Nonlinear elliptic and parabolic equations

NONLINEAR Yu.

ELLIPTIC A.

AND

PARABOLIC

EQUATIONS LrDC 517.956

Dubinskii

Results of recent years are presented on the theory of nonlinear elliptic and parabolic equations of any order including equations of infinite order. PRE

FACE

In the present paper a number of basic results are systematically presented from the theory of nonlinear elliptic and parabolic problems which have been obtained in recent years by both Soviet and foreign mathematicians. The work consists of four chapters, and each chapter begins with an introduction. Because of this, we shall not relate the content of the individual chapters in the preface but rather characterize the material as a whole. The theory of nonlinear boundary-value problems for elliptic and parabolic equations expounded here has formed in the last 10-12 years and pertains to the theory of nonlinear equations which have received the name of equations of divergence type or equations in divergence form. Such equations arise primarily in the study of nonlinear variational problems. Analysis of specific problems shows that a number of the most important variational problems lead to the consideration of two classes of nonlinear equations; one of these classes is naturally called the class of weakly nonlinear equations, while the other is the class of equations with strong nonlinearities. The equations of the first class have, roughly speaking, a linear principal part and contain a nonlinear perturbation in the lower order terms; equations of the second class are essentially nonlinear. Chapter I is devoted to weakly nonlinear problems, linear problems.

while Chaps. II and HI are devoted to strongly non-

Actually, the theory presented is more general than simply the theory of nonlinear boundary-value problems for elliptic and parabolic equations of divergence type. This is the theory of operator and differentialoperator equations of first order in Banaeh spaces satisfying well-known properties of coercivity and monotonieity. The solvability of boundary-value problems themselves is an application of the corresponding operator theory. It should be mentioned that the operator approach makes it possible to simplify to maximum exposition of the theory of nonlinear problems for partial differential equations.

extent the

The main results of the first three chapters are obtained by various modifications of an approximate Galerkin method which consists of a suitable approximation of the problem and subsequent closure. In applications to weakly nonlinear problems this method has received the name of the compactness method, while in application to essentially nonlinear problems it is called the monetonlcity method or the method of monotone operators. In the last chapter, Chap. IV, nonlinear equations in d i v e r g e n c e f o r m of infinite o r d e r a r e studied as a r e q u e s tions which h e r e b y a r i s e : nontriviality of Sobolev s p a c e s of infinite o r d e r , the behavior of solutions of nonlinear b o u n d a r y - v a l u e p r o b l e m s of o r d e r 2 m as m -- ~, etc. It is found that nonlinear problems of infinite o r d e r p o s s e s s a n u m b e r of p r o p e r t i e s both of weakly nonlinear and s t r o n g l y nonlinear equations. The w o r k concludes with a b r i e f s u r v e y of the h i s t o r y of the question and of papers bearing a d i r e c t r e l a tion to the m a t e r i a l presented. To a l a r g e extent this is w o r k which has been reviewed in RZhMat during the period f r o m 1963 to 1974.

T r a n s l a t e d f r o m Itogi Nauki i Tekhniki, S o v r e m e n n y e P r o b l e m y Matematiki, Vol. 9, pp. 5-130, 1976.

0096-4104/79/1205-0475507.50

9

Plenum Publishing Corporation

475

CHAPTER WEAKLY

NONLINEAR

OPERATOR

I

EQUATIONS

TO BOUNDARY-VALUE

AND T H E I R

APPLICATIONS

PROBLEMS

Introduction In the present chapter equations of the form

,4 (u) = h

(o. 1)

are studied in a Banach space; here A(u) is (generally speaking) a nonlinear operator which has the property of weak compactness. Coercive equations a r e studied in Sec. 1. In this case the equation is solvable for any right side h. Noncoercive equations a r e studied in Secs. 2 and 3. Here solvability is established for any right side h up to the kernel of some auxiliary operator. In the f i r s t case applications of the theorems on the solvability of Eq. (0.1) lead to the solvability of c o r r e c t l y posed problems for a number of partial differential equations, equations of the mechanics of continuous media, etc. In the second case applications lead to a number of results in the theory of i n c o r r e c t problems, e.g., the Dirichlet problems for parabolic or hyperbolic equations of second o r d e r , etc. We note that the main tool for investigating Eq. (0.1) is the Galerkin method or its modifications. As already mentioned, this method consists in a finite-dimensional approximation of Eq. (0.1) and subsequent closure. The solvability of the approximating Galerkin equations, regardl ess of the c h a r a c t e r of the nonlinearity of operator A(u), follows f r om the lemma "on the acute angle," which is simpl e but extraordinary for its importance (see Sec. 1, Part1~ The most appreciable difficulties a r i s e in closure, i.e., in passing to the limit. Here the c h a r a c t e r of the nonlinearity of the operator A(u) pays a decisive role. In the present chapter in the study of the abstract equations (0.1) this difficulty is palliated by the requirement of weak compactness of the operator, and therefore the simplest a priori estimates suffice to justify passing to the limit. In applications the verification of the property of weak compactness is occasionally very nontrivial. 1. C o e r c i v e Nonlinear

Equations. Elliptic

Dirichlet

Problem

for

a Weakly

Equation

In this section we study operator equations of the type

A (u) =h,

(1.1)*

where the o p er ato r A(u) (which is generally speaking nonlinear) possesses two properties: coerciveness and weak compactness. We call such equations weakly nonlinear equations. The proof of the solvability of Eq. (1.1) r e s t s on the lemma "on the acute angle,, which, as has already been noted, plays an extremely important role throughout P a r t 1. Our exposition begins with this lemma. We then consider applications to the Dirichiet problem for nonlinear elliptic equations. 1~ Lemma Won the Acute Angle." Let E n be n-dimensional Euclidean (real or complex) space, and let ( . , 9) denote the scal ar product in E n. We consider the continuous mapping

