Higher order nonlinear elliptic equations

HIGHER ORDER NONLINEAR ELLIPTIC EQUATIONS UDC 517.956.25

I. V. Skrypnik

Methods of solution of boundary problems for divergent and nondivergent higher order nonlinear elliptic equations are described. Applications of topological methods to the study of general nonlinear boundary problems are given. Results on a priori estimates and properties of generalized solutions are cited.

PREFACE The theory of higher order quasilinear elliptic equations is one of the most actively developing directions of the theory of partial differential equations at the present time. The goal of the present paper is a survey of the results on the solvability of nonlinear ellitic boundary problems. The author aimed at describing the basic methods of proof of theorems on the existence of solutions of boundary problems for various classes of equations, from divergent quasilinear equations to essentially nonlinear equations of arbitrary order. The study of second order quasilinear elliptic equations has an almost age-old history and the basic directions of study, regularity of solutions, solvability of boundary problems, are specifically the nineteenth and twentieth problems of Hilbert. The research of S. N. Bernshtein, Leray, Schauder, Morrey, de Giorgi, O. Ao Ladyzhenskaya, N. N. Ural'tseva, and other authors led not only to the solution of Hilbert's problems, but also to the creation of many methods which play a fundamental role both in the theory of differential equations and in adjacent areas of mathematics. There is a survey of these studies in [37]. In solving problems of regularity of equations and systems of arbitrary order, basic results were obtained by Petrovskii [47] who singled out the class of systems now called Petrovskii elliptic, all of whose sufficiently smooth solutions are analytic. A broad range of research on quasilinear higher order elliptic equations has appeared since the beginning of the sixties. The first results were obtained by Vishik [11-13] who, by a modifiction of Galerkin's method proved the solvability of boundary problems. Subsequent progress is connected with the application of the theory of monotone and more general operators. The methods of the theory of monotone operators applied to nonlinear boundary problems are described in the first chapter of the survey. We note the solvability of problems with coercive or noncoercive operators, of boundary problems for equations with strongly growing coefficients, of higher order equations. The chapter contains a survey of the results of M. I. Vishik, Browder, Leray, Lions, Yu. A. Dubinskii, S. I. Pokhozhaev, Gossez, and other authors. Essential progress in the study of nonlinear boundary problems is connected with the creation of topological methods for mappings of monotone type. The foundations of these methods are laid in the papers of Browder, Petryshyn for powers of A-proper maps, of the author for powers [in different terminology, rotation (curl) of a field] of maps of class (S)+ of topological characteristics for essentially nonlinear boundary problems. The solvability of boundary problems for operators with linear noninvertible principal part is also considered in the second chapter. Results on the estimation of the number of solutions based on the method of spherical fibration and application of the theory of powers of maps are also cited in this same chapter. At the same time, due to the restricted size of the paper we are unable to include some questions which are closely related to the theme under discussion. In particular, this Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 37, pp. 3-87, 1990.

0090-4104/91/5604-2505512.50

9 1991 Plenum Publishing Corporation

2505

concerns results on regularity of solutions, ramification of solutions, and averaging nonlinear problems in perforated domains. The basic publications in this direction are included in the literature cited, which facilitates the reader's search for them. To a considerable degree, results on the regularity of generalized solutions are reflected in [58]. In conclusion we note that in citing the literature we sometimes use double references of the form [a:b]. This denotes the paper cited as number b in the literature cited at the end of paper a cited at the end of the survey. CHAPTER I SOLVABILITY OF BOUNDARY PROBLEMS FOR QUASILINEAR ELLIPTIC EQUATIONS OF DIVERGENT FORM In this chapter we give a survey of the basic results concerning the solvability of quasilinear variational and nonvariational problems for divergent elliptic equations. Here, as a rule, on the boundary of the domain Dirichlet conditions will be given although most of the results are valid for other boundary conditions, in particular, nonlinear Neumann conditions which arise n a t u r a l l y . The first studies of the solvability of boundary problems for quasilinear elliptic equations of order 2m were made by Vishik [11-12] on the basis of a modification of Galerkin's method. In proving the convergence of the Galerkin approximations strong Lp-convergence in subdomains of the m-th order derivatives which follows from the a priori estimates of the (m + l)-st derivatives obtained by Vishik and the compactness of the imbedding of Sobolev spaces was established. In the present survey the results on the solvability of boundary problems obtained by Vishik and following him by Browder, Yu. A. Duhinskii, Leray and Lions, S. I. Pokhozhaev, Petryshyn, and many other authors are described on the basis of the theory of monotone operators. Vishik's research, although based on the compactness of the Galerkin approximations, on the whole seemed fundamental in the study of quasilinear problems thanks to the development for such problems of Galerkin's method and thanks to the proof of the a priori estimates with which we shall be concerned later. The reduction of boundary problems for divergent elliptic equations to nonlinear operator equations with operators satisfying specific monotonicity conditions suggested by Browder and developed by a large number of authors is now the simplest and longest range route for studying these problems. The fact is that for equations with monotone operators in applying Galerkin's method it is not necessary to establish the strong convergence of the approximate solutions since it is proved that the limit of weakly convergent approximations is a solution of the original equation. In application to differential problems this removes the need for laboriously finding a priori estimates which are connected with additional restrictions. In this connection basic attention is given to the study of operator equations with operators of monotonic type, the creation of general methods of study of operator equations, and on this basis finding results for differential problems. i.

Formulation of Boundary Problems and Their Reduction to

Operator Equations i.I. Reduction of the Problem to an Operator Equation. Throughout the paper we use the following notation: ~ is a bounded open set in n-dimensional Euclidean space Rn with boundary 0~, x='~1 ..... xn)~Rn; ~=(al ..... ~n) is a multi-index with nonnegative integral components ~i, lal

= =1 + . . .

+ ~n

O~u(x)= (\Oxd ~ ""kOxnl ( ~ u(x), Dku={D~u:l~l=k}. For simplicity f u n c t i o n s w i l l be a s s u m e d t o be r e a l - v a l u e d , Lp(~) is the Banach space of functions which are summable to the p-th power, i < p < ~ over ~. Functions from Lp(~) are defined up to sets of measure zero. For a natural number m, W~(~) is the Sobolev space consisting of functions belonging to Lp(~) and having all generalized derivatives to order m inclusive, summable over ~ to the power p; the norm in W~(~) is defined by

II~ II,~,p=

2506

{!

~

l,,l<m

11

ID~u (x)lp dx ~.

0m

Wp(fl) is the subspace of the space W~(~) obtained by taking the closure in the norm ![Oilm,p of the set of all infinitely differentiable functions with supports in ~. In the presence of specific smoothness of 8~ (cf. imbeddings of the Sobolev spaces

Wpm(~)cW~(Q),

if

w~(~)cw$(~), wy(f~)cc,~,~(6),

i>

q

1

m--~>O,

p

q< ~,

~f

[39]) and w h e n 0 ~ . ~ < m - - I there are

n

]- = m--k ,

p

(~)

n

n-<m-Ck+~), o<~<1.

~f

P

We say that the Banach space X is imbedded in the Banach space Y (X c y) if each element of X is an element of Y and the imbedding operator i:X + Y defined by i(x)=x, x~X, is a one-toone continuous map of X into Y. If i is a compact operator, then we say that the imbedding i is compac!. The last two imbeddings in (i) are compact, the first imbedding is compact for I/q > I/p - (m - k)/n. Imbeddings of Sobolev spaces are used systematically in the study of partial differential equations. Later, for an arbitrary Banach space X we shall denote by X* the dual space. For heX* and u~X we denote by the action of the functional h on the element u. We shall denote strong and weak convergence, respectively, by § and -~. In this point we give the reduction of boundary problems for divergent elliptic equations to operator equations and we clarify the properties of the operators which arise here~ It is assumed that the imbedding theorems hold for the Sobolev space ~4(~) for ~. We denote by M = M(m, n) and M' = M'(m, n), respectively, the number of different multi-indices = (~i ..... ~n) of length at most m, m - I, and let M0 = M - M' Let us assume that for x ~ , ~ = { ~ = : [ ~ [ ~ m } ~ R ~ there are defined functionslA~(x,~), [~[~m, s a t i s f y i n g the following conditions (which are similar to the conditions of Leray and Lions [131]): A I) Caratheodory condition: A=(x,~), l~[<~m, are measurable continuous in $ for almost all x. A 2) For a continuous,

positive, nondecreasing

in x for all values of $ and

function gl one has:

A~(x.~)l..
1~, [r~'), ' ~ [-<m.

(2)

m--~lYl<m

Here

{

n} ,

~0= ~ , : t c z [ < r a - - 7

l
for

n ra--7..<[?l..
r~=p--1, if tal=['ft=m, r~v
l ~ [ < m ---~,

if

and np P=~--n..(m--lcz I)P l
fo~ i ~ ] > m - -

n

7"

for [ ~ [ = m ~ ;

where

q~l p~

for I c ~ I < m _ ~ -n, for

m--n--..<[~z[.~m:. 2507

A3) For a r b i t r a r y

A~) For x ~ ,

xe~

~={~:I=I=t~}6RM,, (3)

IA= (x, n, 0 - - A~ (x, ~, ~')] ( ~ - - ~'=)> 0 for ~+ U. lal=m ~6R~', ~6Rm0o n e has

(4)

Z

I=I=m

l~i=m

m--~<[Vl
with ry < py and continuous, positive, respectively, nonincreasing and nondecreasing functions g2, gs. 0

Let /=(x)ELq=(f~):and V be an arbitrary closed subspace of W~p(~) such that Wf(~)~VcW~(f~). Under these conditions we shall study the boundary problem for the equation

(--I)I=ID=A=(x, u ..... Dmu) = ~ I"[<m

(--l)'='D=f=(x)

(5)

Ioq-<m

corresponding to the subspace V. Definition i.I. By a generalized solution of (5) with respect to the space V we mean a function u~V, such that for a l l ~ 6 V one has the integral identity

e I"l<:m

~ lal<m

When the boundary of the domain ~ is sufficiently smooth, we get functions A~ and a 9

0m

generalized solution u(x) for V = Wp(~) such that u(x) is a solution of (5) in the usual sense satisfying the following boundary conditions:

D " u ( x ) = 0 , I = l < m - - 1 , x~O~,

(7)

i.e., u(x) is a solution of the Dirichlet problem. Also, if certain smoothness conditions hold and V = W~(~2), from (6) we get fortp(x)C~Y~(fl) m--I

k~-] q ~ ( x ) ' N i ( x , u . . . . . D~m-J-'u)ds----O, 1=0 o~ where v ( x ) i s t h e o u t e r n o r m a l t o ~ a t t h e p o i n t x , Nj a r e known f u n c t i o n s of A~, f~. In this case we arrive at the boundary conditions

N j ( x , u, . . . . D2m-J-lu)=0, 7-----0,. . . . m - - l ,

defined

i n terms

x6Of~.

(8)

By analogy with the corresponding problem for a second order equation the problem (fi), (8) is c a l l e d the Neumann problem. It follows immediately from the definition of a generalized solution that the solvability corresponding to the subspace V of the boundary problem for (5) is equivalent to the solvability of the operator equation Au = 0, where the nonlinear operator A.'V + V* is defined for u, ~V by

( Au, r > = ~ lal-<m

S [A= (x, u ..... Dmu) -- f= (x)]D ~ d x .

(9)

e

It follows from conditions At), A 2) the imbedding theorems, and the theorems on the continuity of a Nemytskii operator [34] that (9) defines a bounded continuous operator A:V + V*. The key property of the operator A which is the basis for the operator method of study of boundary problems is that a certain generalized monotonicity property holds. We give the definition of the most frequently used classes of monotone maps. Definition 1.2.

Let X be a Banach space, D be a subset of X, A be a map of D into X*.

Then: a) A is called a monotone map if for arbitrary u, rOD one has >~-0; b) we say that A belongs to the class (S)+ if for an arbitrary sequence unED from Un-~U o and |i--m ~<0 the strong convergence of u n to u 0 follows;

2508

c) A is called a pseudomonotone map if for an arbitrary sequence un@D

from un-~u0 and

li--m( Aun, sn--u 0 } ~<0 it follows that linl< Aun, un--ao > -+0 and if uo~D , then n-~oo

Au=-~Auo;

n-~oo

d) A is called a map with semibounded variation if for arbitrary u, v6X such that 11uU, llvIi~-~R one has

>~--c(R, where

It'll ~ i s

a norm which

is

compact

llu'oll'),

compared with

li-lt t h e

function

c(R,

t)

is

con-

in t and such that I__c(R, t) 0 as t + +0. t Monotone maps were introduced in the papers of R. I. Kachirovskii, M. M. Vainberg~ Minty, and others [58:14], [24], [58:50], [137]. The definition of the class (S)+ is contained in Browder [85] and the author's paper [56]. A condition of pseudomonotonicity similar to the one defined above is due to Brezis [83]. Maps with semibounded variation were introduced by Dubinskii [15]. tinuous

It is easy to establish the interrelation between the classes of maps named. In particular, under weak continuity conditions monotone maps, maps of the class (S)+, and maps with semibounded variation are pseudomonotone. LEMMA i.i [85]. fies condition (S)+.

If conditions AI)-A4) hold, the operator A:V § V* defined by (9) satis-

It is simple to formulate conditions under which the operator A defined by (9) turns out to be an operator with semibounded variation (cf~ [15]). LEMMA 1.2 [85]. A I)-A s) hold and

The operator A:V § V* defined by (9) is pseudomonotone if conditions

A~) For ~ 6 R m', ~, x, ~ER m~ [Aa(x,~, ~)--~=](~=--ff~)-+~-oo as I~[-->OO 1~[=m

(i0)

and (10) holds uniformly with respect to xG~, ~60, where G is an arbitrary bounded subset of

RM'. To conclude the point we comment on conditions AI)-A 4) for the function A~(x, ~). First of all, conditions Al) , A2) guarantee the existence for arbitrary functions u(x), of the integral on the left side of the integral identity (6). These conditions, in particular, guarantee that the map A~ defined for I~[ ->-m - n/p by

~(x)6W~(Q)

i = (u (x)) = A~ (x, u (x) . . . . . acts from W~(~) to Lq~(~) and acts continuously. continuity of a Nemytskii operator [34]. The inequality

(')

r=v
O~u (x))

This is a consequence of the theorems on

forl~l ,2-1?I<2r/z which is contained in A2) guarantees the

subordinate role of the dependence of A a on low order derivatives. A~:W~(~)X~Z~(f~)-+Lq=(~) for tat >- m - n/p defined by

A,, (u (x), ~ (x)) = A~ (x, tz (x) . . . . .

In particular, the map

DZ-Iu (x), Dmv (x))

is compact with respect to u(x). Conditions A~)-A.) guarantee strong convergence in Lemma i.i of a sequence u n such that ~n-~u0, n~.co H m < Aun, u.--u o ) ~< O.

From condition A~) we get the convergence of D~un(X) to D~u0(x)

almost everywhere for lel = m and it follows from condition A~) that uniformly with respect to n Hm ,.o~e..o~ ~, I~o"u~(x)lpax=o,

l~l=m

Ec~.

We note further that for conditions A3)-A4) to hold it suffices if Aa(x , $) is differentiable with respect to g that for x~f~,~ R M ~6R~ one has P-~

I~l=l~l=m

IYl=m

/

[~l=m

2509

with

A~(x, ~)-~ OA=(x, ~)

and continuous, positive, nonincreasing function g.

(ii) which guarantees the ellipticity of (5) is a strengthened version of the ellipticity condition. 1.2.

Formulation of the Variational Problem.

For x ~ , !~@RM

suppose defined a measurable

function f(x, $) which is continuously differentiable in $ for each XE~.

Let

f~(x, ~)----O/(x.~)

and let us assume that the functions f~(x, $) satisfy conditions Al), A 2) of the preceding point and f(x, 0)6L 1(e). Under these conditions the equality

(12)

F(u}= I / ( x , u(x), .... D"u(x))dx

defines a continuous functional F : V c W ~ ( f O - + R a, which at each point uO.V has a Frechet derivative F'(u). Here

(F'(u), v >

-~

X

S/=(~x'u..... mmu)m%dx, ~V.

(13)

I~l-.~mQ The following proposition give the connection of the problem of finding minimum points of the functional F and the boundary problem corresponding to V for the equation

(-- 1)laID=/=(x, ~ ..., Dmu)=O

(14)

l~l<m Proposition i.I. Let the functional F :X-+RII defined on the Banach space X have Gateau derivative F'(u 0) at the local minimum point F'(uo). Then F'(u 0) = 0. The problem of finding the minimum of integral functionals comes from the proposition asserted by Hilbert (twentieth problem [58:95]) on the solvability of each regular variational problem. Here Hilbert admitted that the need can arise of giving an extended interpretation of the concept of solution. Direct methods of the calculus of variations coming from the works of Lebesgue, Tonelli, Morrey [139], let one prove the existence of a minimum of regular functionals. They are based on the lower semicontinuity of integral functionals with respect to some weak convergence, weak compactness of bounded sets in spaces of functions with generalized derivatives. A broad development of direct methods is obtained in the papers of Vainberg and his students (cf. [9] for a survey of results and literature), Browder [58:159], the Bergers [58:145] and others where various propositions about the existence of a minimum of nonlinear functionals in Banach spaces, the convergence of minimizing sequences are established. One of the results about the solvability of variational problems is given below. To formulate a result about the existence of minimum points we need Definition 1.3. Let X be a Banach space, D be a subset of it. The functional F:D-+R is called weakly lower semicontinuous if for any sequence un~D, which converges weakly to one has u0~D, F (u0) < llm F (u.).

I

/2-+oo

THEOREM 1 . 1 [ 9 ] . L e t X be a r e f l e x i v e Banach s p a c e and F : X - + R 1 be a w e a k l y l o w e r s e m i continuous functional satisfying lira F ( u ) = + oo. ( 15 ) 11ulf-~oo

Then there exists a local minimal point of the functional F. It is simple to get (cf. [9]) the lower semicontinuity of a convex functional F. We indicate the connection of the properties of a functional with the properties of its gradient expressed in terms of monotonicity. C x.

