 3
du d x + u = O,
0 < x < 1,
du u(O) = 1, u ( 1 ) = O, ~xx(1)= 1.
3. Give an approximation of the solution of the following boundary value problem, regarding the deflection of an elastic beam e2dau dx 4
d2u d x 2 = f (x),
O < x < 1.
with the boundary conditions u(O) = d u
du
~x(0) = u(1) = ~x(1) = 0.
Use as well the method of section 2 as that of section 4. 4. Consider the boundary value problem eL2,T,[u(x)] + u ( x ) = h ( x ) ,
with
x e ~ C R,~
~gu On e (x)  O,
g = O, 1, . . . , m  1 ,
xE0~.
The operator L 2 m , the function h ( x ) and the domain fl are as defined in section 3.4.1.
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SUBJECT d'Alembert's paradox 4 asymptotic expansion 13 generalized expansion 22 sequence 12 multiplicable sequence 18 series 13 uniformly 13 asymptoticall~r convergent 13 equal 14 zero 14 BachmannLandau notation 10 barrier function 146, 178 bilinear form 318 boundary layer term 153, 188 parabolic 192 boundary layer flow 3 characteristic time like 211 space like 211 ChikwenduKevorkian approximation 123 coercive 318 strong 3:20 contraction mapping 119 dispersion relation 48 Dirichlet problem generalized 319 homogeneous 318 divergence form 317 energy integrals 140, 215 entrainment of frequency 83
INDEX equation autonomous 283 elliptic 176 quasilinear elliptic 293, 300 hyperbolic 209 quasilinear hyperbolic 250 Duffing 37 Hill 103 Kortewegde Vries 260 Mathieu 105 Van der Pol 79, 247 fixed point theorem 230 Floquet's theory 102 formal adjoint 317 formal approximation 139, 151 Fredholm alternative 322 frequency response curve 83 Gs inequality 318 gauge function 12 Gronwall's lemma 34 KrilovBogoliubovMitropolski theorem (K.B.M) 64, 69, 70 Laplace transformation 199 locking in phenomenon 83 magnetic hydrodynamic flow 199 maximum principle for ordinary differential equations 145 for elliptic partial differential equations 177 for nonlinear elliptic partial differential equations 292
340
Subject Index
method of averaging 61 local averaging 67 multiple scales 92 Lighthill 5O Lagrange 75 Poincar6 2 Temple 55 modulus of ellipticity 317 node attracting 203 repelling 204 operator elliptic 317 strongly elliptic 317 uniform strongly elliptic 317 order symbols 9, 10 oscillation linear 78 linear with damping 78 weakly forced 80 weakly nonlinear 75 perihelium precession 77 perturbation regular 28 singular 1 of boundary layer type 3 of cumulative type 1 perturbed wave equation 123 phase amplitude transformation 75 Poincar~ expansion 2 Poincar~ inequality 321 quasilinear initial value problems 232 boundary value problems 263 regularized boundary layer 194 relaxation oscillation 248 remainder term 142, 151 resonance 164 saddle turning point 205 secular term 45
skock wave 258 singular perturbation of boundary layer type 3 cumulative type 1 elliptic type 175 hyperbolic type 209 Sobolev space 315 solution classical 319 generalized 319 strained coordinate 43 strained parameter 47 theorem of EckhausSanches Palencia 74 HaberLevinson 289 KrilovBogoliubov Mitropolski 64 LaxMilgram 319 Poincar6Bendixson 249 Riesz 319 Wasow 314 time scales 92 transition curve 106 transition layer 274 turning point 152 isolated 202 wave cnoidal 50 travelling 48 wave equation with linear damping 120 with cubic damping 121 of heat conduction 226 W.K.B. approximation 156