(~) ~_ (p~ (c) . . . . . p,, (c)): E . - . . . e . . We ar e interested in the question of the existence of a kernel of this mapping, i.e., the question of the solvability of the s y s t e m of equations

p (c) = o. LEMMA 1.1. Suppose that on the sphere SR = { ~ : IC'I = R} (R > 0 is some number) the "acute-angle condition" is satisfied: Re (P (C), C) > O. Then there exists at least one point C, I C I ~ R, such that

p(c)=o. *Two Eqs. (1.1) a r e denoted in Russian original - Publisher. 476

(1.1)*

P r o o f . W e s u p p o s e o t h e r w i s e : f o r a n y C, I C I < R , w e h a v e the i n e q u a l i t y

~(e)§ I n t h e b a l l V R = { ~ , {~{ --<-R } w e c o n s i d e r t h e m a p p i n g

Ip(C)l O b v i o u s l y , A(C) : V R ~ S R i s a c o n t i n u o u s m a p p i n g , and t h e r e f o r e by B r o u w e r ' s t h e o r e m t h e r e e x i s t s a fixed p o i n t o f t h e m a p p i n g A ( C ) , i . e . , t h e r e e x i s t s a C E SR s u c h t h a t

X(5=cC But t h i s i s i m p o s s i b l e ,

<~.2)

s i n c e by c o n d i t i o n (1.1)

~e (-~ (Ch, ai--

.o%(c), c)a<0 ' IP(C)I

w h i l e (1.2) l e a d s to t h e i n e q u a l i t y

~e (3 (~), 4 = I~ I-~= m > 0 T h i s c o n t r a d i c t i o n p r o v e s the l e m m a . 2 ~. S o l v a b i l i t y o f t h e O p e r a t o r E q u a t i o n . L e t X be a B a n a c h s p a c e w h i c h w e a s s u m e to be s e p a r a b l e and c o m p l e t e . F u r t h e r , l e t X* be t h e d u a l s p a c e , i . e . , t h e s p a c e of l i n e a r f u n c t i o n a l s on X. T h e xialue of y* E X* on a n e l e m e n t x E X w e d e n o t e by ( y * , x>. We consider the equation

A (~) = h,

(1.3)

w h e r e A(u) : X ~ X* is a n o p e r a t o r ( n o n l i n e a r in g e n e r a l ) w h i c h s a t i s f i e s the f o l l o w i n g c o n d i t i o n s : I. C o e r c i v i t y C o n d i t i o m F o r a n y u E X w e h a v e the r e l a t i o n R e

~
r n ~

w h e r e {u m } i s s o m e s u b s e q u e n c e of the s e q u e n c e {Un}. We have the following result. T H E O R E M 1.1. L e t o p e r a t o r A(u) s a t i s f y C o n d i t i o n s I and II. T h e n f o r any e l e m e n t h E X* Eq. (1.1) h a s a t l e a s t one s o l u t i o n u E X. P r o o f . L e t v 1, v 2, . . . be a c o m p l e t e s y s t e m i n X (we r e c a l l t h a t X i s s e p a r a b l e ) . W e se~ u n = c l v ~ + . . o + CnVn (n = 1, 2 , . . . ), w h e r e the c o n s t a n t s c 1. . . . . Cn a r e d e t e r m i n e d f r o m the s y s t e m of ( n o n l i n e a r ) e q u a t i o n s

< A (u,,), vj > = < h, v] >, ] = 1 . . . . .

n.

(1.4)

T h e s o l v a b i l i t y of t h i s s y s t e m f o l l o w s f r o m C o n d i t i o n I a n d t h e " l e m m a on t h e a c u t e a n g l e . " I n d e e d , t h e s o l v a b i l i t y of s y s t e m (1.4) is o b v i o u s l y e q u i v a l e n t to t h e e x i s t e n c e of a s o l u t i o n o f t h e s y s t e m 0=1)]

(C)--

< A (an), ~o] ) - - < h, v I >, ] = I . . . . .

n.

We shall show that the mapping

(5) _= (p, (~) . . . . . p n (~)): c~-, c~ satisfies the "acute-angle" condition

~e(~ (a), ch>0,

(1.5)

p r o v i d e d t h a t I C I = R , w h e r e R = R{h) i s s u f f i c i e n t l y l a r g e . We have

477

whence

(here 11h IIx, is the n o r m in X*). Hence, by the e o e r c i v i t y condition (1.5) holds provided that tl Un[[X is sufficiently l a r g e . Since in a finited i m e n s i o n a l s p a c e all n o r m s a r e equivalent, this m e a n s that inequality (1.1) is s a t i s f i e d a l s o for I C I = R, w h e r e R = R (h) is sufficiently l a r g e . Thus, s y s t e m (1.4) is s o l v a b l e , and a sequence of a p p r o x i m a t e solutions Un has b e e n found. M o r e o v e r , s y s t e m (1.4) i m p l i e s the equality




which again by c o e r e i v i t y i m m e d i a t e l y leads to the e s t i m a t e

Ilu.l,x K(h), n = l , 2

.....

w h e r e K(h) is a c o n s t a n t depending on IIh IIx*. F r o m this e s t i m a t e and the w e a k c o m p a c t n e s s of bounded s e t s in a r e f l e x i v e Banach s p a c e it follows that t h e r e e x i s t s a s u b s e q u e n c e {which we a g a i n denote by u n) which c o n v e r g e s w e a k l y in X to s o m e e l e m e n t u E X. The e l e m e n t found u ~ X is the d e s i r e d solution. Indeed, in (1.2) we fix the index j and let n - - + oo. C h o o s ing if n e c e s s a r y a n o t h e r s u b s e q u e n c e u m, we find that


j:l,2

.....

s i n c e by Condition tI o p e r a t o r A(u) is w e a k l y c o m p a c t . Since the s y s t e m vl, v 2. . . . m e a n s that A (u)-~ a.

is c o m p l e t e in X, this

The p r o o f of the t h e o r e m is c o m p l e t e . R e m a r k . The solution of Eq. (1.3) which has been found i s , g e n e r a l l y speaking, not unique. The condition of s t r i c t monotonicity of o p e r a t o r A(u) is a sufficient condition for the uniqueness of the solution:

ReO, ueX, vGX, u ~ v . Nonlinear o p e r a t o r s s a t i s f y i n g the monotonicity condition will be c o n s i d e r e d in detail below. 3 ~ D i r i c h l e t P r o b l e m f o r a Weakly Nonlinear Elliptic Equation. L e t G c R n be a bounded d o m a i n with p i e c e w i s e s m o o t h boundary F. We c o n s i d e r the D i r i e h l e t p r o b l e m f o r the elliptic equation

A(u)~--L(D)u+R(D)u=h(x), xEG, D~u[r:O, I ~ l ~ < m _ l .