2510

Proposition 1.2. Then :

Let D be an open convex set in X and

F:D-->R*

be a functional of class

a) in order that the functional F be convex it is necessary and sufficient that its gradient F' be a monotone map of D into X*; b) in order that the functional F be weakly lower semicontinuous it suffices that F' be a bounded and pseudomonotone map. We note the further fact of the strong convergence of a minimizing sequence of the functional~F:X-+R 1, i.e., a sequence u~X, such that F(u~)~in[{F(u):uEX}. This point is important in finding minimal points. Definition 1.4. The operator A:X + X* is called demicontinuous if for an arbitrary sequence un6X such that un-~u0, one has Aun-~AUo. LEMMA 1.3 [58:155]. Let X be a reflexive Banach space and F : X - + R I be a Gateau differentiable functional, F(u) + +~ as llull § ~ the gradient F' be demicontinuous and satisfy condition (S)+. Then any weakly convergent minimizing sequence converges strongly. The next theorem follows from the results of the present point and Lemmas 1.2 and 1.3. THEOREM 1.2. Let the functional F : V - + R l be defined by (12) and satisfy (15). Let us assume that f(x,O)~Li(~)and the functions f~(x, $) satisfy conditions AI)-A3), A88 Then the boundary problem for (14) corresponding to the subspace V has at least one generalized solution. If in addition condition A 4) holds for f~(x, $), then this generalized solution can be obtained as the strong limit of a minimizing sequence for the functional F. 1.3. Examples of Nonlinear Problems of Mechanics. We indicate here examples of specific nonlinear problems described by divergent equations satisfying the conditions formulated above. The first such problem is the problem of elasto-plastic bending of a rigidly fastened plate. The flexion of the plate w(x, y) satisfies the equation \ 02

2

02

O~e

02w

(16)

i0o= !0o=~

(17)

02w

1

and the boundary conditions

where

g being a function which is characteristic for the given material. The problem (16), (17) was posed by L. M. Kachanov [56:74] and was considered in [56:70] and [56:87] where under certain conditions the existence of a weak solution is proved. .

.

.

.

0

.

2

Under weak assumptions the exlstence and unlqueness of a generallzed solutlon !n Wp(~) p > 1 is proved in [56] as well as the convergence of Galerkin's method for the problem (16i, (17). It is known that F(H 2) = g(H2)H is an increasing function of H for H ~ 0 .

Let us assume

that: 1) g(t) is a continuous function for t > 0; 2) for t > 0 one has --P-I ao+c6t 2 l ,

ao>~0, a ~ > 0

~ Alt 2

,

and f o r p < 2, A o = 0. 0~

,

0

Problem (16), (17) will be considered in the space W~(~) under the assumption f~[W~(Q)]*. Analogously to point i.i we can reduce the problem to the operator equation Au= I, 0

where the operator

(18)

0

A:W~(f~)-+[W~(f~)]* is

defined by 2511

< au,~ > =

a

g(H2(u))

TZ q 2

Oy, ] ex" + exOy" OxOy + ( ~ v , - P T

One can verify that for the natural choice of the functions point i.i hold for them. Thus the operator A defined by (19) is satisfies condition (S)+. In addition, the operator A satisfies plays a fundamental role in the proof of the existence theorem. dition

lim

II#ll~

Ox= ]-~v~j dxdv"

A a conditions AI)-A 4) of continuous, bounded, and another condition which This is the coercivity con-

= + o o .

(20)

)!ult

As a second example we consider the system of strong flexion of thin membranes. First we note that the construction of point i.i carries over trivially to the case of systems of differential equations. We shall not do this just restricting ourselves to the example of a system describing a problem of nonlinear mechanics. The problem reduces to finding a solution of a boundary problem for the flexion w and the stress function F:

A 2 w _ L ( ~ , F ) = I q(x, 9),

D

(21)

A

2

A2F + L (~, w) =0, (x, y)6~cR 2,

E

wl = ~ IS

(22)

I =FI = oe I On Is [s On'Is = 0 '

where A is the Laplacian,

L(w,F)---- a~

Ox~

O'F + a,~ Oy2 Oy'

O2F --2 a~ O~F Ox2 axe)y" OxOy'

fl is a bounded domain in the plane with boundary S; h, E, and D are positive constants. We consider the Hilbert space

/-/= {u= (~, F): ~,

F

0

6W22(~)}

with scalar product

(u, X)-----l:i~==I~ {D=wD=~+ D=FD=~;} dxdy, where X=(r

4).

By a generalized solution of the problem (21), (22) we mean a vector-function

uEH, which for all x~H satisfies the equation D.AwA~p.-L(w,F)~+TAFAcp. Q

+L(w,w)~

dxdy= T

h

q(x,g)~p(x,y)dxdy. o

We define an operator A:H + H and an element Q 6 H

(Au, x)= S~

such that

(w, F)~ + E2. AFAr+ L(w, w ) , } d x d y , !

(Q, z)---'K- IS q~dxdy. Here i t is assumed that q(x, y)6Lt(~). Analogously to point i.i one can see that A is a bounded continuous operator satisfying condition (S)+.

It is easy to verify that for

Q

From this one gets that (20) holds for the operator considered. The system of strong flexion of thin membranes was considered in [17, 56], [56:39], [56:72], its solvability is guaranteed by the results of the next section. One can consider the complete system of equations of strong flexion of a membrane [56:52], the problem of equilibrium of sloping thin hulls [56:40], and the boundary problems for equations of yon Karman type [56:4] analogously. 2512

2.

Existence of Solutions of Operator Equations and Boundary Problems

2.1. Solvability of Coercive Operator Equations. A large number of papers starting with those of Browder [87], [58:161] are devoted to the study of the solvability of equations with operators satisfying a monotonicity condition or various generalizations of this condition. There are surveys of these results in [14, 58]. Here we cite one of the existence theorems. THEOREM 2.1. Let X be a reflexive, separable Banach space and A:X § X* be a bounded, demicontinuous, pseudomonotone operator satisfying the coercivity condition (201}. Then the equation Au = h has a solution for any h6X*. We note the basic points of the proof. Let {vi} be a complete countable system of the space X. We define Galerkin approximations u n = clv I + ... + CnV n as solutions of the equations

, i~----1. . . . . n.

(23)

The coercivity condition (20) guarantees the solvability of (23) for each n by virtue of the familiar acute angle le~ma (cf., e.g., [34, p. 314]). The boundedness of the sequence un also follows from (20), i.eo, for some number R independent of n,

]]unlt~R, n = l , 2 . . . . .

(24)

By virtue of the reflexivity of the space X the sequence u n contains a weakly convergent subsequence. Without loss of generality we can assume that u n + u 0 . We choose another sequence w n = k~v I + ... knv n such that w n § u~. Then from

=+ it follows that lim ( A~n, :u~--u0 ) =0.

By virtue of the pseudomonotonicity we get

Au~-~Auo.

Passing to the limit in (23) with respect to n for each fixed i, we get Au 0 = h, Remark 2.1. It follows simply from what was said above that if in addition to the hypotheses of Theorem 2.1 the operator A belongs to the class (S)+ then the Galerkin approximations u n converge strongly to a solution u 0 of the equation Au = h. Remark 2.2. It follows from what was said above that Theorem 2.1 remains valid upon replacing the condition of pseudomonotonicity of the operator A by the following condition: for an arbitrary sequence u~EX from u~-~, lim ~---O it follows thatAu~-~Au0. 2.2. Solvability of Coercive Boundary Problems. From the reduction given in point i.I of the boundary problem to an operator equation, and from Theorem 2.1, it follows that the next theorem holds. THEOREM 2.2. Let us assume that the functions A~(x, $) satisfy conditions Al)-Aa), A~) and for the operator defined by (9) let the following coercivity condition hold:

llm [[U[im~

where I[.[[m, p i s t h e norm in W~(~).


[lU]tm,p

=-~-~,

Then f o r a r b i t r a r y

(25)

f~(x)ELq~(~) t h e boundary problem f o r

(5) corresponding to the subspace V has at least one solution belonging to V. 0m

For the coercivity condition (25) to hold it suffices, require that I~[<m

[~[ =m

for example,

for V = Wp(g) to

I~!<m--I

where v, ~ are positive constants, q
If the hypotheses of point 1.3 hold, the problem (16), (17) on elasto-

plastic bending of a rigidly attached plate has a generalized solution u0(x , 9)EW~i~(~) for an 0 2

arbitrary function f@[~/~(~)]*. This solution can be obtained as the limit of strongly convergent Galerkin approximations in W~(~). 2513

THEOREM 2.4.

If the hypotheses of point 1.3 hold, the problem (21), (22) on strong

bending of thin membranes has a generalized solution (~, P)EIW~(~)]~ for an arbitrary function 0 qEIW~(~)]*. T h i s solution can be obtained as the limit of strongly convergent Galerkin approximations in [W~(fl)]2. We note the further possibility of a proof on the basis of Theorem 2.1 of the solvability of a quasilinear problem with stronger growth of subordinate terms than was assumed in point i.i. As an example we consider the differential equation

~(--1)I='D=A=(x, u . . . . . D mu) Jr- ~ I=l<:m

(--1)'~lDf~B~(x,u . . . . . D"-'u)~-

II~I<:m-~

~(--1)'"rD"f.,

(26)

l=l<:ra

where A~(x, $) satisfy the hypotheses of point I.i for p < n. functions Bs(x, q) for x ~ , ~, ~'ER u' one has

Let us assume that for the

LB~(x'rl)'"
(27)

p

[B#2(x,-T[)--B~ (x,

n')l(n#--n;)>O,

(28)

I~l=m-I

if q~ = D~ for ]#'l<m--2;

Z

B~(x,n)n~>/g~(]~lol)~

In=l;-g,!(l,.noq)

in, l

(29)

m--~
--

np

--

p > ~ - - ~ , r ~ , r~ a r e determined, by the same c o n d i t i o n s as r~x, r 7 in c o n d i t i o n A~) but

with p replaced by p,

N0

={~=:;~l<m-~},~(x)~Z~i(fl), gx, g~being

positive, continuous, re-

spectively nonincreasing and nondecreasing functions.

, [wm-'(fl)] * by We d e f i n e an o p e r a t o r ~: Wm--~,~ ~ ~ )-~[ b<Bu,~)----

~

I B~(x,u ..... Dra-'u)D~q)dx"

(30)

We also assume that for the operator B the coercivity condition -I II u lira_,,; <

lira

Bu,[u > = -l-

holds. THEOREM 2.5. Let the f u n c t i o n s Aa(x, $) s a t i s f y the c o n d i t i o n s A1)-A3), A~) the funct i o n s B~(x, q) s a t i s f y AI) and (27) and (28). Let us assume t h a t c o e r c i v i t y c o n d i t i o n s hold for the operators A and B. Then for any functions f~(x) belonging to Lp~(fl) for I~I = m and to L ~ ( ~ ) for l~l-.<m--lthe boundary problem for (26) corresponding to V = V N ~ 7 ~ - I ( ~ ) has at least one solution in V. The assertion of the theorem follows from Theorem 2.1 provided we prove the pseudomonotonicity of the operator A + B. We outline the verification of this property. Let u n be an arbitrary sequence of the space ~ which is weakly convergent to u 0 such that |i-m < AUn-~Bun, un--u0 ) ~ 0 .

Using (27) and (28), from this we have li'-m ( Au n,

Un--Uo>-.
consequently,

o

by virtue of Lemma 1.2, lira < A u n, gn--u 0 )-----0,

A~n~Au o

in

[~72(~)]*. From this and from the

assumption about the sequence u n we get

lira < Bun, un--u o > ...<0.

(31)

l'l " ~ O 0

From the weak convergence of the sequence u n in V and the compactness of the imbedding of W~D(~) in W~-l(~) the convergence of the sequence {D~,(x)}. I ~ l < m - - I in measure follows. From-(28), (29), and (31), we get

2514

lira I [ Dm-~u" IPdx= 0 uniformly with respect to EG~, which guarantees the strong convergence of u n in W~-I(Q) [58:29]).

(cf.

From this we h a v e ( Bun, un--uo )-+0, Bun-+Bu o in [~-I(fl)]% which completes the verification of the pseudomonotonicity of the map A + B. 2.3. Solvability of Equations with Odd Operators. Following the proof of the solvability of equations with coercive operators, existence theorems without coercivity conditions were obtained. The first result here was Pokhozhaev [48] in which the solvability of an equation with an odd operator was proved. Subsequently these questions were studied by Browder, Petryshyn, Necas, the author, etc. (cf., e.g., [56, 85, 157, 144]). These results are obtained naturally using the theory of degrees and we shall be concerned with this in the second chapter. We note only one result similar to that obtained in [48]. Definition 2.1. A(-u) = -An.

The operator A:X § X* is called odd if for arbitrary u6X one has

THEOREM 2.6. Let A:X * X* be a bounded demicontinuous odd operator satisfying (S)+ and let us assume that for some h~X*, R > 0 one has

[Ihll.
for u@X, HuH=R,

Then in the ball

(32)

BR=={ueX :llul[<-~R}there exists a solution of

The proof of this theorem is analogous to the proof of Theorem 2.1 but the existence of the Galerkin approximations is established by application not of the acute angle lemma but by the Lyusternik-Shnirel'man-Borsuk theorem (cf. [34]). The next theorem follows from Theorem 2.6. THEOREM 2.7. Let the functions Ae(x, g) satisfy conditions A~)--A4), A~(x,--~)=--A=(x,~) for ]~]~-~.m, x ~ , ~ R ~ and let us assume that for some continuous functionlK: R+I~-R+ l for an arbitrary solution of (5) u(x)6V corresponding to the subspace V one has

Then for arbitrary

f~(x)~Lq~(Q) there exists a solution of (5) corresponding to V.

2.4. Nonlinear Operators Having Weakly Closed Range. In [49], [58:90] Pokhozhaev singled out the class of normally solvable operators and proved a solvability theorem for nonlinear equations with operators having weakly closed ranges. Application of this result to boundary problems gives the existence of a solution without the assumption of a coercive condition. In what follows X and Y are Banach spaces, A:X § Y is an operator of class C I, A'(u) is the derivative of the operator A at the point u, [A'(u)]* is the dual operator to A'(u), Ker [A'(u)]* is its kernel. THEOREM 2.8 [49]. Let Y be a reflexive Banach space and A:X + Y be an operator of class C z whose range A(X) is weakly closed in Y. Then if Ker [A'(u)]* = {0} for any u~X, then the equation Au = h is solvable for arbitrary h6Y. We note the application of Theorem 2.8 to the proof of the solvability of a boundary problem for a quasilinear elliptic equation of divergent form although in [49] there are also applications to nondivergent equations. We consider the Dirichlet problem

(--I)!~'D~[A,( x , u .... ,OJu)O'ul-k ~ (--1)'~'D~B~(x,u ..... O~-'u) = ~ (--1)'~lO=f~(x), ( 3 3 ) D=u ~ x ) = 0 f o r ] ~[ ~< m - - 1, x6O~ under the assumptions

that

(34)

2m > n, j < m - n / 2 . 2515

In what follows we assume the following conditions hold: i) for xG~, ~6~ My , the functions A=(x,~), I ~ I = m are continuous and have continuous first derivatives with respect to $; here M 3 is the number of different multi-indices of length at most j ; 2) for x ~ , ~]E~m'[ the functions B~(x,I]), I~]..<~ and their first derivatives with respect to q satisfy Caratheodory conditions; 3) the functions B$(x, q) satisfy condition A 2) for p = 2, and their derivatives

OB~

N)=~(X,

x-~

W)' for

'

B~v(x,

--M'

e ,;NeM

, satisfy the inequalities

m---~-<.161<m-2n

S~=n_2(m__!8!)

where for

tz~<2(~--[6[),

f o r t ~ > 2 ( m - - 1 6 1 ) , s~ b e i n g an a r b i t r a r y

S[~,.~,-~-l-- 1

number g r e a t e r

t h a n one

1 b(x~Ll(~); ~ [ o - - - ~ { r ~ : l ~ l < r n - - 2 } .

o o Under these conditions the operator A: ~7~(~)_~ [W~m (~)], , defined for the problem (33), (34) analogously to (9), belongs to the class C ~. We note that now the operator [A'(u)]* 0

0

acts from ~y~(~) to [~/~(~.)]* and is defined as follows:

( [A,('u)l*v, ~ ) =~ l ~

A~(x,u ..... D~u)D~'vD~P+

'~(~=1 m

+ X X

.....

lat=m tvl
where

+

X

A.y(x,~)-~-~ A.(x, ~). THEOREM 2.9 [ 4 9 ] .

Let us assume that

conditions

1)-3)

0

0 have o n l y t h e zero s o l u t i o n f u n c t i o n K:II~-~It~- s u c h t h a t

)

lluil~,~
m

for arbitrary functions 3.

.....

1131<m-1l~l<m

hold.

Let the equation

[A'(u)]*v=

0

in W~(Q) f o r a r b i t r a r y

ufiW~(f~) and l e t 0

f o r any s o l u t i o n u(x)EIIg~'~(a)

there exist

of the problem (33),

a continuous

(34)one

has 0

for t . = s_~_T_l. Then the problem (33), (34) has a solution in W~(Q)

f=(x) ELi=(e).

Solvability of Elliptic Problems with Nonpower Nonlinearities

3.1. Boundary Problems in Sobolev-Orlicz Spaces. We return to the problem (16), (17) of elasto-plastic bending of a rigidly fastened plate without assuming that the function g characterizing the properties of the material of the plate has power growth. In this case the most suitable space for investigating the problem (16), (17) turns out to be the Sobolev0 Orlicz spade W2L~(Q),constructed from the function

O([)=

I'g(~2)~d~.

(35)

0 We give the basic definitions of Sobolev-Orlicz spaces (there are more details given in point 1.4 of Chap. 1 of Yu. A. Dubinskii's survey contained in the present issue). The function O:RI-§ I is called an N-function if it is continuous, convex, even, and satisfies the conditions O(t) > 0 for t ~ 0, 0(t)/t + 0 as t + 0 and r ~ +~ as t + +~. For an arbitrary open (generally unbounded) set ~ c R ~ and N-function O:RI-+R$ we denote by LO(~) the Banach space of functions u(x) having finite Luxemburg norm

The d e f i n i t i o n s skii [35].

2516

and p r o p e r t i e s

of Orlicz

spaces

are studied

in Krasnosel'skii

and R u t i t -

We denote the closure of the set of bounded, measurable functions with compact support in ~ by E0(~). There is an inclusion E ~ ( ~ ) ~ L ~ ( ~ ) , w h i c h is an equality if and only if satisfies a A2-condition , i.e., if for some k > 0 one has(D(20 ~
~i=i<m

The spaces WmL~(~), WmEr can be identified with subspaces of the space NL~(~) which is the product M of the spaces L~(~), where M is the number of different multi-indices of 9

length at most m.

0

We denote by W ~ L ~ (~) the closure of C~(~) in WmL~(~) with respect to the

o(llA~ (~), IIE~(~)) -topology and by W = E ~ (~) the closure of Co(~) in WmEr with respect to the norm ll.llm,~. Here ~ is the N-function associated with ~ and is defined by

~ (0 = sup {st - - ~, (s)}. s>O

,

We give imbedding theorems from Donaldson and Trudinger [98] for the case of sufficiently smooth bounded ~. Let c o be an arbitrary N-function and let us assume that n~2. By varying the value of c o on a bounded subset of R I, one can guarantee that I0(1) < +~, where t S

!