(1.6) (1.7)

Here

is a s t r o n g l y e11iptic d i f f e r e n t i a l o p e r a t o r of o r d e r 2m. O p e r a t o r R(D)u is defined a s follows. L e t F(~T), w h e r e ~T = (~T1. . . . . ~Tn), IT I = r , r -< m - 1 be a non_negative r e a l - v a l u e d function which is homogeneous of o r d e r q + 1 (q > 0): F(X~T) = xq+IF(~T). We a s s u m e , m o r e o v e r , t h a t F(~T) is d i f f e r e n t i a b l e , and the e x p r e s s i o n 1

0

defines the n o r m in the Sobolev s p a c e W~+t(G). The nonlinear o p e r a t o r R(D)u is by definition the g r a d i e n t of the functional/q+~ (u), i . e . ,

R(D)g~_(--1)rZ Dt~FI~(DVu), w h e r e Ffl (~T) = OF / 8 ~fl. 478

l~['----r,

THEOREM 1.2. L e t h(x) be any g e n e r a l i z e d function of the f o r m h(x) = hi(x) + h2(x) , w h e r e hi(x) ~W~t(G),

h2(x)

E

-r

o

W ( q + l ) / q ( G ). Then p r o b l e m (t.6}, (1.7) has at l e a s t one solution u(x) EWm(G) N W~+,(G). Proof. To p r o v e the t h e o r e m we apply the o p e r a t o r T h e o r e m 1.1. To this end we set 0

0

x = w y (o) n we+, (o),

[I u !Ix = lI ~ [1~,~+ 1Iu !1,,~+~-

Obviously, A (~): X - + X * ,

since 0

0

--r " w2--P,$ (o) + w~+x (o) c(w~?rt (o) n wg+~(a))*.

q

Moreover, lira Re(A(u), u)

since [we take account of the s t r o n g ellipticity of L(D)u and the homogeneity of R(D)u] Re < A (u), tt } = Re < L (a) tt, it } -k Re ( R (D) ft, tt ) > a0 ][u j[~,~+ (q-k t) I q§ (t~). It r e m a i n s to c h e e k the weak c o m p a c t n e s s of mapping A(u). The weak compactness of L(D)u is obvious. We shall v e r i f y this p r o p e r t y for the nonlinear t e r m R(D)u. Indeed, tet Un(X) --u(x) weakly in the space X. Since the imbedding Wm(G) c Wr(G) is compact, we see that f r o m any sequence Un(X) converging in X it is possible to e x t r a c t a subsequenee urn(x) converging together with its d e r i v a t i v e s to o r d e r r a l m o s t e v e r y w h e r e in G. oWe have ~=, b

(1.8)

w h e r e v(x) E X. It is w e l l known (Vitali's t h e o r e m ; ef., e.g., Natanson [39]) that a sufficient condition for the possibility" of p a s s i n g to the l i m i t umAer the integral s i g n is the p r o p e r t y of equicontinuity of the f a m i l y of integrals. In our c a s e for any s e t e c G, using the H31der inequality, w e obtain q q+, \~ '_L "

,.

since the sequence Um(X) is bounded in wr+t(G) q

Hence, for any e > 0

P.

if the m e a s u r e of the set e is sufficiently small. This implies the equicontinuity of the family of integrals (1.8). By Vitali's t h e o r e m we obtain

II~l=r G

l131=r O

which we w e r e r e q u i r e d to show. Thus, all the hypotheses of T h e o r e m 1.1 a r e satisfied. The proof of T h e o r e m 1.2 is c o m p l e t e . Example. A(u) - - A u + I u tq-lu, q > 0. 2.

Noneoereive

Problems

for

Equations. Ordinary

Regularization

Differential

of Nonlinear

Equations

In this section we study nonlinear equations A(u) = h, w h e r e the o p e r a t o r A(u) does not satisfy the c o e r civity condition. With the equation A(u) = h, which in general is not solvable for an a r b i t r a r y right side h, t h e r e is r e l a t e d a c e r t a i n equation of the f o r m B*A(u) = h which is always solvable. The original equation is then

479

solvable up to the kernel of the o p e r a t o r B*. This c o n s t r u c t i o n may also be c o n s i d e r e d as a technique for d e s c r i b i n g the range of an o p e r a t o r A(u) c o r r e s p o n d i n g to an i n c o r r e c t problem. 1~ An A b s t r a c t T h e o r e m . Let X be a r e a l (or complex) reflexive Banach space; let V and Y be linear, locally convex topological s p a c e s with V s e p a r a b l e and V c X. Suppose, f u r t h e r , t h e r e a r e two o p e r a t o r s

A:X~Y,

B:V.+Y*,

w h e r e Y* is the dual s p a c e to Y. We a s s u m e that the following conditions a r e satisfied. I. The o p e r a t o r A ; X ~ Y is weakly c o m p a c t in the s e n s e that if u n ~ u weakly in X, then t h e r e exists a subsequence {u m} c {Un} such that for any y* ~ Y*

( A (urn), y* > -+ < A (u), y* >, w h e r e .
lira Re .......... _}_ II'~Iix

~ as

(2.1)

[[V[[x~+oo.

(Here ]Iv IIX is the norm of the element v E V in the space X.) THEOREM 2.1. Let Conditions I-III be satisfied. Suppose further that h ~ Y is such that sup I < oo. ~(!v [Ivllx

(2.2)

Then t h e r e exist elements u ~ X, p E Ker B* fl Y such that A(a)=h-Fp. The proof of this theorem is similar to that of Theorem 1.1; we shall therefore be brief, Let vl, v2. . . . be a complete system in the space V. We set un = clvI + . . . constants ej a r e determined from the following system of nonlinear equations

( A (~,), B y I > = ( h, B~,j >, ] = 1. . . . . n.