4(t)= c;'(~)~

_1__

"d~.

0

If I0(+=) = +~ we define a new N-function c I by c~l(t) = 10(t). Continuing this process we get a finite sequence of N-functions co, cl, .... Cq, whereq=q(co)<.n is a number such that +=

-I-L

+~=

I

_,_L

I

For N-functions r ~ we shall write (1)-~lr if for some k > 0 and sufficiently large t, ~(0<~F(/~t),and we shall write ~<<~F, if for each l>0q)(0[q~(~t)]-~-~0 as t + ~. Further, for an N-function r we shall assume that I

d * < + =o. 0

(36)

~

For such a function we define (1)=~-c~-t=l for finite sequence constructed above for c o = ~.

ra--q(fD):~Io~l~m , where

Remark 3.1 [120]. I f f o r an N - f u n c t i o n r (36) h o l d s a n d m " q ( ffO ..< l o~ ] ..< m ~<<~=.

+oo

c o ..... Cq(c0 ) is the l

~-~(x)~

~d~----+~, then for

LEMMA 3.1 [98]. Let 9 be a bounded open subset of R ~ with locally Lipschitz boundary. For some N-function r satisfying (36) let ~(x)~W~L~ (Q). For m--q(q~)~lai~m we define r = c=-;=I, where c i is the finite sequence constructeo above for c o = ~. Then: a) for m - - q ( ( D ) ~ [ c ~ i : ~ m , O=u(x)~i~=(~) compact imbedding if ~l~<
D=~(x)~C(~)

with continuous imbedding and~D=~(x)(~P-7~(f2)with

with compact imbedding.

Under the hypotheses of Theorem 3.1 also for u(x)6W~Ee (f~) and m - q(.(D).~
+ l~l=m

(37)

=-q(~)
lN=m

m-q(~)~IN<m

Here ~o-----{~=:]r c is a positive constant, ~ is positive, continuous, nondecreasing function, h=(x)~E~=(~) for l ~ [ = m , h=(~)@L~=(~) for [e[ < m. In [37] ~-~,_ ~ are the functions dual, respectively, to ~=, F; ~[z, ~-z are the inverse functions to ~ , F. Under these conditions we define a solution of (5) under homogeneous Dirichlet conditions (7) assuming that for I ~ I < m .

fa(x)EE~=(O)

Definition 3.1. By a generalized solution of the problem (5), (7) under conditions At), A2) is meant a function u(x)6W=L~(~), such that

A=(x, ufx) ..... Dmufx)~L~=(O)

for' [ ~ J ~ m

(38)

0

and for a l l ~ ( X ) 6 W ~ E ~ (~)

the integral identity (6) holds.

By analogy with (9) we can define a nonlinear operator A defined on the set D(A) con0

sisting of those functions

u(x)6~L~ (~),for which (38) holds and defined by (9) for u(x)~D(A),

o

~(x)~E~

0

(~)., The operator A acts here from D(A) to D(A) B [W=E~ (~)]"

A specific character of the case considered is that if a A2-condition holds for the 0

function r the operator A just defined is not generally defined on ~ L ~ (fl). Because of this it is necessary to introduce special supplementary systems and consider operators in these systems satisfying monotonicity conditions. The following point is devoted to these questions. 3.2. Equations with Pseudomonotone Operators in Supplementary Systems. Let Y and Z be Banach spaces with norms ll'i]y, ll-ilZ which are dual with respect to a continuous bilinear form <., -> on Y x Z and let us assume that for y~Y, z~Z .~
the linear

maps

The quadruple of spaces (Y, Y0; Z, Z 0) is called a supplementary system

i:Z'-*'Yo*, j:

Y - ~ Z o * , d e f i n e d by

(iz)(yo]=,

~0~Y0, are homeomorphisms of Z and Y, respectively,

z~Z, zo~Z~,y6Y,

(]y)(Zo) = f o r

onto Y~, Z~. 0

0

0

An example of a supplementary system is the quadruple (~/mLe(~), %VmE~(~), [%[/m~e(O)]*, 0

[WmL~(Q)] *, where we have used the notation of the preceding point.

[~g~Eo(Q)]*----If---- ~ t o

l=l~m

Here one can identify

( - - I ) '=' D=f=tx):f=(x)EL$(a),

Ial<m}.

(x)ee6 (a),

!= l ~< m]-

- - 1 )' , D f = (x): f =

In the present case the bilinear form <., .> with which we are concerned in the definition of 9

0

0

of a supplementary system is defined for u6W~L~ (~), [~[W=Ee (~)]* by Ir Cf.

[98,

120] w i t h r e s p e c t

to supplementary systems.

Definition 3.3. L e t (Y, Y0; Z, Z 0) be a s u p p l e m e n t a r y s y s t e m and l e t A:D(A)cY-+Z be a n o n l i n e a r map w i t h domain o f d e f i n i t i o n D(A). We s h a l l s a y t h a t t h e o p e r a t o r h i s p s e u d Om o n o t o n e w i t h r e s p e c t t o t h e d e n s e s u b s p a c e V o f t h e s p a c e V0 i f : 1) h a c t s c o n t i n u o u s l y f r o m e a c h f i n i t e - d i m e n s i o n a l s i d e r e d with o(Z, V)-convergence;

subspace F c V into the space Z con-

2) for each sequence u n in V, which is o(Y, Z0)-convergent to ~6Y, it follows from the o(Z, V)-convergence of Au n to k6Z and [ ~ ..< that Au = h and + . "~ We formulate a result obtained by Gossez about the solvability of an operator equation.

2518

THEOREM 3.1. Let (Y, Y0; Z, Z 0) be a supplementary system and A:D(A) be an operator which is pseudomonotone with respect to a dense subspace V of the space Y0. Let us assume that Y0 c D(A) c y and:

{u~D(A) :

a) for each h6Z0 there exists a neighborhood O(h) in Z such that the set

Au40(h)}cY is bounded; b) ~O for a l l u~Y0 with I]u]Jy s u f f i c i e n t l y large. Then the equation Au = h has a solution for arbitrary

h~Zo~

3.3. Solvability of Boundary Problems The next theorem assures the applicability of Theorem 3.1 to the solvability of the problem (5), (7) under conditions Al), A3), A~), ~A2)THEOREM 3.2 [120].

Let fl be a bounded open subset of R n with locally Lipschitz boundary 0

!

and let us assume that conditions At) , Aa) , A4), A2) hold. Then the operator 0

[WmEo(~)]*

A:D(A)cWmLo{~)-+ 0

is pseudomonotone with respect to a dense subspace V of the space ~WmEo(~).

Under the hypotheses of Theorem 3.2 the operator A does not satisfy assumptions a) and b) of Theorem 3.1. We can guarantee the latter assumption by the following condition: there exist functions b=(x)OE~(Q) for [~[~<m and b(x)~Ll(~) and positive constants d I and d~ such that

(39) for xs

~fiRM.

THEOREM 3.3. Let ~ be a bounded open subset of Rn with locally Lipschitz boundary and let us assume that conditions At) , Aa) , A2), and (39) hold. Then for arbitrary f=(x)GE~, (~), 0

l~I~<~ the problem (5), (7) has at least one solution in

WmLo(e).

We give examples of solvable problems for more specific choice of the functions A=(x, ~) and the previous conditions about the domain g. Let:p: R*-+~ l, be a nondecreasing continuous odd function such that p(t) > 0 for t > 0, p(t) § +~ as t § +~ and we define t

~(t)= l p(*)d~. 0

We consider the Dirichlet problem for the equation

(-- |)I='D=p(D=u)= ~ ( _ l=l~m

I)'=IO~f=(x).

(40)

]alUm

On the basis of simple inequalities relating the functions r and ~ (cf. [35, Chap~ i, Sec. 2]), one can verify that for the problem (40), (7) conditions Al), A~), ~2), and (39) hold. So the next theorem follows from Theorem 3.3. THEOREM 3.4.

Under the conditions formulated above for the function p:RI-+R I and arbi0

trary functions Lr

f=(x)6E~(~), [~I-.<~the problem

If in particular examples of solvable

(40), (7) has a solution u(x) belonging to wm x

one takes as p(t) in (40) the function [eltl~l]signt, 2~e~,then problems with strongly growing coefficients. For the choice

p(t)___--[I n ( l + l t l ) ~ 'I +t~l ]--

we get in (40)

of

slgntwe have an example of an equation with slowly growing coeffi-

cients. One gets the next theorem analogously. THEOREM 3.5. In (16) let the function g(t) be such that r(H =) = g(H=)H is a positive increasing function which is continuous for H > 0 and satisfies the conditions, lim F([)= 0, t~+O

llm r(t)= + ~.

Also let ~(t) be defined g(T) by (35).

Daf~(x,Y), fa~E~(~)

Then for

arbitrary,f(X,y)~ Z

(--l)~

o

the problem (16), (17) has a unique solution

u(x,y)~W~L~(~).

2519

4.

Solvability of Boundary Problems for Equations of Infinite Order

4.1. Formulation of Boundary Problems. The theory of nonlinear equations of infinite order as developed in Dubinskii (cf. [19, ]4]) shows the natural connection of the classes of equations of infinite order introduced by him and the boundary problems for them with some problems of nonlinear mechanics. The consideration of nonlinear equations of infinite order by functional methods requires the introduction and study of Sobolev spaces of infinite order. The determination of conditions under which these spaces do not reduce to the zero element alone was a nontraditional question. Exhaustive answers for various domains (bounded domains, the spaces R ~, the torus) were obtained by Yu. A. Dubinskii in the form of corresponding criteria. I n addition important connections were discovered with such questions of analysis as the analyticity of functions of several complex variables, Hadamard classes of one real variable. Let ~ be a bounded domain in R ~ with boundary 8~ of class C~ and in ~ we consider the boundary problem

2 (--I)'~ID=A~(x,a. . . . . D~)-~ Y, (--l)I~la~D=h=(x,),

m=0 lal=m

(41)

I~I=0

D=u(x)=O, xeOf~, I,~l=O, 1. . . .

(42)

We shall assume that A~(x,~0 ..... ~m), [=l=m, m = 0, 1 .... are defined for xfi~, ~-----{~: IyI=/}~R N<~), ]= 0, .... m are real-valued functions satisfying condition A~) of Sec. 1 [here N(j) is the number of different multi-indices Y of length j ]. In addition, for xO~, ~ 6 R Nu), ]---0..... rn,~]~l~N~m~, m = 0, i,... let the following inequalities hold

I ~ A=(x,~o..... ~)n,,l-..
(43)

I"1=m

A~(x,~o

~ a~lhl P~t--b,~,

~,~)~>~,

. . . . .

l~l=m

(44)

I"I=m

with constants aa, Pa, bm, c, ~ satisfying the conditions a=~>0, p~>1, b=~0, r w>0. In addition we assume that the sequence Pa is bounded, b I + b 2 +... < ~ the functions ha(x) in (41) belong to 6,=(~), t

a,, ll h= (x) lt%,a < ~ , ,

I~I=0

where

I1"11/. is the norm

(45)

in L ,(~). P~

Corresponding to the problem (41), (42) we define the space

W o o {a~, p=}----

'

a,,tID~ttlt

oo

(x)fiCD (f~): p ( u ) - 151=0

oo

--

< oo,

(46)

'fl

~

where CD(~ ) i s t h e space o f f u n c t i o n s u ( x ) w h i c h a r e i n f i n i t e l y differentiable i n ~ and satisfy (42). I f among t h e numbers a a t h e r e a r e i n f i n i t e l y many p o s i t i v e ones, t h e n t h e 0

question arises of the nontriviality of the space zero functionu(x)fiCD=(~), for which p(u) < ~.

W~{a~,p~}, i.e.,

the existence of a non-

There is a criterion for the nontriviality of

0

W={a=,p=}i in [19, 14] (cf. also Dubinskii's survey in the present collection, Sec. 3 of Chap. 0

i), and we shall assume that W={a=,p=}=~{0} 9 0 Definition 4.1. The function u(x~%~'~{aa, Pa) is called a solution of the problem (41), 0

(42) if for an arbitrary function-o{x~V/~{a=, p=}

one has the integra ! identity c~

~

m=Ol"l=m

~A~(x, tt. . . . ,D~u)D"~(x)dx = ~ aa~ h,,(x)D=~(x)dx. ~

(47)

I=1=o

We consider another boundary problem for (41) on the torus T n obtained from the cube Kn-----{xs i=| .....n) by identification of points x'-----(x~ ....,x',),x"=(x~.....x~)6OK" such that x: = x m o d 2 w for j = 1 ..... n. 3 J 2520

We denote by C~(T n) the restriction to K n of the space of infinitely differentiab!e 2~periodic functions in R n and we consider the problem of finding in C~(T n) a solution of (41) under the assumption that for x6T n, ~j~R~(~),]=0 .... ,m, the functions A=(x, ~o.... , ~m), [~I =m, m = 0, i, 2 .... satisfy Caratheodory conditions and the inequalities (43) and (44), and ha(x) belong to Lp~(T n) and satisfy (45) for fl = T n. We define the corresponding Sobolev space of infinite order

a=IL1:)=ull, < oo}

I=(=0 and we shall assume its nontriviality.

Definition 4.2. The function u(x)EW~{a=,P=} is called a solution of the periodic boundary problem for (41) if for an arbitrary function v(x)~W~{a=,p=} the integral identity (47) holds with ~ = K n. 4.2. Existence of Solutions of Boundary Problems. We formulate results on the solvability of the problem (41), (42) assuming the following conditions abou t {a=,p=} hold: 0

W I) there exists an integral sequence rj tending to infinity as j § ~ such that for sufficiently large j, from an arbitrary sequence Uk(X) satisfying the conditions

IIp=,~"-
o
(48)

with constant c independent of k, one can extract a subsequence which is convergent in crJ(~); 0

0

W2) the spaceW~{a=, p=} is nontrivial,

i.e., does not reduce to the zero element.

THEOREM 4.1. Let us assume that the functions As(x , $0 ..... gm) for !a I = 0, i,..~ satisfy a Caratheodory condition and the inequalities (43) and (44), and let conditions W 2) hold.

Then for arbitrary functions

(~),satisfying (45), the problem (41),

0

(42) has at least one solution in RT={a=, p=}, The proof is based on the construction of a solvable approximating boundary problem for an equation of finite order and subsequent passage to the limit as the order of the equation tends to infinity. We approximate the problem (41), (42) by the following sequence of problems: k

~'~ (--l)'=la=O=[ID=u!P=-2D=u]~~., ~ (--I)=D={A=(x,u, . . . . D~u)--h=(x)}=O, ja~=k+,

(49)

(50)

D=u(x) =0, !at
Analogously to Sec. i one can define a solution and prove the solvability of the problem (49), (50). We denote by Uk(X) one of its solutions. Then from (44) for ]6Z and sufficiently large k we have P~,~

with constant c independent of k.

0

Using condition Wl) we can ~se a diagonal process to choose

a subsequence ilk(X) of Uk(X) such that for some function g0{x)@VY~{a=, P=}~O~k(x) converges uniformly to D~u0(x) for each multi-index ~ [here we consider D ~ k ( X ) starting from some sufficiently large number k depending on ~]. One proves that the limit function u0(x) is the solution of the problem (41), (42) sought. The periodic problem for (41) can be considered analogously. following conditions hold:

Let us assume that the

W I) there exists an integral sequence rj tending to infinity as j + ~ such that for sufficiently large j, from an arbitrary sequence Uk(X ) defined on Tn, satisfying the condition ~='~n< =o IIO=uk IL=,_

c,

O~l=l<J

with constant c independent of k, one can extract a subsequence which converges in crj(Tn); 2521

W 2) the space

W~{a=,p=} is dense in L2(Tn).

THEOREM 4.2. Let us assume that the functions A=(x, ~o..... ~m), [=I=0, |..... are defined for x~T '~, ~j6RIvCj), j--0,;..,m, satisfy a Caratheodory condition and (43) and (44), with p=>/p>l and let conditions WI) , W2) hold. Then for arbitrary functions h=(x)6Lp,=~Tn),satisfying (45) with ~ = T n the periodic problem for (41) has at least one solution in w| 4.3. Behavior of Solutions of Nonlinear Elliptic Equations of Order 2k as k + ~. The passage to the limit in the problem (49), (50) described above is a model for the more general situation. In the domain f~cR n we consider the family of Dirichlet problems k

oo

~ (--1)l"lO"A=,k(x, u ..... Omu)= ~(--1)l"lO~h,,,~(x), m~O lal=m

(51)

I~.!=0

D=u(x)----O, Iczl
(52)

and we are interested in the behavior of the solutions of the problem (51), (52) as k ~ assuming these solutions are known. Dubinskii (cf. [14]) studies the question of convergence of such a sequence of solutions Uk(X). If the limit equation has infinite order, then if inequalities of the form (43) and (44) hold for the functions A~,k(X , g0 .... ,gm) with constants a~,k, P~,k depending on k, one proves under the convergence of A~, k to the functions Aa and the numbers a~,k, Pa,k to as, Pk and the corresponding convergence of h~,k(X) that the sequence Uk(X) has a limit point u(x) 0

W~

p=}, which is a solution of the problem (41), (42). Just as in the limit passage in the problem (49), (50) one uses an analog of condition

0

W I) which assures for the subsequence of {Uk(X) } the uniform convergence of any derivatives. The latter situation does not hold if the limit equation has finite order. In this case to prove the limit passage one uses ideas of the theory of monotone maps which requires an additional monotonicity condition for A~(x, ~0 .... ,~m)" CHAPTER 2 TOPOLOGICAL METHODS IN THE THEORY OF NONLINEAR ELLIPTIC BOUNDARY PROBLEMS In the papers of Vishik, Browder, Leray-Lions, of which we spoke in the first chapter, not only are specific classes of nonlinear boundary problems, but proaches to the study of these problems on the basis

Petryshyn, Pokhozhaev, Dubinskii, etc. solvability theorems established for even more importantly, general apof operator methods are found.

Subsequent progress in the study of nonlinear boundary problems is connected with the creation of topological methods for the study of new classes of operator equations. The origin of topological methods in nonlinear analysis based on the theory of degrees of maps is found in the famous paper of Leray and Schauder [132] which is devoted to maps of the form "identity plus completely continuous." The methods of Leray-Schauder which are the basis for the study of the solvability of the Dirichlet problem for quasilinear elliptic equations of second order turned out to be insufficient for the consideration of other classes of problems. This was connected with the absence of the a priori estimates needed which play a decisive role in the application of topological methods and with the difficulty of reducing boundary problems to operator equations studied by Leray and Schauder. The difficulties indicated are removed upon reducing the boundary problems to equations with monotone operators and the creation for such operator equations of topological methods of study. The foundation for the creation of these methods appeared in the papers of Browder, Petryshyn [92], [58:168], [58:173] and the author [56], [58:112], [58:113], in which the concept of degree is defined for various classes of maps independently and simultaneously. In the papers of Browder, Petryshyn [92], [58:173] the degree of A-proper maps is defined, and its values are subsets of Z ' = Z U { -oo, q-oo}, where Z is the set of all integers. The theory of degree of A-proper maps is extended in Browder [85], Fitzpatrick [58:187] to maps which are limits of A-proper ones, in particular, the multivalued degree of a pseudomonotone map is defined. There is a survey of these results in [58, 158]. At the same time it turned out that the classes of maps which arise in the reduction of nonlinear elliptic boundary problems have a single-valued degree which has all the natural 2522

properties of the degree of finite-dimensional maps. This is done in [56], [58:112], [58: 113] in which in particular the degree of maps of class (S)+ (maps satisfying condition ~ in the terminology of [56]) and various generalizations of them is defined. We note that under broad conditions it is proved that maps of the same degree are homotopic. In recent years Browder [89, 90] proved that the degree of maps of class (S)+ introduced by the author in [56] are uniquely defined by natural axiomatic properties. Topological methods first applied to divergent boundary problems let one prove existence theorems on the basis of homotopization of the problems to simpler ones in the presence of a priori estimates in the energy spaces. These same methods turned out to be quite fruitful for estimating the number of solutions of boundary problems, in studying the branching of solutions, and in a number of other questions. An important new feature was the possibility of using the degree of maps of monotone type in the study of general nonlinear elliptic boundary problems reduced in [58, 60] to equations with operators of class (S)+. We note further that topological methods in nonlinear elliptic boundary problems can also be developed on the basis of the degree of Fredholm maps. This approach, whose foundations were laid by Elworthy and Tromba (cf. [i00]), was developed by Nirenberg, Yu. G. Borisovich, etc. These methods applied to boundary problems give an assertion similar to that obtained by the methods of monotone operators, but they require the differentiability of the nonlinear maps which arise. We shall not treat them here since they are summarized in detail in [8, 147]. i.

Degree of Maps of Monotone Type

i.i. Definition of the Degree of a Map of a Separable Space. In the present point X is a real, separable, reflexive Banach space, X* is its dual. We denote strong and weak convergence, respectively, by + and -~; the question of the convergence in the sense of which space will be clear. For arbitrary elements u~X and h~X* we denote by in what follows the action of the functional h on the element u. X*.

We consider an operator A, generally nonlinear, defined on a set D c X with values in Let F be an arbitrary subset of the set D.

Definition i.i. We shall say that the operator A satisfies condition ~0(F_~) if for an arbitrary sequence un~F from un-~Uo, Au~ --~0 and

li-m < Au~, u~-- Uo ) .-.<0

(i)

the strong convergence of u n to u 0 follows. The concept introduced is similar to the condition of pseudomonotonicity of Brezis [83], the condition introduced by the author and Browder in [58:113], [88], condition (S)+ from Browder [85], and condition ~) from [56]. The concept of maps of class (S)+ and pseudomonotone is given in Chap. i. We give another definition of operators satisfying condition ~). Definition 1.2. We say that the operator A satisfies condition ~(F) if for an arbitrary sequence un~F fromu~-~u0 and (I) the strong convergence of u n to u 0 follows. Condition (S)+ coincides with the condition ~(D). We note the definitions of monotone and more general maps which preceded the appearance of the classes of operators named above. We set BR={u~X:IIulI~.R}. Definition 1.3. The operator A is called an operator of semibounded variation if there exists a nonnegative continuous function c(R, t) defined for R,:t>~0 and satisfying the condition c(R, t)/t § 0 as t + 0 such that for any R > 0 and any elements u, v6DNB ~ one has

~--c(R, Ilu--~ll'), where I1"11' is a norm which is compact compared with It'it. When c(R, t) ~ 0 an operator A satisfying Definition 1.3 is called monotone and we were concerned with them in the first chapter. The definition of semiboundedness of variation of an operator is due to Dubinskii [16].

2523

An example of an operator satisfying condition (S)+ can be a monotone operator A satisfying the following stronger condition than the monotonicity condition:

>>-c(llu-vll), where

c(r)>~O i s a c o n t i n u o u s n o n d e c r e a s i n g f u n c t i o n w h i c h i s e q u a l t o z e r o o n l y f o r r = O.

Numerous examples of operators satisfying the cited definitions are given by boundary problems for nonlinear elliptic operators. Similar examples were considered in the first chapter. We return to an operator A satisfying the condition ~0(SD) and for it we introduce the concept of degree. In addition we shall assume that this operator is demicontinuous. Definition 1.4. The operator A:D + X* is called demicontinuous if for an arbitrary sequence uo~D, which converges strongly to un~D, one has Aun~Auo. Let {vi} , i = l, 2,... be an arbitrary complete system of the space X and let us assume that for each n the elements vl,...,v n are linearly independent. We denote by F n the linear span of the elements vl,...,v n. In what follows D is an arbitrary bounded open set of the space X and we denote by 3D the boundary of the set D. We introduce the classes of maps for which the degree will be defined. Definition 1.5. For F c D we denote by Aa(D, F) [respectively A(D, F)] the set of bounded demicontinuous maps A:D + X* satisfying condition s0(F) [respectively, condition ~(F)]. If F = D, we shall write A0(D), A(D) instead of A0(D, D), A(D, D). We shall define Deg (A, D, 0), the degree of the map A of the set D with respect to the zero of the space X* under the following conditions:

a)

A~Ao(D, OD);

b) f o r an a r b i t r a r y F o r e a c h n = 1, 2 , . . . follows :

element

u~OD Au=/=O.

we d e f i n e

finite-dimensional

approximations

An o f t h e map A as

A n u = s
(2)

t~1

THEOREM i.i. such that for n ~ N

Let the operator A satisfy conditions a) and b). the following assertions hold:

Then there exists an N

i) the equation Anu = 0 has no solutions belonging to 8Dn; 2) the Brouwer degree deg (An, D n, 0) of the map A n of the set D n with respect to 0@F~ is defined and is independent of n. The existence of the limit liradeg(A,, m,, 0), which we denote by D{vi} follows from Theorem i.i.

n-.-~

THEOREM 1.2.

Let conditions a) and b) hold. Then the limit

D {vi}= lim deg(A., Dn, 0)

(3)

n-~

is independent of the choice of the system of functions. Theorems i. 1 and 1.2 make the following definition natural. Definition 1.6. For an operator A satisfying conditions a) and b), by the degree of the map of the set ~ with respect to the point 0~X*, we mean the number

lira deg (An, D~, 0), w h e r e An, Dn a r e d e f i n e d

according

to (2).

We s h a l l

d e n o t e t h e d e g r e e so d e f i n e d

~, 0). Further,

for

hEX*\A(OD)we d e f i n e ;Deg(A, f), ! h) by t h e f o l l o w i n g e q u a l i t y : Deg(A, D, k)=Deg(A--h, L), 0),

where A - h is the operator which acts according to the rule (A - h)u = Au - h. 2524

by Deg (A,

1.2. Definition of the Degree for a Nonseparable Space. The constructions of the preceding point used the existence of a countable complete sequence of the space X. At the same time, for an operator A satisfying the conditions of point i.I, one can introduce the concept of degree without the requirement of separability of X. In this point X is a real, reflexive banach space and we denote by F(X) the set of all finite-dimensional subspaces of X. Let FoP(X) and v I .... ,vl be a basis of F. We define the finite-dimens ional map

Ap (it)= ~ vi for uEOe= D fqF.

(4)

i=l

THEOREM 1.3. Let the demicontinuous operator A:D F § X* satisfy condition ~(3D), 3D being the boundary of a bounded open set D c D F and Au ~ 0 for u6aD. Then there exists a subspace P00f(X) such that for any subspace F belonging to F(X) and containing F 0 the following properties hold: I) the equation AF(U) = 0 has no solutions belonging to 8DF; 2) deg (AF, DF, 0) = deg (AF0, DF0, 0), where deg is the degree of the finite-dimensional map. The theorem justifies the introduction of the following definition: Definition 1.7. If the hypotheses of Theorem 1.3 hold, by the ~ the set D with respect to the point 0GX* we mean the number

Deg(A, s

of the map A of

O ) = d e g ( A p , , Dr, o, 0),

where AF, D F are defined according to (4), F 0 is the finite-dimensional fined by Theorem 1.3.

subspace of X de-

One can also introduce the degree of a map under weaker requirements on the operator. We note, for example, the degree of maps of the form A + T, where A is an operator with semibounded variation and T is compact. Let D be a bounded open set in X and let us assume that on the space X there exists an operator Ao~A(D, aD), where A(D, 8D) is introduced in Definition 1.5. For uniformly convex spaces X and X* a dual operator can be chosen as such an operator [15]. Let A:D § X* be a demicontinuous operator with semibounded variation, T:D + X* be a completely continuous operator. Let us assume that 0 ~ (A + T)(3D) where the dash denotes the strong closure and 8D is the boundary of the set D. Under these conditions we define the degree of the map A + T. By the hyupotheses one has

IIAu+Tull.>~& with

some p o s i t i v e We i n t r o d u c e

number 5.

for

u~OD

(5)

H e r e tl'll i s t h e norm i n X*.

t h e map EA0u + Au + Tu f o r

f o r a map EA 0 + A + T s a t i s f y i n g degree

all

~619.

the hypotheses

Let

ll4=supllAotlll.. Then f o r 0 < s < ~/M

of Theorem 1.3,

there

is defined

the

Deg(eAoq-Aq-T,/5, 0). One c a n show t h a t f o r 0 < ~ < 5/M t h e d e g r e e s the following limit exists: lim

defined

(6) by ( 6 ) do n o t d e p e n d on E and h e n c e

Deg(sAo+ A-}-T, D, 0}.

8-+0

One proves that this limit is independent of the choice of the map A 0. introduce the following concept:

Thus one can

For a demicontinuous operator A with semibounded variation and a completely continuous operator T by the degree of the map A + T of the set D with respect to the point 00X* we mean the following number:

DEQ(A-~T,

D, O)=limD~g(sAo+ Aq-T, D, 0), 840

where

AofiA(D, OD),and Deg i s t h e d e g r e e i n t r o d u c e d

by D e f i n i t i o n

1.7. 2525

One can also define the degree of pseudomonotone maps analogously. 1.3. Properties of the Degree of Generalized Monotone M@ps. The degre of a map introduced above in various cases has all the natural properties of the degree of finite-dimensional maps. We show this on the example of the degree defined in point i.i. In the present point X is a separable reflexive Banach space, D is an arbitrary bounded open set in X. Let ~,I]={t0R~:0~t~.~I}, At:D-','X*, tE [0, I] be a parametric family of nonlinear maps. Definition 1.8. sequences

We say that the family A t satisfies condition

u.EOD, t~s

aoC~

if for arbitrary

1] froml u=-~.uo, Atn(u~)--*O

and

1T~ < At.(u.)u.--uo > < 0 the strong convergence of u n to u 0 follows. Definition 1.9. Let A', A":D § X* be maps of class A0(D, 3D) and let A'u ~ 0, A"u ; 0 for uOaD. We call the maps A' and A" homotopic on D if there exists a parametric family of maps At:D.-+X*, t~, I], satisfying condition ao(~ such that:

a) Atu ~ 0 f o r ueOD, t0[0, 1];

Ao=A', A,=A";

b) f o r any s e q u e n c e s t n, u n s u c h t h a t /.G[0, 1],U.fib, t.-+to, un-+Uo, t h e s e q u e n c e AtnUn converges weakly to At0u 0THEOREM 1.4.a. Let A r : D _ + X *, A~:D-+X * be maps of class A0(D, 8D). Let us assume that A'u r 0, A"u ~ 0 for u00D and the maps A' and A" are homotopic on D. Then

Deg(A', s

0)-----Deg(A u,/9, 0).

(7)

The degree of a map defined in point i.i is, under certain conditions, the unique homotopy invariant. The theorem cited below generalizes the classical Hopf theorem for finite-dimensional maps. In the case of maps which are the sums of an identity and completely continuous operators the corresponding theorem was proved by Krasnosel'skii [34]. THEOREM 1.5. Let D be a convex bounded open set in the space X and Ao:D-+X *, A,: ~)-+X" be maps of class A(D, 3D) such that A0u ~ 0, Azu ~ 0 for u~aD and Deg(A0, D, 0) = D e K ( A I, D, 0). Let us assume that the spaces X and X* are uniformly convex. Then the maps A 0 and A~ are homotopic on D. Remark i.i. The assumption of uniform convexity of the spaces X and X* in the formulation of Theorem 1.5 can be replaced by the condition that there exists a demicontinuous operator A0:D + X* satisfying condition ~(SD) and such that > 0 for u ; 0. Numerous applications of the theory of the degree of maps to the study of the solvability of nonlinear operator equations and nonlinear boundary problems are based on applications of Theorem 1.4 and the following assertion. LEMMA i.I. Let A:D + X* be a map of class A0(D) and let Au ~ 0 for u~OD. In order that the equation

Au=O

(8) have a solution in D it suffices that Deg (A, D, 0) ~ 0. We cite two frequently used tests for the difference of the degree of a map from zero; other similar tests are connected with the calculation of the index of a critical point with which we shall be concerned in the next point. THEOREM 1.6. Let A:D + X* be a map of class A0(D, 8D). that for uOaD one has

Let us assume that

OOD\aD and

~O.

(9)

Then Deg(A, D, 0) = 1. THEOREM 1 . 7 . L e t B g = {u~X : [[u]]~R} and A : BR(O)--+X* be a map of class A(B R, 8BR). us assume t h a t Au ~ 0 f o r u6OBR and 2526

Let

Au A(--u) for u~0BR. il Au % =/=HA (--u) II,

(i0)

Then Deg (A, BR, 0) i s an odd number. 1.4. Calculation a separable, reflexive Definition 1.5. Definition i.i0.

of the Index of a Critical Point. L e t D be a b o u n d e d open s e t i n Banach s p a c e X, A:D + X* be a map o f t h e c l a s s A0(D) i n t r o d u c e d i n The point uo~D

is called a critical point of the map A if Au 0 = 0.

The term "critical point," which is natural in the case of a potential operator A, which is the gradient of a functional, is applied in this paper for general nonpotential maps also. Let u 0 be an isolated critical point of the map A, i.e., there exists a ball B~~ {u~X: which contains no other critical points than u 0 of the map A. One can show that for 0 < r < r 0 one has

Ilu-uoH<~ro},

Deg(A, B~ (uo), 0)= D~g(A, B r (Uo), 0). This makes the following definition natural. Definition i.ii.

By the index of an isolated critical point u 0 of the map A is meant

lira Deg(A, B r (~o), 0). r~0

We shall denote the limit indicated by Ind (h, u0). THEOREM 1.8. Let us assume that the map A of class A0(D) has only isolated critical points in D and Au ~ 0 for u~OD. Then there are finitely many critical points and f

D e g ( A , / 3 , 0 ) - - Z Ind(A, u~),

(11)

t=t

where ui, i = i, .... i are all the critical points of the map A in D. Later, under certain conditions we shall calculate the index of a critical point of the map A. For simplicity we assume that the critical point studied coincides with the zero of the space X. Let us assume that the Frechet derivative of the operator A at zero exists, ioe., that there is a bounded linear operator A':X + X* for which

Au--AO==A'u-~o ( u),

(12)

where ,,~(u)l% +0 as lluil § 0, ll'll~,being the norm in the space X*. I!u I! THEOREM 1.9. Let U be a neighborhood of zero of the space X, A:U + X* be an operator of class A(U) having Frechet derivative A' at zero. Let us assume that the following conditions hold: I) the equation A'u = 0 has only the zero solution; 2) there exists a completely continuous linear operator F:X § X* such that

<(A'+r)u,u>>0

for u=~0

(13)

and the operator L = (A' + F)-zF:X + X is defined and completely continuous; 3) for sufficiently small e > 0 the weak closure of the set

{

~

z~= v=~:Au=---

,,Aui,, T Tu:If, --A'u, i[ A

o
}

does not contain zero. Then zero is an isolated critical point of the map A and the index of zero is equal to (-i) ~, where ~ is the sum of the multiplicities of the characteristic values of the operator L lying in the interval (0, i). Remark 1.2. We note the essential nature of all the hypotheses of Theorem 1.9. In particular, an example of a map satisfying all the hypotheses of the theorem except 3) is constructed in [56] for which zero turns out to be a nonisolated critical point.

2527

In the case of a Hilbert space the hypotheses of the preceding theorem can be relaxed. THEOREM i.i0. Let H be a real separable Hilbert space, U be a neighborhood of zero of the spao'$ H, A:U + H* be an operator of class A(U) having Gateau derivative A'(u) at each point u ~ . Let us assume that the following conditions hold: i) the equation A'(0)u = 0 has only the zero Solution; 2) for an arbitrary element o~H

it follows from u n + 0 that

[A" (u.) ]*v--~[A'(O)I*v, where [A'(u)]* is the dual operator to g'(u); 3) there exists a completely continuous linear operator F0:H + H* such that for some r > 0 and positive constant c one has

<(A" (v) + ro) u, u>>~c[lull~ for ueH, lvll<- 0 one has ~(0)~.~'(u) for u~B,-----{u~XiI[u[[~
1.12. Let~': X-~R l be a functional which at each point ufiX has Gateau deThe functional ~- is called nondegenerate if for some R>0$r'(u)=/=0 for

ilull~>RJ Definition 1.13. The functional ,9": X--~R l is called increasin~ if for any C~Rl the set {u : u0X, ~'(u)~.c} i is bounded in X. THEOREM 1.12 [27]. Let ~ ' : X - + R 1 be a continuous functional which at each point u~X has Gateau derivative ~"(u). Let us assume that ~ is a nondegenerate increasing functional and the map ~ " belongs to A(B R) for each R. Then

lira Deg(9 r', B~, 0 ) ~ 1 . We note another generalization of Theorem i.ii obtained in [7] by Bobylev which relates to the case when the functional realizes a local minimum on a finite-dimensional manifold. Let H be a real, separable Hilbert space, D be a bounded open set in H, and ~ : D - + R I be a functional of class C I. THEOREM 1.13. Let us assume that the set of critical points of the map 8r' contained in is a finite-dimensional compact connected smooth manifold M without boundary. If~" belongs to the class A(D), M realizes a local minimum of the functional ~', and~4NdD-----~," then Deg(~",\D,O) =X(M),~ where x(M) is the Euler-Poincar6 characteristic of the manifold M. 1.6. Uniqueness of the Degree of Maps of Monotone Type. In recent papers of Browder [89-91] conditions defining the degree of a map uniquely are discussed. We restrict ourselves to consideration of maps of class (S)+ in the case of uniformly convex spaces X and X*. Let D be a bounded open set of the space X and we consider the family of maps A(D) defined in point i.i. A dual operator J which makes correspond to an element u~X a functional luOX*,satisfying the conditions

2528

<]u, u> = Itull 2, IIYuII.= Ilull. belongs to this family for uniformly convex spaces X and X*. Let us assume that for arbitraryA~A(D), open subsets G c D andhEX*, h~A~OO) there is defined an integer-valued function d(A, G, h) satisfying the following conditions: I) normalization: if d(A, G, h) ~ 0 then where J is a dual operator;

h6A(G);

for each h ~ 7 ( G ) \ ( d ~

d(f,G,h)=l,

2) additivity with respect to the domain: let Gl, G 2 be disjoint open subsets G c and let hCA(G\(GIUG=), then d(A, G, h) = d(A, G~, h) + d(A, G 2, h); 3) invariance with respect to homotopy: if A t , ht,t~,l] are, respectively, parametric families of maps from A(D)_and elements of X* such that the map (A t - ht)(u) = Atu - h t is a homotopy on G c D in the sense of Definition 1.9, then d(A t, G, h t) has constant value for t~, I]. THEOREM 1.14.