+ enVn, where the unknown

(2.3)

The solvability of s y s t e m (2.3) follows f r o m the " a c u t e - a n g l e " condition, which in t u r n follows f r o m r e l a tions (2.1) and (2.2). We m o r e o v e r have the obvious e s t i m a t e 11U. [Ix < K ,

(2.4)

w h e r e K > 0 is a constant (depending on h e Y). By the reflexivity of the s p a c e X the sequence Un c o n v e r g e s weakly to s o m e e l e m e n t u ~ X (strictly speaking, a f t e r choice of a subsequence). In this c a s e f r o m {2.3) as n ~ ~o we have (cf. Condition I) (A(u},Bv 1>~

, j

1,2 . . . . .

whence it follows that (cf. Condition II)

( A (a), B y > ~ < h, B y } , "o6V. This means that

A(u)=h+p, w h e r e p E Ker B* Q Y. The p r o o f of the t h e o r e m is complete. 2 ~ R e g u l a r i z a t i o n of Nonlinear P r o b l e m s for Ordinary D!fferential Equations. We shall now c o n s i d e r applications of the t h e o r e m proved.

480

1. Cauchy P r o b l e m on the S e m i a x i s [0, + ~o). We c o n s i d e r the p r o b l e m

u'~f(u)~-h(t),

(2.5) (2.6)

t>0,

u(O)=O, w h e r e f(u) is an a r b i t r a r y continuous function. B e f o r e f o r m u l a t i n g the t h e o r e m , we introduce the following notation: a) t~ 1 = { u ( t ) : u ' ( t ) ~ L2(0 , ~), u(0) = 0}. The n o r m in the s p a c e ~t I is defined by co

U~ !)o = i! u' (0)~et; 0

b) L2,comp = {u(t) i suppu(t) < ~o, u(t) ~ L~(0, :o)}. The topology in L2,comp is the topology of the inductive l i m i t of the family L2(0, a), a > 0, i.e., uv - * u in L2,comp if t h e r e e x i s t s a c o m p a c t s e t K c 1~1+such that suppuv cK, v= 1,2 .... and I l u - u v t i L 2 ( K )--* 0 a s v ~ o % c) We s e t L2,io c = (L2,comp)*. Obviously, L2,10 c = {u(t) : u(t)v(t) ELl(0 , r

v v ( t ) EL2,comt ~.

0

d) H ct o m p = {u(t) : suppu(t) < :% u'(t) ~ L2(0, r162 The topology of Hcompl is that of the inductive l i m i t of the family Hi(0, a), a > O. THEOREM 2.2. F o r any function h(t) E L2(0 , oo) t h e r e exists a function u(t) ~ ~I1 and a r e a l constant p such that ~" (t) + f (u) = k (t) + p. [In o t h e r w o r d s , p r o b l e m (2.5), (2.6) is solvable up to an additive constant.] Proof. We shall apply T h e o r e m 2.1; to this end we s e t

A ( u ) ~ u " + f (u), B t t ~ t t ' , o

O

t X ~ t t t, Y ~ L 2 . Ior V~Hcomp"

Obviot~sly, if u e V, then (A(u), Bu ) -= IIu II~. F u r t h e r , if h(t) ~ L 2(0, :r

u(t) ~ V, then sup ) < h (t), Bu >I

Thus, all the h y p o t h e s e s * of T h e o r e m 2.1 a r e s a t i s f i e d , whence we obtain the r e q u i r e d result. The p r o o f of the t h e o r e m is c o m p l e t e . + :r

We shall p r e s e n t a g e n e r a l r e s u l t r e g a r d i n g the solvability of the Cauchy p r o b l e m on the s e m i a x i s (0, (el. Dubinskii [24, 25]). Let 0

He(0, r

{u (t):u(s)(t)EL2(0, ~ ) , u ( 0 ) = 0 . . . . . ~('-~)(0)=0},s> t.

THEOREM 2.3. L e t f(t, ~0, . . . . ~s-1) be a continuous function, while for any function u(t) c o m p a c t l y s u p p o r t e d on the s e m i a x i s t h e r e is the inequality

~'f (t, u . . . . u (~-I)) u(~)dt >i --no ~l u(~)12dt-- K, o

o

0

w h e r e a 0 < t, K > 0 a r e constant. Then for any hmction h(t) e L2(0 , m) t h e r e e x i s t a function u(t) 6Hs(o~ ~) and a polynomial

~s-i (t) _~ Po+ pit + . . . + p,_Its-l such that u(s) + fit, u, . . . .

u(s-l)) = h(t) + :~s-~ (t).

E x a m p l e . u" + f(uq + g(n) = h(t), w h e r e f(~) is an a r b i t r a r y continuous function and g{$) is an a r b i t r a r y d i f f e r e n t i a b l e , n o n i n c r e a s i n g function. *The v e r i f i c a t i o n of the p r o p e r t i e s of the s p a c e s X, Y, and V and of the o p e r a t o r s A and B is obvious h e r e and below and is omitted.

481

2. A B o u n d a r y - V a l u e P r o b l e m on the S e m i a x i s . We c o n s i d e r the p r o b l e m

u"+g(u')+f(u)----h(t),

t>0,

u(0)=0. THEOREM 2.3. L e t the continuous functions f(~) and g(~) be such that /t

g(~)~<0,

If(~) d~> --K, 0

w h e r e u ~ R 1 is a r b i t r a r y and K > 0 is a constant. Then for any function hit) for which th(t) e L2(0 , oo) t h e r e 0 e x i s t s a function u(t) ~ H i and a constant p such that

u"+g(u')+f(u)=h(t)+#-',

t>0.