If the integer=valued function d(A, G, h) defined for A@A([)), OCl), 1)-3), then d(A, G, h) D e g ( A - h, G, 0), where D e g ( A - h, G, 0) is the degree of the map (A - h)u = Au - h introduced in point 1.2.

h6X*\A(OG),satisfies

Remark 1.3. The assumption of the invariance of the function d(A, H, h) with respect to homotopies in the sense of definition 1.9 can be relaxed, replacing it by invariance with respect only to linear homotopies A t = tA 0 + (i - t)A I. 2.

Application of the Theory of the Degree of a Map to the Proof of

the Solvability of Nonlinear Equations 2.1. Solvability of General Operator Equationsm The application to operator equations of topological methods based on the concept of degree of a map let one simply generalize the results of the first chapter on the solvability of equations with coercive or odd operators and get new existence theorems. The topological approach lets one, in particular, include operator equations in a parametric family of equations of the same form and reduce the study of solvability to establishing the difference from zero of the degree of simpler maps and getting a priori estimates. First we note a simple corollary of Theorem 1.4 and Lemma i.I. THEOREM 2.1. Let X be a reflexive, separable Banach space, D be a bounded domain in X with boundary 8D, A:D • [0, i] + X* be a bounded demicontinuous operator and let us a s s ~ e that: i) the family of operators A t = A(., t) satisfies condition A I satisfies condition ~0(D); 2) A(u, t) ~ 0 for

uE#D, t~,

ao~~

and the operator

i];

3) Deg(A0, D, 0) ~ 0. Then the equation A1u = 0 has at least one solution in D. The next theorem follows directly from Theorems 2.1, 1.6, and 1.7. THEOREM 2.2. Let X be a reflexive, separable Banach space, t6[0, |]~ As: %-+%* be a family of bounded demicontinuous operators satisfying condition ~(X) and such that for an arbitrary bounded set BcXAt~u) depends continuously on t, uniformly with respect to u~B. Let us assume that condition i) and one of the two conditions 2), 3) hold: i) for any h6X* there exists an R = R(h) such that from

Atu-~h,t~[O, I]

it follows that

2) the operator A 0 is coercive; 3) the operator A 0 is odd. Then the equation A1u = h is solvable for arbitrary

h~X*.

We note another corollary of Theorem 2.1 which can be useful in proving the solvability of boundary problems. Definition 2.1. The operator A:X + X* is called asymptotically homogeneous if it can be represented in the form A = A 0 + Al, where A 0 is a positively homogeneous operator of order 2529

k, i . e . ,

A0(tu) = TkAou f o r t > 0 and lira

r[A,ul[,ll[lull~k=o.

Definition 2.2. An asymptotically homogeneous operator A is called regular at infinity if k > 0 and the equation A0u = 0 has only the zero solution. THEOREM 2.3. Let A:X § X* be an asymptotically homogeneous operator which is regular at infinity. Let us assume that A and A 0 are operators of class A(X) and that the index of zero of the map A 0 is nonzero. Then the equation Au = h is solvable for any h~X*. We note one of the consequences relating to the solvability of equations with odd homogeneous operators. COROLLARY 2.1. Let A:X § X* and B:X + X* be an odd completely Then for arbitrary %6R ], for which equation Au + %Bu = h is solvable

be an odd positive operator of order k > 0 of class A(X) continuous positively homogeneous operator of order k > 0. the equation Au + XBu = 0 has only the zero solution, the for any A6X*

The first such result was obtained by Pokhazhaev [48]. Subsequently similar results were established by Browder, Petryshyn, Necas, the author, etc. (cf. [166]). Remark 2.1. In applying the degree of pseudomonotone maps, the condition that maps belong to the class ~(X) can naturally be replaced by the condition of pseudomonotonicity. We show that the concept of degree permits one to prove the invariance of domain for the maps considered. The first such results for operators of the form "identity plus completely continuous" were obtained by Schauder [58:271]. For locally A-proper maps having a special local A-proper homotopy these questions were studied by Petryshyn [58:257]. Below we give an improvement of a theorem of Petryshyn obtained in [58]. Definition 2.3. We say an operator A defined on an open set D of a Banach space X satisfies condition ~) locally and is locally one-to-one if for each point u0~D there exists a ball B % (u0)~---{u:[]U--uoll<~ro} such that B,, (u0)cD and A is one,to-one on Br0(U0) and satisfies condition ~(Br0(U0)). THEOREM 2.4. Let X be a separable, reflexive Banach space, D be an open set in X, A:D § A* be a continuous locally one-to-one operator locally satisfying condition ~). Then the set A(D) is open in X*. 2.2. Conditional Solvability of Boundary Problems. We give applications of the general rseults of the second chapter to the question of finding solutions of a boundary problem for (5) of Chap. I. First we formulate a general conditional existence theorem for a solution of the boundary problem and afterwards we get some unconditional theorems as corollaries of it. Let the equation (_l)Ist OSA=(x, u ..... mmu) --- ~ I=[<m

(--l)'s'msfs(x) ,

(14)

}~l<m

considered under the conditions of point i.i of Chap. 1 be included in a parametric family of equations of the same form

(--l)'='D~As(t, x, u . . . . . Dmu) = ~ Islam

(--l)ts'DS/s(t, x),

(15)

l~l<m

where As(l, x, ~)=As(x, ~), fa(1, x ) = f ~ ( x ) ; t h e f u n c t i o n s A~(t, x, r~)) fs(t, ~) f o r each t6[0, 1] s a t i s f y c o n d i t i o n s A1)-A 4) o f Chap. 1.

Let V be an a r b i t r a r y

c l o s e d subspace of W~p(~) such t h a t

0

W~(O)CVCW~(f~)

A generalized solution of (15) for fixed t 9 [0, I] is understood in accord with Definition i.i of Chap. I. Analogously to (9) we define a family of operators At:V ~ V*: < Atu, ~ > = ~ .I [As(t, x, tt . . . . . Dmu)--fs( t,x)l m ~ a x " (16) Is1<m g

As in Chap. 1 in what follows the norm in W~(O) is denoted by It-llm,p and l e t / g R = { u E W ~ •

II u

e}.

THEOREM 2.5. Let t h e f u n c t i o n s A=(t,x,~), ~ ( t , x ) , and satisfy the following conditions:

I = l ~ m , be d e f i n e d f o r (t, x, ~)6~, I ] X ~ X R K

a) As(t , x, ~), f=(t, x) are uniformly continuous in t for x ~ , bounded set in RE; 2530

~O,

O being an arbitrary

b) A~(t, x, g) satisfy assumptions AI)-A4) of Chap. 1 with constants r ~ , r.r independent of t and functions gl, g~, gs;f~(t, x)6L~=(~), where the qe have the same value as in point i.I of Chap. i. Let us assume that there exists a constant K such that for all solutions of (15) corresponding to the subspace V for t6[0, l] one has llu(x)Ik~ 0 let

/~L~(~)

~> for u ~ B a where A is defined by (9) of Chap. i. satisfying the inequality II~[I~,~R.

Then there exists at least one solution u~V of (44)

The assertion follows from Theorem i~6 and Lemma I.i. From Theorem 2.6 we get the solvability of the boundary problem for arbitrary [~@L~(~),directly if the operator A satisfies a coercivity condition. Remark 2.2. From Theorem 2.6 under certain conditions on A~ one can get the solvability of the Dirichlet problem for (i) in a domain of sufficiently small measure. Let us now assume that the functions A~ satisfy the following conditions:

1)

A=(x, ~)-~--A~)(x, $)+A~)(x, $),

where

A~)(x, ~)

a r e d e f i n e d f o r ( x , $)@~XR ~, a r e m e a s u r a b l e

in x for all values of ~, continuous in ~ for almost all x, and satisfy the inequalities

IA~(x, Dl
A$l(x,

--$)----

--A$)(x,

3) for arbitrary xEQ, ~ ~' one has

~); ~={~:]a]___m}CR m0,

[A,,(x, ~1, r

~r-----{~i:!~[-----m}@RM~ n@{n~:l~l<m}eR M' for

A=(x, ~l, r162162

0,

lai=m

[A~ ) (x, n, ~)-- AS ~(x, n, U)I ( ~ - - ~'~)> 0; laJ=m 4) f o r XC~, ~lGRM', ~CRM~ one h a s I~l~m

l~[=m

[Y(
Here cl, c 2 are positive constants. THEOREM 2.7. Let us assume that conditions i)-4) of the present point hold and the boundary problem for the equation Z(--I)I~ID:A$

la!<:m

~ (x, u ....

, Dmu)----O

(17)

corresponding to V has only the zero solution. Then (14) has at least one solution corresponding to the subspace V for arbitrary functions [~@Lq~(~). The assertion follows from Lemma i.i of Chap. 1 and Theorem 2.3. 2531

THEOREM 2.8. Lq~(~), and let

Let the functions A~(x,

~) satisfy conditions AI)-A4) of Chap. i, fa(x)


[IA~[Iv*-~ ilT!!~o -->-[- oo as

(18)

[lu[j.,,p._>oo,

where A is the operator defined by (9) of Chap. I. Then there exists at least one solution u(x) of (14) corresponding to the subspace V for arbitrary functions f=(x)~t~=(~). One gets this theorem from the conditional Theorem 2.5 for a special choice of homotopy. We define the operator A 0:V § V* by

( Aou,~ ) =I~l<~S 1D~u ]p-2D~uDC'~dx and the family of operators At = tA + (1 - t)A 0. One v e r i f i e s t h a t i f the numberK~R' i s such t h a t

][Aullv*+ ,-~V-- > l [ f H v , ,,

for [lu[[,~,o>K,

(19)

, m,p

then for solutions of the equation Atu = tf one has the a priori estimate

llullm,p < K.

Existence theorems under conditions of the form (18) were obtained by many authors Browder, Fitzpatrick, Petryshyn, etc. Cf. [155] about these results with the corresponding bibliographical references. Remark 2.3. The assertions of points 2.2 and 2.3 can be proved under relaxed hypotheses on the coefficients As(x , ~), namely replacing condition A 4) by condition A~) of Chap. 1 which is connected with the application of the theory of degrees of pseudomonotone maps. 2.4. Convergence of Galerkin Approximations. Here we show that the theory of degrees of generalized monotone maps lets one establish the strong convergence of the Galerkin approximations for nonlinear problems. We start with theorems of convergence for Galerkin approximations of an operator equation. Let X be separable, reflexive real Banach space and let vi, i = i, 2,... be a system of elements in X such that UF,~=X, where F n is the n-dimensional linear span of the elements v~,...,v n and the dash denotes closure. By an approximate solution of the equation Au = 0 we shall mean an element

u,~F~

such

that

----0, i = 1 . . . . . n.

(20)

THEOREM 2 . 9 . L e t D be a b o u n d e d domain o f t h e s p a c e X and A:D + X* be a b o u n d e d d e m i continuous operator satisfying condition a0(D). L e t u s a s s u m e t h a t i n t h e domain D t h e map A has a unique critical point, Au ~ 0 for Deg (A, D, 0) ~ 0. T h e n approximate solutions u n of the equation Au = 0 exist for n larger than some n o and as n + ~ the approximate solutions u n converge strongly to a solution of the equation Au = 0. Remark 2.4. Convergence of Galerkin's method for an equation of the form u + Fu = 0, where F is a completely continuous operator in a Banach space, is obtained in Krasnosel'skii [34]. R. I. Kachurovskii and M. M. Vainberg were concerned with the justification of Galerkin's method for equations with monotone operators. In [166:77] equations with uniformly monotone operators were considered. The results contained in this paper follow directly from Theorem 2.9. We note that many papers of Petryshyn are devoted to questions of convergence of Galerkin approximations (cf., in particular [158]). These same questions are discussed in Landes [1281. As an example we cite the application of Theorem 2.9 to establishing the convergence of the Galerkin approximations for (14). THEOREM 2.10. 1 and for ~, ~ ' ~ M

2532

Let the functions

A=(x, ~), lal~m

satisfy conditions Al), Ai), A~) of Chap.

~ [A=(x,~)--A=(x,

~ ' ) ] ( ~ = - - ~ ) > O.

Let us assume that f=(x)~L~=(~) and (18) holds. Then the boundary problem for (14)corresponding to the subspace V has strongly convergent Galerkin approximations Un(X) defined according

to (20). .

Solvability of Weakly Nonlinear Boundary Problem~

In this section we shall consider the Dirichlet problem for a divergent elliptic equation although on the whole the results also hold for nondivergent equations and genera] linear boundary conditions subject to the condition of Ya. B. Lopatinskii. We shall consider the solvability of the Dirichlet boundary problem J

(--l)'=ID~[a=~(x)D~u]-r - ~ (--1)'='D~b~(x, u. . . . . D~u)i= ~(--1)I=!Daf~(x), r I,ID]<m

J=l
Ioq<m

,

xCQ,

(21)

D=u(x) =0, xEOf~, laI~<m--I

(22)

for a bounded domain QczR n, p~m. In this section the starting point is the study of the solvability of the problem (21), (22) on the basis of comparison of it with the linear problem obtained for b=------0,[=[~PLet us assume the following conditions hold: a l) are bounded measurable functions for lal, I~I~m, a=~(x) are continuous functions ~ for I~I =

I~I

=

m;

a 2) ellipticity condition: for some positive constant v > 0 one has

a=~(x)~=~>~,j~? '' fo,:

l~l=l131=,n

b z) the functions b=(x, to. . . . . ~m), [ ~ [ ~ p and the estimates

xe~,

~R";

for xEf~, ~ R N(~I satisfy Caratheodory conditions m

j=0 with positive constant ~, (1610,I), f~(x)EL~(~). 0zn~

We shall establish the solvability of the problem (21), (22) for f=(x)6L2(Q) in ~ 2 (Q), by reducing it to the solvability of the operator equation

Lou + Nou --~f (x),

( 24 )

where the operators Lo, No:l~"~"(n)-+~7~n(fl ), the functionsfeW~(Q) the conditions

are defined, respectively,

(Lou, (p)~= X I a='~(x)D~uD=q~dx' f"l,l 31<:m (Nou, ~)m= ~. I b=(x, u,..., Orau)D'~dx,

by

(25)

(f, q~),n'----X S./'=(x) D"~dx, Ial~m Q

where ( . ,

")m d e n o t e s t h e s c a l a r p r o d u c t in ~ ( ~ ) .

By G a r d i n g ' s

inequality

[ 1 1 2 ] , we have, f o r u(x)El~'~(f2)

(Lou, u)m > el II u where c l ,

c 2 are positive

,2

=-

C

=

constant independent of u(x),

g

2

0,

I1.11m i s t h e norm in

(26)

~ (~q).

3. I. Perturbation of a Uniquely Solvable Linear Boundary Problem. The simplest question about the solvability of the problem (21), (22) is solved in the case of a compact map N o which one gets for p < m and invertible operator L 0. THEOREM 3.1. Let conditions az), a~), bz) hold and the equation L0u = 0 have only the zero solution in ~m(fl). Then the problem (21), (22) is solvable for p < m for arbitrary functions f~(~)6L~(f~). 2533

Under the hypotheses of Theorem 3.1 the solvability of the problem (21), (22) can be obtained from Theorem 2.2 by the inclusion of (25) in the family of equations Atu = f, where 0

Atu = L0u + tN0u.

Verification of condition ~(W$~)I assures the inequalities 0

(23), (26),

0

p < m and the compactness of the imbedding W~(~)-~W~(~). To verify condition i) of Theorem 2.2 it suffices to use (23) and the estimateIIul]m~cllL0u]l~, which follows from (26) and the hypotheses of Theorem 3.1. Remark 3.1. Obviously to prove Theorem 3.1 it suffices to assume the asymptotic linearity of the operator L 0 + No, i.e., one can relax condition b I) so that llN0ullm=o(IIullm) as We note that the solvability of asymptotically by Necas [58:249].

linear divergent equations was considered

3.2. Solvability of Weakly Nonlinear Operator Equations. In this point, following the scheme proposed by Petryshyn [153], we formulate a theorem on the solvability of an abstract equation. We shall speak here about maps of class (S)+ which suffices from the point of view of the applications to differential boundary-value problems considered. In Petryshyn's papers (cf. [153]) devoted to weakly nonlinear equations, one uses considerations based on the theory of A-proper maps. Let H be a Hilbert space and L be a linear Fredholm map in H of index zero. This means that KerL={u~H:Lu-----O}iis a finite-dimensional s p a c e , R ( L ) = { L u : u ~ H } is a closed subspace of finite codimension of Ker L. We denote by Hi, F I orthogonal complements to H in the subspaces Ker (L), R(L), respectively. We define a arbitrary linear isomorphism M of Ker L onto F l and let

C = M P : H-~FI=[R (L) ~

(27)

where P is the orthogonal projection of H onto Ker L. THEOREM 3.2. Let L:H + H be a linear map of class (S)+ of index zero and N:H + H be a demicontinuous map such that i) Nu = o(iluJl) as 1[u[[ + ~ the map L + N is pseudomonotone; 2) for some sequence E k + +0 one of the sets Z +, Z- where

Z• and C i s t h e o p e r a t o r

{u~H:Luh-+-ehCuhq-Nuh--[=O}

defined

by ( 2 7 ) ,

is bounded.

Then t h e e q u a t i o n

Lu+Nu=[ is solvable

(28)

i n H.

To p r o v e Theorem 3 . 2 i t s u f f i c e s t o u s e t h e r e s u l t s o f p o i n t o f t h e t h e o r e m L i s a F r e d h o l m map and f o r any ~ ~ 0 one h a s

[]Lou+eCul].>~K~llu]]

2.1.

Under t h e h y p o t h e s e s (29)

with K E > 0. Introducing the homotopy A t = L 0 + EC + tN we get the solvability for any E ~ 0 of the equation Lou+sCu+tNu =f. Passing to the limit as ~ § 0 using condition 2), we arrive at the solvability of (28). In [153] sufficient conditions are given for assumption 2) of the preceding theorem to hold. THEOREM 3.3. Let L:H ~ H be a linear map of class (S)+ of index zero and let the demicontinuous map N:H + H be such that L + N is pseudomonotone. In order that the set Z + be bounded under the hypotheses of Theorem 3.2 it is sufficient that one of the following three conditions hold: I) Nu = o(lluii) as L[u[l + ~ and for an arbitrary sequence uk~H such that [luhl]-+oo, I]Uk]l--lUk-+ vEKerL one h a s l i m ( N ~ , M y ) > ( / , 2dr); (30)

2534

2) one has

llNull<~alluY+b,

u~H, o~[o, 1)

(31)

with positive constants a and b and for arbitrary sequences t~6R+ i, v~Ker that V~---~, []v[[= 1, t~-+oo, ]]u~ll~-~C
L, u~6H~

such

Jim (N(t~w~+t~u~), M~a)>(/, My); 3)

one has (31) and for arbitrary sequences

v~KerL

u~H

(32)

such that

llPu~II-+ooiand:v~=IIpu~II-',pu~-~

one has lira ( N u ~ , J 4 ~ ) >

( f , .4if:o).

(33)

Here M and P are the same operators as above in the definition of the operator C by

(27). Remark 3.2. To get boundedness conditions for the set Z- in Theorem 3.2 it suffices to replace the operator M in (27) by -M. In this case by replacing the inequality signs by their opposites in (30)-(32) and replacing lim by lim on the left sides of these inequalities one gets sufficient conditions from Theorem 3.3. 3.3. Solvability Conditions for Weakly Nonlinear Differential Problems. Now, following [15317 we give applications of the preceding results and the solvability of problems of the form (21), (22). The research of many authors, starting with the basic paper of L a n d e s m a n and Lazer [129] is devoted to these questions. It being impossible to list all the publications here we indicate only some of them: Williams [173], Nirenberg [148], Schechter [163], Shaw [165], Hess [123], Ambrosetti, Mancini [79], Fucik [I08], de Figueiredo [i01], Necas [145], Brezis, Nirenberg [84], Petryshyn [153]. First we consider the solvability of the equation Z (--I)I=;D~ [a~(x) D~u[+htx, u(xJ)i-----j=~,(-l~l,IBl~m

l)l~ID=f=(x),

with the boundary condition (22) when conditions al), a2) , and a~) hold for h) The function for

h0(x)@L2(Q ).

k:~XR~-~R

~

satisfies a Caratheodory condition for

(34)

i=I, Ipl <m

and

x~, s~R 1 [ /%(x, s) l -.<~0 (x )

We define functions

~x (x)---~-lird h (x, s), h• ( x ) = lira h (x, s). s--,-+ oo

--

~•

Further, L 0 is the operator defined by (25). THEOREM 3.4. Let conditions al)-a~) , h) hold and let us assume that for an arbitrary function w(x)~KerL0* w(x)~/=O one of the following conditions holds:

Im<m

lal~,m f~

where f~+f~)-~-{x~:_+w(x)>O}. for arbitrary /~(x)EL2([2).

~+(~)

~+(~)

Q-(w)

Then the boundary problem (34), (22) has a solution in W~(f~)

Theorem 3.4 is a corollary of Theorems 3.2 and 3.3. W~"(hq) by @

~-(~)

0

We define an operator gl 99W2m ( e ) - +

(N,u,~)~= I a (x, u(x)) ~ (x)dx

(37)

and we verify that under the hypotheses of Theorem 3.4 condition I) of Theorem 3.3 holds. Since L 0 is now assumed to be self-adjoint, one can choose the operator M to be the identity.

2535

0

(30) is proved by contradiction. If (30) did not hold, then for some sequence such that [] ak []m,2-~ ~ J] U~N~liuk-.vQKer L0 one would have

lim I k-~oo ~

u~(x)~W~m(~)

h(x,u~(x))v(x)dx~< X Sf~(x)D~v(x) dx" Ir

From this we get

lira

hCx, u ~ ( x ) ) v ( x ) d x > l i m

k ~ o- v)

I h(x,u~(x))v(x)dx+

o§

~-(v)

which contradicts (36) and proves (30) in the case considered. Thus, Theorem 3.2 is applicable to the operator equation L0u + N1u = f and from it the solvability of the problem (34), (22) follows. COROLLARY 3.1. Let us assume that all the hypotheses of Theorem 3.4 hold and in addition that for xGQ, $s 1

(38)

h_(x)=a__(x)
o

(22) to be s o l v a b l e in Wmi(fl) i t i s n e c e s s a r y and s u f f i c i e n t

f=l~m fl

~;§

for an arbitrary function ~)(x)6Ker L,

that

o-(~)

ii~ 11-----I, h+(x) = h + (x)=h+ (x).

The sufficiency of (39) follows from Theorem 3.4. The necessity follows from the fact that for a solution u(x) of the problem (33), (22) one has

~.~ ! f ~(x)D~w(x)dx = S h(x, u(x))w(x)dx Ir

for

w(x)6KerLo

(40)

m

and i t i s f u r t h e r n e c e s s a r y to e s t i m a t e t h e r i g h t s i d e of (40) u s i n g (38).

The general theorems of point 3.2 can be applied to a large circle of problems of the form (21), (22). We restrict ourselves in the following point to the most complicated case p = m, of interest for the noncompactness of the perturbation. 3.4. Noncompact Perturbations of a Linear Problem with Nonzero Kernel. The solvability of the problem (21), (22) for p = m was considered in [124, 125, 153]. Subsequently we give one of the results of Petryshyn [153]. Let us assume that conditions ax), ai), b l) hold as well as b 2) there exists a constant c'O[0,c,) such that for x6~, n~I~ mhi, k=0, l..... m--l, ~m,~m'%~ N(~) one has

2~ tb~(~, no. . . . .

|r

n~_,, ~ ) - O ~ ( x , % . . . . , n~_~, ~'~)l(~-

~'~) > - ~ ' ~ I~-~'~ I~, lal~ra

b a) there exist functions h~(x,~o ..... ~m), l~l--<m, defined for xfi6, ~fiR g(~), k = O ..... m, satisfying a Caratheodory condition, homogeneous with respect to $0,.-.,$m of degree o such that lh=(x, ~0..... ~m)l..
~(n)6RN(m), rn6R I from ~(n)_+~,

k = O ..... n*, rn-+ + oo it follows that

lira l b,,(x, rA(fl), . . . , r . ~ ) ) = h , , ( x ,

~o. . . . .

U~).

(41)

for x6 ~.~ Here e has the same value as in (23). 0

For an arbitrary function ~)(x)6W~m(f~)we define

H(~)= X I h'(x' ~ ..... D'~)D%z(x)dx. Ictl<:rn

2536

(42)

THEOREM 3.5.

Let us assume that conditions ai)-aa), bl)-b 3) hold and let K e r L z 0.

Then: i) if oE[0,I) and H(w) > 0 for w~KerL0, Hw[Im,2=l, then for f~(x) - 0 the problem (21), 0

(22) has a solution in wm(~); 2) if ~6(0,|) and H(w) > 0 for ~Ker(L0), 0

able in wm(~) for arbitrary functions

ll~[Im,2=|, then the problem (21), (22) is solv-

f=(x)GLf(f~);

3) if o = 0 and H(w) > (f, w) m for ~ K e r L 0, [l~OlIm,2-----l, then the problem (21), (22) has a 0

solution in wm(~). Like Theorem 3.4, Theorem 3.5 follows from Theorem 3.2. Here, in each of the cases considered one verifies that condition i) of Theorem 3.3 holds. 3.5. Existence of Multiple Solutions of Weakly Nonlinear Boundary Problems-- Here we give one of the results of Ambrosetti and Mancini [79] which gives the existence of at least two solutions of the equation I~l,l~]<:tn

(--1)l~lD"[a~,(x)D~u]-~Xku+c(x, u ) = ~

['=[<m

(--I)I~'D~f~(x), xEa

(43)

under the boundary condition (22) for a bounded function c(x, u). Let us assume the following conditions in addition to al)-aa): c I) the function c(x, s) is bounded for xEQ, s6R i, measurable in x for each s, and coniinuously differentiable in s for almost all x; c 2) for some constants at, i = I, 2, 3, 4 one has

~'k-l
Oc(x, s) OS

, Oe (x, s)

"~ a2 ~ ),k+l,

if

k>],

~<

where I i are the eigenvalues of the operator L0:IF2 (~)-+~2 (Q), defined by (26) THEOREM 3.6. Let conditions al)-a3) , ci) , c2) hold, and let the function c(x, s) be independent of x, c(0) = 0, c'(0) < 0 and the limits |im c~g)=c• exist. Let us assume that s-++oo

Ik is a simple eigenvalue of the operator L0, v k is an eigenfunction corresponding to it, and let

where Qf={xfi~2: +v~(x)>O}. ferent nonzero solutions 0

(25).

Then t h e problem ( 4 3 ) . 0

(22) f o r f a ( x ) = 0 has a t l e a s t two d i f -

ua(x), u~(x)O.W~(f~l.

F u r t h e r , f (x)GW~(ff~) i s t h e f u n c t i o n c o r r e s p o n d i n g to {f.(x)fiL~(~?),

t ~]<m}

a c c o r d i n g to

THEOREM 3 . 7 . Let conditions al)-aa) , cl) , c2)hold and l e t ;kk be a s i m p l e e i g e n v a l u e of the operator L0, v k be an eigenfunction corresponding to it, and let us assume that

lira sc(x, s ) - - ~ > O . --

(45)

S-~• 0

0

Then there exist functionals a,a:W~(ff)-~l{~, such that for

f(x)@W~(Q), a(f)<0<~(f):

i) the problem (43), (22) has a solution if and only if

a(f)~<(f, v,~)m-.
i f ( f , v~)~fi(a_(f),O)U(O, a ( f ) ) , t h e n t h e problem ( 4 3 ) , 0

(22) has a t l e a s t two . d i f f e r e n t

solutions in W~(V2), In the proof of Theorems 3.6 and 3.7, as well as in the majority of other papers devoted to weakly nonlinear differential boundary problems, the study of the solvability of the problem (44).~ (22) is reduced by the Lyapunov-Schmidt method to the study of the solvability of a finite number of algebraic equations (in the present case one). 2537

0 m

We define operators

0

Lk, C:W2 (~)-~W~(~2) by o

(L,u, ~),n=(Lou, ~),nq-~k I u(x)~p(x)dx,

(46)

(Cu, %,,= l c (x, ~ (x))~(x) dx. n

Then t h e problem ( 4 3 ) ,

(22) i s e q u i v a l e n t t o t h e o p e r a t o r e q u a t i o n

L~u-l-Cu-~-/.

(47) 0 m

In the present case due to the self-adjointness of L0 we have the decomposition I[/~(~)=Ker Lo@R(Lo) and let P and Q, respectively, be the operators of projection o f ~ ( f l ) onto R(L0) and KerL0. (47) is equivalent to the system

L~w+PC(tv~+w) =Pf,

OC(tv~+w) = Q f

with respect to fORI, w~R(Lo). Under the conditions considered one proves the unique solvability with respect to w of the first equation of this system for arbitrary t@R *. We denote the solution obtained by wf,t(x). Then the solution of the problem reduces to finding t from the equation

Fy(t)----(f,ok).,, where F/(t)----Ic(x, tv~+wt,~)v~(x)dx. The a s s e r t i o n s

of Theorems 3 . 6 and 3.7 f o l l o w from c o n s i d e r a t i o n

of the last equation.

Here the functionals ~, a w i t h which one i s c o n c e r n e d in Theorem 3.7 a r e d e f i n e d , spectively, by the equations

re-

a_(f)-----inf {r/(t): tERI}, a ( f ) = sup {F! (t): tER1}. 3.6. Perturbations by Subordinate Summands of Linear Growth. Here we formulate some results on the solvability of a problem of the form (21), (22) when (23) holds with o = i. First we consider (34) under conditions al)-a a) and h') the function h(x, s) satisfies Caratheodory conditions for x ~ , s6R I and

Ih(x, s)l<~(s)+h~(x), sh(x, s)~h2(x) Is l--h3(x) with a positive constant ~ and nonnegative functions hI(x)EL~(Q), ha(x)@L1(~), h2(x)ELp(fl), where p = 1 for m > n/2, p > 1 for m = n/2, p = 2n/(n + 2m) for m < n/2. THEOREM 3.8 [164]. trary

Let conditions al)-a~), h') hold and let us assume that for an arbiThen the problem (34), (22) has a solution u(x)

functionw(x)~KerLo, w=/=0 (35) holds.

~m(~). F o l l o w i n g [135, 156] we g i v e a r e s u l t for the equation

Z

about the solvability

of t h e D i r i c h l e t

( -1)I=ID~[a~(x)Dpu]=g(x' u . . . . . Dmu)u+f(x).

problem (48)

lal,iPl.~m

We preserve conditions al)-a~) and we denote by Xj the eigenvalues of the operator L 0 numbered in increasing order. Let ~(x), ~(x) be measurable functions such that for some k

mes {xO~ : tz(x) >~,h} > 0 ,

(49)

mes{xOfl : [~(x) 0 . THEOREM 3.9. Let conditions al)-aa) hold, the function theodory condition for x6~j@l~NCJ~, ]=0, ..., m and

g(X,~o . . . . . ~m)

satisfy a Cara-

,~(x)~g(x, ~o. . . . . ~ ) ~ ( x ) with functions ~(x), ~(x) satisfying (49). 2538

Then the problem (48), (22) has a solution ~m(~).

The proof is based (cf. [156]) on a linear homotopy of Eq. (48) to the equation obtained from (48) by replacing g(x, u ..... Dmu) by g(x, 0 ..... 0). We note that in [156] Theorem 3.9 is obtained for larger classes of equations obtained, in particular, by replacing g(x, u ..... Dmu), f(x), respectively, in (48) by g(x, u, .... D2m-~u), f(x, u ..... D=mu). 4.

Topological Characteristics of General Nonlinear El_liptic Problems

4.1. Reduction of General Nonlinear Elliptic Problems to Operator Equations. Let ~ be a bounded domain inRn with infinitely differentiable boundary 8~, n o = [n/2] + i, m, ml, .... m m be nonnegative integers, m > i~ We denote by M(q) the number of different multi-indices = (~i ..... ~n) with nonnegative integral coordinates ~i of length ]~[=~l-~...q-~., at most q. Let s be an integer no less than max(2m, m I + 1/2,...,m m + 1/2) and let us assmne that there are defined functions

/~':~XRM(2"~

~.i:~XRM(m:)-+RL

7---1 . . . . . m,

having continuous derivatives with respect to all arguments, respectively, to orders s 2m + i, s - m j + i, where ~ is an integer satisfying l>~i0q-n0. We shall also represent the functions ~-(x,~), Gj(X,N), ~={~=: l=i~.2m}, N={N~: l}l!~mj} in the form

~qx, ~)=~r(x, ~0, .... ~=~), Oj(x, n)=Oi(x, no..... %@ w h e r e ~k={~= : {=[ =k}, ~tk= {r[= :

t~z[

=k}.

We shall use the previously introduced notation D~u, Dku and let

_~__ OGI(x, ~i) o~(.,~) o~ ' G ,~(x,u~-~.

~r.(x,g)= In the present

section

we s h a l l

reduce the boundary problem

,..~-(X, U. . . . .

D2mu)=[(X),

O~(x, u, . . . . D ~ u)=g~(x), t o an o p e r a t o r

(50)

j=l

(51)

x6~d,

. . . . . m, xEOQ

(52)

equation.

L e t u s assume t h e f o l l o w i n g

conditions

hold:

I) for an arbitrary function v(x)6Ht(~) = W ~t( Q) t h e o p e r a t o r I

I

u(~):m(~)~m(a, oa) =H,-~Ca)x~-~'-r(oa)x... XH

m

(oa),

defined by

(53)

U(v)u=(L(v)u, & (v)u . . . . . B.,(v)u), where

L(v)u(x)=

~.d ~=(x, v . . . . , D~mv)D=u(x), xfiQ, I=]~2m

Bj(v) u ( x ) =

~

O1,~(x, v , . . . , D'~Jv)Dl~u(x), xeO~,

t={<mj

is elliptic

and h a s i n d e x z e r o ;

2) t h e r e e x i s t s a f u n c t i o n H:~XRM(=m+I)-+R 1 o f c l a s s function vfHt(fi) the problem

L(v)u+M(v)u=O, B~(v)u=0, has only the zero solution in Hs

] = 1. . . . .

C t-2=§

x~-fa, m,

such that for an arbitrary

(54)

x6afa

Here

H~ (x, v . . . . . D 2'~-I'o) OVtt, IVl.<:2m--I

2539

H~(x,~)----

OH(x, ~) I

1

k----

~----

H~II

~ w

0

where the

infimum is taken

over all

1

k--

clarify the concepts used. For any natural number k , H 2 (~)~W2 2(0~) and W p (8~)i, is the space of functions which are boundary values of functions u(x) belonging to W~(~). In this space we define the norm

P(0~)

We

=in~IJ~lt w

P (oQ)

functions

u(x)

from wk(~) equal

t o ~ on t h e b o u n d a r y .

1

One can give an equivalent definition of the norm in W p of ~(x) to the domain ~ (cf. [166:24]).

P (Of~),without using the extension

We call the operator U(v) elliptic if L(v) is a regular elliptic operator and the system of boundary operators Bj(v) is connected with L(v) by the condition of Ya. B. Lopatinskii (cf. [38, 75]). The ellipticity of the equation L(v) means that for ~=(~i .... , ~n)~Rn the inequality

X ~r~'(x ' v . . . . . D2mv)~">~A(v)I~[ 2ra, I"t=2tn

holds

with positive

constant

A ( v ) d e p e n d i n g on v .

Remark 4.1. Making simple changes one could use, instead of the operator M(v) from condition 2), a completely continuous operator F(v):[/l(f2)-+Hz(~,0~) such that the operator H(v)+r(v) for each uECz.8~) established an isomorphism of the corresponding spaces. If condition i) holds, one can construct a similar operator F(v) and hence the class of problems for which a topological characteristic is introduced below is actually determined by condition i) alone. Remark 4.2. The constructions of the present section can also be made when the operator U(u) has negative index. Now if the index of this operator is positive then there has not been established a one-to-one correspondence between the boundary problem (51), (52) and the operator equation to which it is reduced. In this case one can introduce additional nonlinear functionals so that solutions of the operator equation satisfying nonlinear analogs of orthogonality conditions are solutions of the boundary problem. 1 l-mj---

Let f(x), gj(x) be fixed elements of the spaces fft-2m(~), f-f

2 (0~).