Proof. We s e t

A(u)~u"+g(u')+f(u), 0

Bu=--tu',

o 9

X = H I, V=Hcomp, Y ~ H - I +L2,1oo o

Obviously, for any function uit) e H 2c o m p

( A (u), Bu ) > 1[u II~x--K, I ( h (tJ, Bu ) [ ~ 0. F o r t h e s e e x a m p l e s all the conditions of T h e o r e m 2.3 a r e satisfied. 3. A B o u n d a r y - V a l u e P r o b l e m on the Semiaxis /Another Version). We c o n s i d e r the p r o b l e m

u"+f(a)=h(t),

t>O,

(2.7) (2.8)

u(O)=O, w h e r e f(u) is a positive homogeneous function of o r d e r q > 1.

0

THEOREM 2.4. F o r any function h(t) ~ L2,~,, 7 > 0 t h e r e exist a function u(t) E H~,~and a constant p such that

t~j. u " + f ( u ) = h ( t ) + p e X p t q + 2vq l A R e m a r k R e g a r d i n g Notation. If X is a s p a c e of a b s t r a c t functions u(t) on I c a l, then X T is the s p a c e of functions u(t) such that u(t) e x p { - T t } EX; h e r e T is a r e a l p a r a m e t e r . F o r e x a m p l e , if H s is the Sobolev s p a c e , then H~ is the S o b o l e v - S l o b o d e t s k i i s p a c e with weight e x p { - T t ) . Proof. We m a k e the change of v a r i a b l e u = v exp {--q2~_vlt} 9 We then a r r i v e at the p r o b l e m 47' ~ + e x p {27t} f (v).=h (t)exp {-- q2--~v 1 t}, A ( v ) - - v " + q4V--l v" q (q_02 v (o) = o. 02

0 i

With an eye to T h e o r e m 2.1, we s e t By = v' e x p { - 2 T t ~ , V = Hcomp, X = HT, Y = H~ 1 + L2,1oc. It is not hard to 0

s e e that for any function v(t) E H2omp (A(v), By) -> 3/I[ v IIX. M o r e o v e r , if h(t)~ L q+l, then -v q+I t II Hence, t h e r e e x i s t functions v(t) E I~I~,and a constant p such that

R e t u r n i n g to function u(t) and r e p l a c i n g 3'[(q + 1 ) / ( q - 1)1 by 3', we obtain the a s s e r t i o n of the t h e o r e m .

482

Remark. (ue-Tt),.

F o r q = 1 the t h e o r e m is a l s o true.

F o r the p r o o f it is n e c e s s a r y to s e t d i r e c t l y Bu = e - 7 t •

4. A B o u n d a r y - V a l u e P r o b l e m on a Finite I n t e r v a l . We c o n s i d e r the p r o b l e m

u" + f (u) = h (t), u(a)=u(b)=O, O
(2.9) (2.10)

oo,

w h e r e flu) is a p o s i t i v e - h o m o g e n e o u s f u n c t i o n of o r d e r q > 1. We note t h a t b e c a u s e of the i n v a r i a n c c of p r o b l e m (2.9), (2.10) with r e s p e c t to t r a n s l a t i o n it is p o s s i b l e to a s s u m e that a > 0 is s u f f i c i e n t l y l a r g e . We s e t u = vt ~ , w h e r e ~ = - l / ( q -

1). We then obtain

A (v) =--v" + 2~t-lv ' + ~ (~ -- 1) t-2v + t - i f (v) = h (t) t-% v (o) = O. o

We let Bv = - t v ' ,

V = H2(a, b), X = ~Ia(a, b), Y = H - l ( a , b). It is not h a r d to s e e t h a t

( A ( v ) , By> >ao[Iv[]~, a 0 > 0 . M o r e o v e r , if h(t) E L2(a, b), then I l -< K ' t l h IlL2 .ll v IIX, K > 0. T h u s , by T h e o r e m 2.1 t h e r e e x i s t s a 0

function v(t) E Hi(a, b) and a c o n s t a n t p s u c h that A (v) = h Ct) t - ~ + ,~t -~"

o r , w h a t is the s a m e ,

u"+f(u)=h(t)+pt-P,

p=q/(q--1).

We have thus e s t a b l i s h e d the following r e s u l t . 0 T H E O R E M 2.5. F o r any function h(t) E L2(a , b) p r o b l e m (2.9), (2.10) is s o l v a b l e in Hi(a, b) up to an a d d i tive t e r m p t - P . R e m a r k . It is obvious that the b o u n d a r y - v a l u e p r o b l e m (2.9), (2.10) c a n a l s o be studied by the s a m e t e c h n i q u e as the a n a l o g o u s p r o b l e m (2.7), (2.8) on the s e m i a x i s so that the r e s u l t s of P a r t 3 a r e a l s o valid f o r a finite i n t e r v a l . In o t h e r w o r d s , in the c a s e of a finite i n t e r v a l as well it is p o s s i b l e f o r Eq. (2.7) o r (2.9) to set [ E x a m p l e . u" • 3. for

Noncoercive Partial

q=h(t), Equations.

Differential

q+l

~' ~ -- ~ ' 1 "

q-> 1. Regularization

of Nonlinear

Problems

Equations

In this s e c t i o n s o m e i n c o r r e c t p r o b l e m s f o r p a r t i a l d i f f e r e n t i a l equations a r e studied on the b a s i s of the a b s t r a c t t h e o r e m of Sec. 2; t h e s e a r e equations of the t h r e e c l a s s i c a l t y p e s : elliptic, p a r a b o l i c , and h y p e r b o l i c . 1~ R e g u l a r i z a t i o n of the D i r i c h l e t P r o b l e m f o r Some N o n c o e r c i v e Q u a s i l i n e a r E l l i p t i c Equations. L e t G c R n be an a r b i t r a r y d o m a i n with b o u n d a r y F w h i c h s a t i s f i e s the w e l l - k n o w n cone condition* (it is p o s s i b l e that F = O, i.e., G = Rn). L e t F(~T), ~/ = ( ~ 1 . . . . . ~Tn), I TI = r , r -< m - 1 be a n o n n e g a t i v e , r e a l - v a l u e d function w h i c h s a t i s f i e s a h o m o g e n e i t y condition of o r d e r q + 1 (q > 0) : F(k~ 7) = kq+lF(~T), ~ > 0. We a s s u m e that F(~/) is d i f f e r e n t i a b l e with r e s p e c t to ~7" In the d o m a i n G w e c o n s i d e r the D i r i c h l e t p r o b l e m

A(u)~L(D)u--R(D)u=h(x), D~tt ! r = 0 ,

xEG,

]~l~<m--1,

*This c o n d i t i o n m e a n s t h a t to any point x E F it is p o s s i b l e to affix the v e r t e x of s o m e finite c o n e w h i c h lies e n t i r e l y in the d o m a i n G.