We define a non-

iinear operator A I:f-it(Q) _+ [f-it(~)]*i ( Atu, ~ > = ( F ( x ,

u. . . . .

D ~ u ) - - f ( x ) , L(tOq~+M(u){p)~_~m,n +

m

+ X~=~(O~(x,u. . . . . D m;tt)--g~(x), Bs(tz)~)~_m~_~_ o~, w h e r e (., .)~, n,

(',')~, on a r e ,

respectively,

the scalar

products

(55)

in the spaces

One can show that the right side of (55) is a continuous linear functional on the space Hs which makes the definition of the operator A~ proper. The proof of the boundedness of the functional so arising is based on the Nirenberg-Galliardo inequalities. The operator A~ introduced lets one reduce the study of the solvability of the nonlinear boundary problem (51), (52) to the solvability of the nonlinear operator equation A~u = 0. THEOREM 4.1. In order that the function u(x)~HZ(f~)'~ be a solution of the problem (51), (52) it is necessary and sufficient that Amu = 0. Questions of the solvability of a nonlinear problem (51), (52) will be studied on the basis of the degree of the map A m which we can define by virtue of the following theorem. THEOREM 4.2. If conditions i) and 2) hold and l>~10~-n~, the operator A~ defined by (55) is continuous, bounded, and satisfies condition =(Ht(~)). The a priori estimates of linear elliptic problems [75] are key in the verification of condition ~. If now D is an arbitrary bounded domain in the space H~(~) such that Alu ~ 0 for u~OD then Deg(Al, D, 0) defined in Sec. i, is the topological characteristic of the problem (5i), (52) sought. 2540

The operator AI introduced by (55) can be defined for sufficiently high smoothness of the functions ~r(X, ~), Oj(x,~j). One can relax the assumption about the differentiability of these functions by introducing into consideration corresponding operators in " V/pt,+0(~) for p>n. Let S be a bounded domain in R n with boundary 8~ of class Cl,+ 2 and we define a finite number of open sets U l ..... U I covering ~ and diffeomorphisms 9~:UI-+R n of class C t0+=,for which

cpz(U~)= B 1= {g~R" :[ Y I < 1}, i f U~ c ~ , and ~Pt (Ul)A ~)-----B1+~-- {vERn:I 9' [ < 1, 9n > 0}, ~ (U~ N 0 ~ ) = B(--{V~ Rn:]y,[ 1 and a natural number k as a norm in the space of traces of functions from wk(~). Namely, a ;

norm in W #

P(O$q), equivalent to the one introduced above is

1

+

~[D~ IX, (y)u~ (y)] --D~ [k, (z)ut (z)] lP ly'--z'ln+P -~

'

where ui(y)=u(c~i-l(y)), I ' i s t h e s e t of i n d i c e s i , l<~i<~I, f o r w h i c h U~]O~---~, t h e summation s i g n Z' d e n o t e s summation over t h o s e ~ f o r which t h e l a s t c o o r d i n a t e ~n i s e q u a l to z e r o . Let t h e f u n c t i o n s 8 r : ~ X R ~ t ~ - ~ R', O / ~ X R ~ f ~ D - ~ R ', ] = 1 . . . . . m belong, respectively, to the spaces

C~o+~-2m C~,+2Lm[and

satisfy the following conditions:

a) for an arbitrary function ~(x)6~F//p,+1(~)

u(~):w~'+'tn)-~w';+'(n,

lo+l-mm_-

lo+l--ra~----

~(0~)x ... xu%

on) =w~'+'-~(~)xw~

d e f i n e d by (53) i s e l l i p t i c

~(~n),

and of index z e r o ;

b) there exists a function H:~XRmt~m-~)-~R~ of class C 2+~~ such that for an arbitrary function ~_.w,,,+1(~/) w~ the problem (54) has only the zero solution in ~/~~

.... For _f(x)~W~~ gj~,X)~.W p',+~-~:-i# A 9Iyflo+I "'2.," p (n)-+ [~7~'+' (~)] * by ( A~u, ~ > ----

X

(~),

j = i, ... ,m we define a nonlinear operator

i'*a{D=I~r(x, u . . . . . D ~ & ) - - f ] }

D=[L ( u ) ~ l + M ( u ) n ] d x +

?/z

]=1 [ [~[
+

B'I

X'

....

DZ~&,)-- g:~]}

D" IBm,, (u~) n,] dr' +

.....

t~l=te-rni B1 El , --D~ IX, (V) (Oj,~ (z, u~, . . . , h 2rn: ,u,~--gin (z~)l}X

ay'az',,,+,,-, X {Dr~ [Xt(g) Bm (u,) ~h] -- O~ [~,, (z) B],, (u3 n,1} ~~,_~,

Here ,p(t)=ltlp-~t, are, respectively, stitution y=~i(x).

2).

}.

(56)

u~(D=u(~r'(D), n,(D=n(~r'CD).gJ.(v)=gl(~r'(D), o j . ( e , u, . . . . . D2'~:~3, Bj,, t h e f u n c t i o n and o p e r a t o r o b t a i n e d from Gj(x, u . . . . . D2mJu), Bj:by t h e subIn (56) L ( u ) , M(u), B j ( u ) a r e t h e same o p e r a t o r s as in c o n d i t i o n s 1) and

THEOREM 4.3. Let us assume that p > n and conditions a) and b) hold. In order that a function u(x)6~[/~~ be a solution of the problem (51), (52) it is necessary and sufficient that A2u = 0. The operator A 2 defined by (56) is bounded, continuous, and satisfies condition ~). 2541

Theorem 4.3 lets one associate with the boundary problem (51), (52) when conditions a) and b) hold, a topological characteristic, the degree of the map A 2 of an arbitrary domain

D~Wt~ +i (f~).,if 0~A~(0,f~). The degrees of the maps Al, A 2 introduced in the present point lets one develop topological methods of studying general boundary problems (51), (52), and to study questions of solvability, of ramification of solutions, of estimating the number of solutions of nonlinear problems on the basis of these methods. 4.2. Coercive A Priori Estimates for Pairs of Linear Elliptic Operators. The possibility of concrete calculation of the topological characteristic introduced in the preceding section depends on the construction of maps and hence it is desirable to have a simpler representation for the corresponding operators. In the case of Dirichlet boundary conditions coercive a priori estimates of the form

Re(Lu,

Mg)t>c, tl" It~m+,--c.II ullL u(x)ew~m+'(f~)n~Tg'(f~)

(57)

for second order linear elliptic operators L and M of order 2m with smooth coefficients simplify the construction of the nonlinear map corresponding to the differential problem. In (57) and later ci, c 2 are positive constants independent of u(x), s is a nonnegative integer, (., -)s II'IIs respectively, are the scalar product and norm in the Sobolev space W2s (57) for L = M follows from a priori L2-estimates for elliptic linear operators. When = 0, m = i, (57) is obtained for arbitrary second order elliptic operators L and M with real coefficients in Ladyzhenskaya [36] and Sobolevskii [66] and the importance of this inequality in the theory of differential equations is noted. It was shown by the author [61] that for ~ > 0 or m > 1 (57) may not hold. It follows from the examples given in [166:65] that there exist fourth order linear operators L and M with real coefficients for which (57) is invalid for s = 0, there exist second order operators L and M for which (57) does not hold for s = i. It is also shown in this same paper that (57) may not hold for m = i, s = 0 for operators L and M with complex-valued coefficients. In connection with the counterexamples indicated the following problem is solved in [61]: construct for a given family of elliptic linear operators {L} a linear operator M with special scalar product [., .]s in W2s such that an inequality of the form (57) holds with (Lu, Mu)s replaced by [Lu, Mu]z and such that the norm generated by [', "]s is equivalent with II"IIs These results are given in the present point. These results are recounted in detail in [166]. Subsequently in this point ~ is a bounded domain in the Euclidean space R ~ with boundary 3~ of class C ~ although it is obvious that the infinite differentiability of the boundary can be replaced by a specific smoothness condition. The positive number A is called an ellipticity constant of the linear operator

L(X, D)----

if for

X6~, ~=(~t . . . . .

X a=(x)D=, I= l<2rn ~ . ) c R n one has

x----:(x~. . . .

, x~)~cR 5

( -1)~ X a~(x)~>Al~T ~". lal=2~ Here as(x) the coefficients of the operator L(x, D), are real-valued functions, ~ = = ~

... ~=~.

For a nonnegative integer s and 0 < I < 1 we denote by Sfl,X am t ,A , B, f~) a family of regularly elliptic linear operators L(x, D) of order 2m with one ellipticity constant A, and coefficients as(x) of class CI,X(~) satisfying the conditions

IIa,, (x) Ilc~,~(~) --< B,

I c~l -< 2ra,

where ll'llc,.~(Q)denotes the norm in the space Ct'~(~) 9 THEOREM 4.4.

Let A and B be arbitrary positive numbers, 0 < ~ < i.

operator 7%4(x,m)----- X

be(x) m = w i t h

lal<2m

There are a linear

infinitely-differentiable real coefficients b~(x) and

positive constants cl, c 2 depending only on A, B, ~, m, ~ such thatil4(x, D) 6Se2~ o,x (I, c2, ~2) and for L(x,D)~ ~0,~ 2m(A, 19,~) one has 2542

f L(x, D ) u M ( x , D ) u d x > c l

(SS)

II ~ll~-c2llul[~

fl

0m

for any function

u(x)~W~m(f~)nW2 (f~).

Definition 4.1. We call a linear operator M(x, D) satisfying the conditions of Theorem 4.4, directing for the family ~0,zla2m ~ , B, fl). We proceed to a coercive estimate for pairs of elliptic linear operators for s > 0. For zA , B,~) we shall indicate a special choice of the scalar product in W~(~) for a family ~r 2 ~, ~,~ which a coercive estimate holds for an arbitrary operator L(x, D)6 2~ ( , B, Q) and directing operator M(x, D) defined by Theorem 4.4. Examples show the necessity of special choice of the scalar product. THEOREM 4.5. For arbitrary positive numbers A and B, nonnegative integer s and ~6(0, |) there are infinitely differentiable real-valued functions C~8(x), l~I, l~I~
~,, ~,.

(A, B, ~), v(x)6W'2(~),

.

0

.,~W2"+ uc ~ ' (Q)~ W ~m(Q) one has

[Lu, mul, > Kx II~ IlL+z-- K2 IIu 15

(59)

Ka It~' II~~< [v, vl, < K= [Iv lift,

(6o)

where the operator M(x, D) is defined by Theorem 4.4, the scalar product [v x, v2] ~ for v~(x), ~=(X)6V/~(~) is defined as follows:

[v~, v2],= 4.3. Reduction of Operator Equation. Now differential problem in strict ourselves to the

X

i C~p(x)D=vt(x)D~v2(x)dx"

(61)

the Dirichlet Problem for a General Nonlinear E l l ~ u a t i o n to an we give a simple construction of a nonlinear map corresponding to the the case of a Dirichlet boundary condition. For simplicity we recase of a problem with homogeneous boundary data.

Let ~ be a bounded domain in R ,~ with infinitely differentiable boundary 8Q and let us assume that the function fr : ~ X R ~ 2 m u + R ~ has continuous derivatives with respect to all its arguments to order s + i, s > n o . Here n o , M(2m) mean the same thing as in point 4.1. We shall reduce the boundary problem

Or(x, u . . . . .

Dmu)=f(x),

x~g2, (62)

D~'u(x)=O,

[~[~m--1,

x00~,

to an o p e r a t o r e q u a t i o n under t h e assumption t h a t f(x)~HZ(~) and t h e f u n c t i o n or(x, ~) s a t i s fies the condition. A) t h e 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 A such t h a t f o r a r b i t r a r y

~ R ~r

2~ ~r~(x, ~ ) n ~ > A l ~ j ~

~ R n one has

(63)

]al=2ra

where Sz'~(x, ~) is d e f i n e d by (50). o

Let D be an arbitrary bounded domain in the space:X=H~m+z(Q)NHm(f~). We assume the norm in X coincides with the norm in H2m+z(f~) and let B R be a ball in X with center at zero containing D. We consider a family of uniformly elliptic operators

2~-----{Lv=i=,<2m ~ gr~(x' v . . . . . D2~v) D= :v6B~}" The family so defined is contained for some constant B in the family '~2~ ~t.2(A, 23, ~2), consisting of regularly elliptic linear operators L(x, D) of order 2m with ellipticity constant A and coefficients a~(x) of class W2Z(~) satisfying the conditions

IIa~ (x)I1~(~ < B for I~ I< 2~. One verifies simply [166] that Theorem 4.5 remains valid upon replacing the family of

operators

2m~ , B, ~2) by t h e f a m i l y ~ t , a t a

B, ~2), f o r 2s > n 2543

For the constants A and B, the numbers m and s ing operator

AJ(x, D ) =

Z

and the domain ~ one can define a direct-

ba(x) D=

(64)

["[<2m

and a s c a l a r product in Hs l:hli~[
I

(65)

with infinitely differentiable real functionsba(x), ~B~,

us

C:~(x0), C:fl(x)=C~:(x), such that for

w ~ H ~ ( ~ ) o n e has

ul,>K, Ilull 22m+l --Xzllullg,

IL(v)u, ,

(66) (67)

l"

Here KI, K z are positive constants independent of v, u, and w. We can further subject the directing operator M(x, D) to the following condition:

(68)

(--1)~ I M ( x , D ) u . u d x > K~

For t h i s i t s u f f i c e s to r e p l a c e M(x, D) by the o p e r a t o r M(x, D) + (-l)mK with sufficiently l a r g e p o s i t i v e c o n s t a n t K. In t h i s case too, (68) is a simple consequence of Garding's ine q u a l i t y [112]. 0

We define a nonlinear operator A:D-+X*,

X=H>+t(f~)Nf-fm(f~)

( Au, cp ) = [ i f ( x , u . . . . . D~nu)--f~x), M ( x , D)~]t, ~6X. THEOREM 4.6. C[ +I.

(69)

Let l > t t 0 and c o n d i t i o n A) hold f o r t h e f u n c t i o n ff:~XRM(2m)-~R I of c l a s s

Then the operator A defined by (69) is bounded, continuous, and satisfies condition 0

In order that the function u(x)6H2~+l(~)f]Hm(f 0 be a solution of the problem (62) it is necessary and sufficient that Au = 0. Like the preceding theorems of this section, Theorem 4.6 lets one define a topological characteristic of the problem (62), the degree of the map A. 4.4. Conditional Solvability of General Nonlinear Boundary Problems. The reduction of differential problems defined in the preceding points to operator equations of the type considered in Sec. 1 lets one apply topological methods to the study of boundary problems, in particular, to homotopize the problem to a simpler one is the presence of specific a priori estimates. We restrict ourselves to the formulations of conditional existence theorems for general boundary problems and the nonlinear Dirichlet problem which one gets on the basis of the reduction made above of boundary problems to operator equations and Theorem 2.2. Some unconditional existence theorems can be obtained analogously to the corresponding assertions of Sec. 2. THEOREM 4.7. Let ~ be a bounded domain in R n with infinitely differentiable boundary 3~, n o = [n/2], m, ml,...,m m be nonnegative integers, ~ I , 10 be an integer no less than max (2m, m I + 1/2 .... ,mm + 1/2) and let the functions if(t, X, ~), 07(t, x~ ~j), j=| ..... m, be defined and continuous for t~[0,I], X 6 ~ ~ER ~(~m), ~yERM(mj)- Let us assume that there exists a continuous function If(t, x, ~ defined and continuous for t6[0, I], xE~, ~ER M(2m-~),such that f o r each t@[0,I] the functions 8z't(x,~)=~-(t,x,~), O~,t(x,~)=Oj(t,x, nj) , Ht(x,~)=H(t,x,~)are continuously differentiable with respect to their arguments, respectively, to orders Z - 2m + i, |--mj~-l, l ' 2 m + | , l>~lo-~-noand satisfy conditions i) and 2) of point 4.1. Let us assume in addition that the following conditions hold: a) there exists a positive function K: RI-+R I such that for t6~, ]] and u0HI(~) from

~art(x, u . . . . . m2mu) =#(x), O j , , ( x , u . . . . . D~lu)=tg~(x), it follows that 2544

x6e,

/ = 1 . . . . . m, x 6 ~

(70) (71)

[[u[[t.~
(72)

b) 8r0(x, --~}-------~0(X, ~), Ot,o(X, --~ll)-------Ol,o(X, TIj),Ho(x, --;)-----Ho(X' Then for t = 1 the problem (70), (71) has at l e a s t one s o l u t i o n in Hs Remark 4.3.

;)"

One can r e l a x condition a) of Theorem 4.7, r e q u i r i n g t h a t

Ilullc;,,~,~a)4K [//II.z-2,~,o+~

(73)

instead of (72). In this case (72) follows from (73) on the basis of a priori estimates of solutions of elliptic linear problems. Analogously to Theorem 4.7 one can formulate an assertion about the solvability of the problem (70), (71) in Wt~ We note the application of Theorem 4.6 to the study of the solvability of the Dirichlet problem. THEOREM 4.8. Let ~ be a bounded domain in R n with boundary 3~ of class C ~ the function ~r(t, X , ~ ) b e defined and continuous for [6.[0,I],;xG~, ~CR M(2m) and for each /~[0,I] ~r(~,~x,~) be]~-|. long to the class C t+* in the variables x,~=,/>t~0=[ on--" L~A conditions hold:

Let us assume that the following

I) there exists a %fi(0,I) and a positive constant K such that for t~[0,I] from 0

F (t, X, v . . . . .

it follows that

D ~ m ~ ) = 0, ~ 6 X = H m ( e ) ~

H 2ra+t(e),

( 74 )

HvHc2m,~,(n).~
1~l=2m

O~~

'

2) 8r(0,-X,--~)=--~(0, x,~) for mQfi, ~6/~M(~m). Then the problem (74) for t = i h a s at least one solution in X. Remark solvability we see that quires less

4.4. Comparing Theorem 4.8 with Browder's theorem [58:165] on the conditional of the problem (74), proved on the basis of the Leray-Schauder degree theory, the use of the theory of the degree of operators satisfying conditions ~) rerestrictive assumptions.