483

where

L (D) u =--~ (-- 1)I~ID~(a~Oeu)(i~,l, Ifll < m) is an elliptic o p e r a t o r of o r d e r 2m, aafl ~ R 1, aafl = aria. If the d o m a i n G is unbounded, then o p e r a t o r L(D)u is a s s u m e d to be positive definite; if the domain G is bounded, then it suffices that the o p e r a t o r L(D)u be lower semibounded. The o p e r a t o r R(D)u has the f o r m R(D)u~-(--I)" ~

D~Fg(D~u),

I~l=r

w h e r e Ffl(~T) = 0 F / ~ r. If the d o m a i n G is bounded, then the o p e r a t o r R(D)u is obviously the F r e e h e t derivative of the functional

ro (u)= ~~F (D~u) dx,

u (x)d:I~ (0),

G

under the condition that q < (n + 2(m - r ) ) / ( n - 2 ( m - r)) in the c a s e n > 2(m - r) [in the c a s e n --< 2 ( m - r) the n u m b e r q > 0 is a r b i t r a r y ] . THEOREM 3.1. Let G be a bounded domain, let h(x) E L2(G) , and suppose that the conditions on q just formulated a r e satisfied, Then f o r any n u m b e r ~ ~ ( n / 2 - m, n / ( q + 1) - r) t h e r e exist a function u(x) E 0

Hm(G) and a function p(x) E H-m(G) which is homogeneous of o r d e r ~ - n such that

L (D) tt-- R (D) u = h (x) + p (x). Proof. We set Bu = r(0u/0r) + ~u,

X = fir, (O), Y = H-r"(6),

V = / 4 " + ' (G), w h e r e r = Ix I.

We shall v e r i f y that the o p e r a t o r s A and B s a t i s f y the condition lira Ilullx = + ~'

(3.1)

if u E V, IIu IIx -* + ~o. Indeed, noting that ?l -

8u

r--= ~ Or

Ou

~_j x i ~I Oxi '

we find (after integration by parts) that

< A (u), 7 -o~ ~ ) = ( - - n + 2m) a ( u ) - - ( - - n + r (q + D)ro(u), where

2.a (u)~ ( L (D) u, u >. On the other hand, it is obvious that

( A ( u ) , tt ) = 2 a ( u ) - - ( q + l)ro(U). Hence

(A(u), Bu ) = [-- n + 2 (rn + x)l a (u)-b [n-- (r + x) (q -~. l )] ro (u), so that if q satisfies the conditions of the t h e o r e m , then

( A ( u ) , Bu > >~ao[a(u)-[-ro(tt)], since - n + 2(m + -~) > 0, n - (r + • since h(x) E L2(G) , it follows that

a0>0,

+ 1) > 0. This inequality obviously implies condition (3.1). M o r e o v e r ,

I
(3.2)

E x a m p l e . A(u) = Au + I u Iqu, q > 0. 0S

Suppose now that G is an unbounded d o m a i n in R n. We d e n o t e by H c o m p (s -> 0) the inductive l i m i t of the f a m i l y ~ s (GR) ' w h e r e G R = G N { x : Ix I -< R }, R --~o. Suppose f u r t h e r that W~+I(G) is the S o b o l e v s p a c e d e t e r m i n e d by the m e t r i c r~/(q +1) (u). T H E O R E M 3.2. L e t h(x) E L2(G) , xih(x) ~ L2(G) , i = 1 . . . . .

n. T h e n u n d e r the h y p o t h e s e s of T h e o r e m 3 . i

0

t h e r e e x i s t functions u(x) ~ Hm(G) N W~+t(G) , p(x) E H-m(G) + w~r(G) [p = (q + 1)/q] s u c h that (D) u - - R (D) u = h (x) + p (x). H e r e p(x) is a h o m o g e n e o u s function of o r d e r x - n. P r o o f . We s e t 0

0

,~ (G) . . .~/X ~ - t t m (G) Q Wq+l . . . . . r4m+t p, y = t - i - , n ( G ) + W ; r

(O),

B u - - r Oa + xtt. Or Further,

as in the p r o o f of T h e o r e m 3.1, we obtain the e s t i m a t e (3.2) and h e n c e the a s s e r t i o n of the t h e o r e m .

2 ~ . D i r i c h l e t P r o b l e m f o r a P a r a b o l i c Equation. L e t Q c R n+t be an a r b i t r a r y d o m a i n with b o u n d a r y S. In the d o m a i n Q we c o n s i d e r the " D i r i c h l e t p r o b l e m " f o r the p a r a b o l i c e q u a t i o n

Ou jr L (x, D) a + a (x) f (tt) = h (t, x), ot D~uls = 0 ,

(3.3)

(t, x) EQ,

itol~
(3.4)

where

L(x, D ) u - -

~.a

(--1)l~lD~(a~t~(x)D~u)'

a:,f~(x)=a~,(x),

is an elliptic, l o w e r s e m i b o u n d e d o p e r a t o r . It m a y be a s s u m e d with no l o s s of g e n e r a l i t y that L(x, D)u is p o s i t i v e definite. The function a(x) is bounded in the d o m a i n Q. F u r t h e r , let f(u) be a p o s i t i v e - h o m o g e n e o u s function of o r d e r q > 1 , w h e r e q < 2(n + m ) / ( n - m) if n > m. I f n -< m, t h e n q > 1 is a r b i t r a r y . Notation. L e t HI,s(Q) (l, s E 1~1) be the S o b o l e v - S l o b o d e t s l d i s p a c e of functions u(t, x) defined in the d o m a i n Q; H / # ( Q ) a r e the c o r r e s p o n d i n g s p a c e s with the w e i g h t e x p { - y t } , T E R t. 0

We denote by H~,m(Q) the s u b s p a c e of functions u(t, x) ~ H~,m(Q) s a t i s f y i n g the conditions (3.4). F i n a l l y , let G be the p r o j e c t i o n of the d o m a i n Q onto the s p a c e R n of the v a r i a b l e s x~ . . . . .

x n atong t.

0 T H E O R E M 3.3. L e t h(t, x) E L2,T(Q) , T > 0. T h e n t h e r e e x i s t functions u(t, x) 6 H~,m(Q) and p(x) s u c h that

. (.)_=

+,(.

o). +o(+)i(.)=+(,

H e r e p(x) ~ H-re(G) + Lp,loc(G) w h e r e p = 2(n + m ) / q ( n -

(+)| ox,

m) if n > m and p > 1 is a r b i t r a r y if n -< m.

P r o o f . We s e t 02s V=Heomp (s>max(m, 0,--m

Y=H v

B =oxp(_

n)),

0

X - - H t'mffr~

(Q) +La,lor (Q),

,)[.

w h e r e , a s u s u a l , the s y m b o l , d e n o t e s d i f f e r e n t i a t i o n with r e s p e c t to t. A c o m p u t a t i o n s h o w s that f o r u E V q--I

1

( A ( u ) , Bu ) = ( u" e-vt, u" e-~t ) + T ~--4-.f ( mue-v~, m u e - v t ) -- 2"~2~--~T ( ue-Vt, ue-vt ) , w h e r e Mu -- x/'-L-u. 485

Since ( u e - T t , u e - T t ) _ T -2 ( u , e - T t , u , e - T t ) , for q > 1 we obtain the c o e r c i v e n e s s inequality q--I ( A (u), Bu ) > ~ (7 ( Mue-vt, Mue-~ 't ) + ( u'e-vt, u'e-vt ) ) > ao J]u I]2x, a0 > 0. On the other hand, it is obvious that if h(t, x) E L2,T(Q) , then

sup t < h, 8u > t ~< It k [[ L2,,~q~"

Ilutlx

T h e o r e m 3.3 now follows f r o m the a b s t r a c t t h e o r e m 2.1. The proof of the t h e o r e m is c o m p l e t e . R e m a r k . S t r i c t l y speaking, the function p(x) is not univalent and a s s u m e s as m a n y values as the i n t e r s e c t i o n of the line x = c o n s t with the domain Q has connected components. E x a m p l e . L e t G c R n be a d o m a i n w i t h boundary r ; Q = [0, T] x G is a cylinder with l a t e r a l s u r f a c e S = [0, T] x F. If T = ~o, then we c o n s i d e r the f i r s t b o u n d a r y - v a l u e p r o b l e m in the cylinder:

Ou Ot t_L (x, D) u -{- a (x) f (u) = h (t, x), u(O, x).=O,

(3.5)

DOu]s-----O, [o~[.<.m--l.

(3.6)

If T < 0% then in the c y l i n d e r Q we c o n s i d e r the D i r i c h l e t p r o b l e m :

Ou Ot ~-L (x, D) u + a (x) f (u) = h ( t, x), u(0, x ) = 0 ( T , x ) = 0 , D~

(3.7)

[~o[..<m--l.

(3.8)

It follows f r o m T h e o r e m 3.3 that p r o b l e m s (3.5), (3.6) and (3.7), (3.8) a r e solvable up to a t e r m p ( x ) | exp { 2 T q t / ( q + 1)}, w h e r e p(x) E H-re(G) + Lp,loc(G). 3 ~ Another V e r s i o n of " A l m o s t - S o l v a b i l i t y " of a P a r a b o l i c Equation. We c o n s i d e r p r o b l e m (3.3), (3.4), w h e r e the function f(u) s a t i s f i e s the following conditions:

f ( k u ) ~ k qf (u), ~ > 0 ;

f (u) u > O,

(3.9)

w h e r e q > 1 is an a r b i t r a r y n u m b e r (the c a s e q = 1 is trivial). We introduce the following notation: W = {u(t, x) :u E H~?m(Q), f(u)u E LI,T(Q) }THEOREM 3.4. Suppose that f(u) s a t i s f i e s conditions (3.9), a(x) <- a 0 < 0, and the function h(t, x) E L2,T(Q). Then t h e r e e x i s t a function u(t, x) E W [u(t, x) s a t i s f i e s conditions (3.4)] and a function p(x) E H-m(G) + Lp(G) (p = (q + 1) / q ) such that o_u + L (x, D) u + a (x) f (u)----h (t, x) + p (x) exp {(7 + B) t}, dt w h e r e T > 0, 6 > 0 a r e a r b i t r a r y p r e s c r i b e d n u m b e r s connected by the inequality 5 < T(q - 1 ) / ( q + 1). Proof. We m a k e the change of v a r i a b l e u = ve-Tt, T > 0. We then obtain

A (v) ~- 00 ~t- + "fv = L (x, D) v + er Do,v l s = O ,

(x) f (v) = h (t, x) e-vt, teeth<m-- 1.

We now s e t O.

V = H 2, s r r Olllp x

n)),

0

X = {v (t, x): vEH 1, m (Q), f ( v ) v~L~, ~1-q~ (Q)},

I n t e g r a t i n g by p a r t s and noting that f ( v ) v = ( q +

1)~f(~)d~, we find that for v(t, x) E V 0

vevO-q)~ ) ( A ( v ) , B y ) = ( . ' , ~," ) - - ' ~q( a- ( 1x ) f ( ~ ) , + ~g ( V, V ) +~ ( MY, M Y ) + ( f (v),.vevO-q)t ) , w h e r e Mv = ,/-L--v.