4.5. Solvability of th e Nonlinear Dirichlet Problem in a Narrow Strip. In this point we establish the existence of a classical solution of the Dirichlet problem for a general nonlinear elliptic equation in a narrow strip. We note that the condition of narrowness of the strip does not assume that the measure of the strip is small since the measure of the strip can be arbitrary. The results of this point are obtained in [59] and they are proved in [166]. By {~h, O < / z ~ l } we shall denote a family of domains in R n with infinitely differentiable boundaries such that: a)

~n,Cf~, for

hl


b) there exist open coverings {U i, i = i ..... I} of the set ~l and diffeomorphisms ~:UtC]~I-+~ ~ of class C ~ for which

~ (u~s~) =s~ = { x ~

: Ix'l < ~, 0<~x~<~},

x'= (x~, .... xn-~). Let {~i(x)}, i = i ..... I be a partition of unity subordinate to the covering {Ui}. For nonnegative integers m, s and k, and arbitrary p > I we denote by V/~'n't't~(f~n)the closure of the set of infinitely differentiable functions on ~hwith respect to the norm I

2545

where p

~ h)

l~l~2m

I~l~t

Here E' denotes summation over all those multi-indices whose last coordinate is equal to zero. The space cs,r,k(~h) is defined for nonnegative integers s and r and for % belonging to the segment [0, i] as the set of functions with finite norm

where

and If" II~ '~ is the usual norm in the space of functions satisfying a HSlder condition with exponent I. We note an auxiliary estimate for functions belonging to the spaces just defined: 0

I

For an arbitrary function U(X)@~'~m't'O(~h)NW$(~h)

for I > n n u|,

O
has

I

where K is a constant depending only on n, m, s and the constants characterizing the differential properties of the functions ~i, ~, which are assumed fixed. In what follows we give a coercive estimate for pairs of elliptic linear operators in S h. The difference of this estimate from the estimates given in point 4.2 is on the one hand the uniformity in h6(0, I] and on the other, the specifics of the choice of function space for whose elements the estimate is established. These results are basic both for getting estimates of the solution of a nonlinear problem and for proving the existence theorem. As before, ~I'XtA, --2m- B, Sh), for ators defined in point 4.2.

~(O,l], A, B > 0

denotes the family of regularly elliptic oper-

THEOREM 4.7. There exist constants el, c2 depending only on A, B, m, n, %, s such that for q > c I, p > qc I an arbitrary operator L{x,D)E~(A,B, Sh) and an arbitrary function u(x)@

W~2ra't"(Sta)N~2ra 2 ~t S hi,

equal to zero for

I x'{

c l o s e t o one, one has

s!{L(x'D)D:u'3~(D)D:u+P,~ L(x'D)D~u'AJ(D)D~u}dx> >"7-

iD%m+'ul=dx

s,,i,,l
-q-c,pq~' ii~l~m~'I[D"Df~Drtt[,dx--c(p,q) I ~ la]
where

=

I'?l=mSh

3'I(D)=D2nm+q[D~+...+D~_~] m i s

[D=ttlgdx:

(76)

Sh /~j<2m

a directing

operator,

c ( p , q) i s a c o n s t a n t

depending

on A, B, m, n, X, p, q. In what follows the number s is considered fixed, l>t~q-l. in W ==,t,1( ~ of the nonlinear Dirichlet problem

ff (x,.tt,~.... ,Dgmtt)-=f~x), xE~h; D=u=O,

xEo~h,I=l< m - i "

We consider the solvability

(77) (78)

I t i s assumed t h a t t h e f u n c t i o n ~ ' ( x , ~ ) i s d e f i n e d f o r x ~ , ~ = { ~ ' = : 1~]~<2m}~R~,has cont i n u o u s derivatives with respect to all a r g u m e n t s to order l>~-~-!l and for some positive constant v for N = (N~..... Nn)~R ~ one has

0%~x, ~) ~l. > v i ~l ]~m"

(79)

1=l=2m

We s h a l l and

2546

assume t h a t ~r(x,O)--O, t h e f u n c t i o n 11f

ilw~,t,~ta)'-~-I1f

f(x) belongs to the space ~'t'~(~OflC~'X~O Ilc~.X(~o~
First we give a priori estimates for solutions of the parametric problem

tSr~x, u, . . . , D~mu)+i 1 - t ) L ~ D~u=O, x~O~, [ ~ 1<~m - l , where

Lo(D)-----Za~D~

XO~h;

(80) (81)

is a fixed elliptic operator with constant coefficients for which

Lo(~)>~l~l == for ~'fiR~. We give an estimate for a solution of the problem (80), (81) satisfying the additional condition

IIu(X)llc~-,,2,x(~)<1.

(82)

For such a solution one gets the estimate

(83) with constant c depending on v, R, g(1), m, n and the functions ~i, ~

g(t)=lt~'(x, $)]tct~,• (82).

simply.

Here

B,----{~ERg:i~ l
THEOREM 4 . 8 . L e t u ( x ) be an a r b i t r a r y s o l u t i o n of t h e problem ( 8 0 ) , (81) s a t i s f y i n g There e x i s t s a c o n s t a n t N depending on m, n, ~, .v, R, g ( c ) mes~ 1 and t h e f u n c t i o n s

~p~,*~, such that

tl u (x)I

"-< N.

( 84 )

The existence of a solution of the problem (77), (78) is established by the topological method based on the degree of a map introduced in Sec. i. 0

1 (Qn) f] W 2 m (Qh) and l e t D={u~X:llullw~ ~,t,~(ah> < N + I We d e n o t e by X t h e Banach s p a c e lV72m,/, w~

},

where N is defined in Theorem 4.8. We pass for xEU~f]~n from the variables x to new variables y by x-----~1(Y), and we denote by Fi, L i the differential operators arising from F and L 0 under the change of variables indicated. We consider a family of linear operators of the form

L(V, D y ) = t X

I~ ,g2m

for

u(x)6D

ff~,~(Y, tt~, .

"'

D 2m v u,)D v + ( 1 - - t ) L , ( y , D,)

and which are solutions of the problem (80), (81), ui(y)=u(~i-1(y)).

One verifies simply that the set of these operators is contained in a family ~ m U X ( A , 13, Sh) and hence by Theorem 4.7 for the set considered one can define an operator M(D) and constants p and q such that (76) holds. Below q and p are assumed to be chosen in precisely this way. We define a nonlinear operator A t :D + X* by I

( Atu, ~) > =

S ~2~(Y){O~ [t$'~(y, u~. . . . . DmuO+ Sh

+ (,1 -- t) L , u , - tf~ (y)] D~Mvi + p t=t,~t THEOREM 4.9.

The operators At,

D ~ [t~'~ (y, ut . . . . . DWuO +(1 -- t) L#~ -6 tf~ (y)] D=Mv~} dy. tE[0,1],satisfy condition a(00 of Sec. i.

The applicability of Theorem 2.1 to the proof of solvability of the operator equation I

A1u = 0 follows from Theorem 4.9. Choosing h so thatA'h 2 (N+|)~
THEOREM 4.10. Let f~h2 (N+l)-.
Then the Dirichlet problem (77), (78) has a solution

Estimate of the Number of Solutions of Nonlinear Boundary Problems

In the last 20 years along with the theory of the degree of maps other methods of global nonlinear analysis, in particular the methods of Lyusternik-Shnirel'man and Morse theory have found broad application to the theory of nonlinear boundary problems. Estimates of the 2547

number of solutions of nonlinear variational problems have been found by these methods in the papers of Browder, Palais, Smale, Necas, Fucik, S. G. Suvorov, the author and others. There is a survey of these results basicaliy in [58]. In what follows we give some recent results connected with estimating the number of solutions. 5.1. Method of Spherical Fibration in Noncoercive Variational Problems. In [51] Pokhozhaev suggested a method of finding solutions of the variational equation

(85)

~" (u) =O

in the form u = rv, where r = Ilull. The problem splits into finding the normalizing factor r and the element v~Sl={uEX :llul]=1[. Finding v is connected here with the problem on a conditional extremum which opens the possibility of using the methods of Lyusternik-Shnirel'man for new classes of functionals. Let X be a real Banach space with differentiable norm on X \ { 0} and Sr be a differentiable functional on X. Let u = rv and we consider the functional

fir(r, o)=~r (rv). (r,o)6R'~(O})XS,. Let (r,v)e(O,q-o6)XSl

d e f i n e d on RI_XX f o r

THEOREM 5.1. be a critical point of the functional ~, considered on (0, +~) • S I. Then the vector u = rv is a critical point of the functional ~-. The realizability of the spherical fibration is determined by the scalar equation

~ r ' / ( r , v) = 0 with respect to r for

vESt.

(87)

'

THEOREM 5.2. Let (87) for any v6SI have a solution r(v) from the class CI(sI). Then to each critical point ve~Sl with r c = r(v c) ~ 0 of the functional ~'{t(v)v),considered on S I corresponds a stationary point Uc = rcv c of the original functional ~'. THEOREM 5.3. Let the functional following maximum exists:

~ - f r o m the class CI(X) be such that for any

~'(~)= r6max ~(rv) > ~(0) R,

v%St the

(88)

Let the functional ~ be differentiable on S I. Then to each critical point v c of the funct i o n a l ~ r, considered on the unit sphere S l corresponds a critical point u c = rcV c of the functional ~- with r c ~ 0 such that ~-(revc)=~'(vc). If under the hypotheses of Theorem 5.3 ~ ( 0 ) = 0 , then the functional ~'(v), defined by (88) turns out to be even and positive on S I. This lets one, using the results of LyusternikShnirel'man (cf. [34]), get existence theorems for many solutions. Thus one gets THEOREM 5.4. Let X be an infinite-dimensional, reflexive Banach space with Schauder basis and with differentiable norm on X \ {0} whose derivative is uniformly continuous on the unit sphere S l and considered on S l as an operator from S l to X*, has continuous inverse. Let the functional 5r from the class CI(X) be such that for any v6S1 there exists an ~(v),defined according to (88), and let ~-(0) = 0. Let us assume that the functional ~ admits an extension ~ to a ball B R = {u@X :llull<~.R}, R > I , such that ~ is a_weakly continuously differentiable functional whose derivative is uniformly continuous on B R and moreover,

r Then t h e f u n c t i o n a l

r

r

O'(u)#0

8z" has a c o u n t a b l e s e t o f d i f f e r e n t

for u G G \ { 0 } . critical

We show f u r t h e r w i t h an example how t h e n o n u n i q u e s o l v a b i l i t y t e n c e of m u l t i p l e s o l u t i o n s of boundary problems. L e t ~ be a bounded domain i n Rn,

n<.7, w i t h smooth b o u n d a r y .

points. o f (87) l e a d s t o t h e e x i s In the domain S we consider

the following problem

A~u (x) " u ~(x):-bh (x)

D=u (x) = O,

2548

x6.c9•,

=

O,

x~,~,

1~ I<- 1,

(89) (90)

0

(87) for the problem (89), (90) has the form

where %(x)~[W~~(~)1..

(9~) for 9(x)6SI----{~(x)~*~(~): :

liD=~[l~---~|}.

Under the

condition

[c;l=~

1

8

2-3 -~

(92)

Eq. (91) has three isolated smooth branches of solutions. On the basis of Theorem 5.2 we get that under the conditions (92) the problem (89), (90) has three different solutions in

5.2. Application of the Degree of Maps to Estimation of the Number of Solutions. One of the methods for gettin Z a lower bound for the number of solutions of nonlinear elliptic problems is the use of Theorem 1.8 which relates the degree of a map and the values of the indices of critical points. In this way one can consider both variational and nonvariational boundary problems including those for general nonlinear equations. We restrict ourselves just to the case of variational problems (cf. [27]). Let Q be a bounded domain in n-dimensional Euclidean space. functional

0m

On Wp(~) we consider the

(93)

8r(u)= I f (x, ~. . . . , Dmu)dx, assuming the following conditions hold:

a) the functions f(x, 0) and g(x, 0) belong to the space L~(E), f~(x, 0), and g~(x, 0) belong to Lq~(~) and for x6Q, $6~ a, ~6~ ~' one has

I I, l r
0if(x, D c)f (x. D , /c;~(x,~)= • ' analogously for g, h I is a posi-

]

Here

~0= ~ : l ~ l < m - - un~

t i r e , continuous, nondecreasing function, 2-.
np n ' P"----h'(m--lo~ !) p" i f //Z--~-
Pc;~= ~

pc;~=l,

P=~ &l,

qc;--~l, iflcz],

l~<m, Pa~=pg~ and they are determined by the eondiPc~

if

i~l=l~l=m, pc;~-----1-- 1 ' qc;= ' Ip---~T--1

l~l<m--p,

O
ifm---P~<]~

Pc; P~-' i f i~zl, I ~ t > m - p ,

n

l~[<m----~,

l~lq-1~l<2m;

b) there exists a continuous, positive, nonincreasing function h 2 such that for ~ER z~, N6RM~,one has

x6~,

t=l=l~l=m

THEOREM 5.5.

Let the function f(x, ~) s a t i s f y conditions a) and b), , ~ be an increasing

functional having two critical points of local minimal type. more critical point. The assertion follows from Sec. i. 0

m

Then there exists at least one

For sufficiently large R > 0

Deg{~, BR, 0)~I

by

~

Theorem 1.12, BR = { u Q W o ( ):ll~llm,0~/~}. iIf the local minimum points u I and u 2 are nonisolated critical points of the map ~', then the assertion being proved is valid. If the local minimal points ul, u 2 are isolated critical points then by Theorem i.ii, Ind(~",ul)-----Ind(SZ",uz)=l and the assertion being proved is a corollary of Theorem 1.8. 2549

Definition 5.1. The critical point u 0 of the functional $r is called nondegenerate the equation ~r#(u0)h=0 has only the zero solution.

if

THEOREM 5.6. Let ~ be a bounded domain in R 2 with boundary 8~ of class C ~ and let conditions a) and b) hold for the functional ~'. Let us assume that for the increasing func0 tional ~ there exists a nondegenerate critical point u I and a function u2(x)EWpm(~), u2=/=ul for which 8r(u2)<~r(ul). Then the functional $r has at least three critical points. Just as in the preceding theorem the assertion being proved follows from Theorem 1.8. In exactly the same way as above, for some R, Deg(~-,Ba, 0)= I. One can verify that under the hypotheses of the theorem the formula for calculating the index of the critical point u I is applicable so that [Ind(Hr, u~)H = I. Moreover, in B R there exists a local minimal point of the functional ~- different from u I. All the result follows from Theorems i.ii and 1.8. As an example we consider the question of the number of solutions of the problem of state of tension of flexible plates. The corresponding boundary problem was already posed in Sec. 1 of Chap. i. Now we write it in the form

a~f (x, y) = - - L (m, ~),

Aiw(x, y)~=~L(Fo,w)+L(~,w),

(96)

(x, g)E~,

~(x, v~=~(x, Ow v)=f(x, v)=~f(x, v)=o, (x, v)~a~,

(97)

where ~ i s a bounded domain in t h e p l a n e w i t h boundary 8~, F0(x, y) i s a known f u n c t i o n of c l a s s W~(fl) i s d e f i n e d in p o i n t 1.3 of Chap. 1. For any ~ the problem (44), (45) has the solution f = w = 0 however the zero solution is not always unique; this corresponds to the familiar fact that plates can have several equilibrium forms. As a rule, only one of the equilibrium forms is desirable. Passage to the other forms may mean destruction of the construction. In this connection the need for predicting such a transition arises. 02 02 The problem (96), (97) is equivalent to a variational problem. Let ~:W2(~)-+W2(~) be the operator which assigns to a function w a generalized solution f = Rw of the equation

Aif=--L(w, m). One proves that the problem (96), (97) is equivalent to the variational problem

s~'(~) =xo'Cw), where

~', O'

are the derivatives of the twice continuously Frechet differentiable functionals

02 ~, O:Wi(~),-~ I, defined by fi

Along with the problem (96), (97) we consider the linearized problem

A2m(~, V)=XL(F o, ~), (x, V)~, g ) : ~ d ( x , y)----O, ( x , y ) ~ ,

(98) (99)

whose solutions satisfy

~-" (0) w--s

(0) w =0,

(i00)

~r't{0), O"(0) being the Frechet derivatives of the operators ~r" G" ar zero. Now we consider the question of the number of equilibrium forms of a plate for different values of k. We shall assume the mechanically justified condition

I S L (., .)p0 (x, y)exey>o fi

h o l d s f o r ~s

2550

0 2

~=/=0.

In this case the problem (98), (99) has a countable number of eigenvalues An, where 0<~i<~.,.~<~<... and An + =. One verifies simply that for A > A l the minimum of the funct i o n a l ~ u ) - - %O(u) on W~(~) is negative9 We denote by A 0 the infimum of the values of A for which the minimum of the functional 0 Sru--%G(u) onW22(Q) is negative. Obviously %0~<11. Analogously to Theorem 5.6 one gets THEOREM 5.7. For any value of A the functionalSr(u) --%O(u) is increasing. Let ~ > A0, A ~ An (n = I, 2 .... ). Then the problem (96), (97) has at least three solutions. LITERATURE CITED i.

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HIGH ORDER NONLINEAR PARABOLIC EQUATIONS

UDC 517.956.45

Yu. A. Dubinskii

The basic results and methods of the theory of high order nonlinear parabolic equations are described. In the first chapter boundary problems for quasilinear parabolic equations having divergent form are considered. In the second chapter nonlinear parabolic equations of general form are considered~ Attention is mainly paid to methods of study of nonlinear parabolic problems~ In particular, the methods of monotonicity and compactness, the method of a priori estimates, the functional-analytic method, etc. are described.

PREFACE The present survey is devoted to one of the actively developing directions of contemporary nonlinear analysis, the theory of high order nonlinear parabolic equations. Just as in the corresponding elliptic theory, the theory of high order nonlinear parabolic problems counts nearly a quarter of a century in its development (M. I. Vishik's first paper on nonlinear parabolic equations of order 2m appeared in 1962.) However in this comparatively short period it has been enriched with such important results, and what is no less important, methods of study, that it has rightfully become one of the fundamental directions of contemporary research. The object of the present paper is the description of the basic achievements of this theory. The paper consists of two chapters. In the first chapter a survey of various results in the theory of nonlinear parabolic equations of divergent form is given. In the second chapter boundary problems for nonlinear parabolic equations of general form are described. Since a brief scientific and historical annotation prefaces each section (cf. also the comments in the "Literature" section) we shall not dwell on the content of the paper by chapters here in the preface but characterize it on the whole. First of all we note that in the choice of material we have chosen papers in which the methods of studying nonlinear boundary problems have been developed. Among such methods are the method of compactness, the method of monotonicity, the method of a priori estimates, the function-analytic method, etc. Further, as is known there are two general approaches to the theory of nonlinear parabolic problems. The first approach is based on the consideration of the initial-boundary value problem for a nonlinear parabolic equation as a Cauchy problem for the corresponding nonlinear differential-operator equation in a Banach space.

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 37, pp. 89-166, 1990.

0090-4104/91/5604-2557512.50

9 1991 Plenum Publishing Corporation

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