486

(3.10) (3.11)

T h u s , f o r 5 < y(q - 1 ) / ( q + 1) we have the i n e q u a l i t y

( A ("o), B"o ) 2>/at ( ( .'o', ,'o" ) nL- ( M'o.., M"o ) -~- /., f (79), ,-oe;'(1-q)t ) ), al ~> 0, i . e . , the o p e r a t o r s A and B f o r m a " c o e r c i v e p a i r " on the s u b s p a c e V. M o r e o v e r , if h(t, x) E L2,T(Q), then o b v i o u s l y

I'(h, B y )I~ 0 is a constant. A p p l y i n g T h e o r e m 2.1, we obtain the s o l v a b i l i t y o f p r o b l e m (3.10), {3.11) up to the t e r m p(x) | e St. R e t u r n i n g to the f u n c t i o n u(t, x), w e a r r i v e at the a s s e r t i o n of the t h e o r e m . The p r o o f of the t h e o r e m is c o m p l e t e . E x a m p l e . A(u) =- 0 u / 0 t - AU -- I U Iqu, w h e r e q > 0 is any n u m b e r . 4 ~ D i r i c h l e t P r o b l e m for a H y p e r b o l i c Equation. In the d o m a i n Q we c o n s i d e r the " D i r i c h l e t p r o b l e m " f o r a h y p e r b o l i c e q u a t i o n (here the n o t a t i o n is the s a m e as in p a r t s 2 ~ and 3~): 02az0~--+ L (x, D) t~ + a (x) f (t~) ~ h (t, x), D~tt[~=O,

(3.12)

[wl -< m - - 1 .

i3.13)

The following r e s u l t holds. T H E O R E M 3.5. L e t f(u) be a p o s i t i v e - h o m o g e n e o u s function of o r d e r q > 1, w h e r e q < 2in + m ) / ( n - m) if n > m and q > 1 is a r b i t r a r y if n -< m. Then for any function hit, x) E L2,y(Q) (~ > 0) t h e r e e x i s t functions

0 u(t, x) E H~,m{Q) and p(x) E H-m(G) + Lp,loc(G) (p = 2(n + m ) / ( n -

c)~uOt ~ 'rL(x, D)u+a(x)f(u)=h(t,

m)) s u c h that

x ) + p (x) | exp [ q2u +I

t~J"

The p r o o f is s i m i l a r to that of T h e o r e m 3.3. We note the s p e c i a l c a s e w h e r e the d o m a i n G is bounded and q < (n + 3 m ) / ( n - m) f o r n > m. In this e a s e Lp,lociQ) c H - I , m ( Q ) , and we obtain the following r e s u l t . T H E O R E M 3.6. L e t 1 < q < in + 3 m ) / ( n - m) if n > m and let q > 1 be a r b i t r a r y if n -< m. T h e n for any 0 f u n c t i o n h(t, x) E L2,y(Q) t h e r e e x i s t functions u(t, x) ~ HI,m(Q) and p(x) E H-m(G) s u c h that 02u

27q

--..

- ~ - - ~ t. (x, D) u-}- a (x) f (u) = h (t, x) + p (x) | exp {-'~-4-f+ 1 t} (7 > 0). 5 ~ A n o t h e r V e r s i o n of R e g u l a r i z a t i o n of the D i r i c h l e t P r o b l e m f o r a H y p e r b o l i c Equation. L e t the d o m a i n Q be bounded. B e c a u s e o f the i n v a r i a n c e o f p r o b l e m {3.12), (3.13) u n d e r t r a n s l a t i o n it is p o s s i b l e to a s s u m e t h a t the d o m a i n Q is s u f f i c i e n t l y d i s t a n t f r o m the o r i g i n along the t axis. T H E O R E M 3.7. L e t f(u) be a p o s i t i v e - h o m o g e n e o u s function of o r d e r q > i, w h e r e q < 2(n + m ) / ( n - m ) if n > m-an-d q ~ i ~s a r b i t r a r y if n - m. T h e n p r o b l e m (3.12), i3.13) is s o l v a b l e up to the t e r m p(x) | t - q / ( q - 0 .

0

M o r e p r e c i s e l y , f o r any function h(t, x) ~ L2(Q) t h e r e e x i s t functions u(t, x) E HI,m(Q), p{x) E H-re(G) + Lp(G) [p = 2(n + m ) / q ( n - m) f o r n > m; p > 1 is a r b i t r a r y f o r n -< m] s u c h that

o~ ~_L (x, D) u + a (x) f (u) = h (t, x) + p (x) | t -q/(q-I). Ot 2 P r o o f . We m a k e the c h a n g e of v a r i a b l e u = vt ~ , ~ = - (q - 1)-1. P r o b l e m (3.12), (3.13) is then t r a n s f o r m e d into the p r o b l e m

A '(v)-" ~V ~ +2~t-1-~i-t +~(~--l)t-2v+L(x,D)v+t-~a(x)f(Q=f~(t,x)t -~,

D~

[~[~<m-1

We i n t r o d u c e the n o t a t i o n

0

0

V=H:'~ (Q)(s>max(m, ~)), X=HI'm(Q), Y=H-X'-~(Q)+L v (Q), Be=--t-~F.

487

It is not hard to see that for any function v(t, x) E V

< A (v), By ) > a o ][v [J~; f u r t h e r , if h(t, x) ~ L2(Q) , then

[ < h, By >i ~ o, 0

Thus, T h e o r e m 2.1 is applicable, i.e., there exist functions v(t, x) E HI,m(Q) and p(x) E H-m(G) + Lp(G) such that

A (v) = h (t, x) t - ~ + p (x) | t-', It r e m a i n s to r e t u r n to the function u(t, x). The proof of the t h e o r e m is complete. In conclusion we note the c a s e of a linear equation

O'u Ot2 t- L (x, D) tt = tt (t, x). Here ~ = 0 (or, what is the s a m e , q = co) and Bu = - t ( 0 u / 0 t ) . Example. A(u) -- ~2u/Ot2 - AU = h(t, x) + p(x) | t -~. CHAPTER STATIONARY

EQUATIONS

MONOTONICITY

WITH

METHOD.

II

STRONG

QUASILINEAR

NONLINEARITIES. ELLIPTIC

THE

PROBLEMS

Introduction In this c h a p t e r we study equations for which the method based on weak c o n v e r g e n c e is, s t r i c t l y speaking, inadequate for establishing their solvability. This is explained by t h e fact that the equations c o n s i d e r e d below have s t r o n g e r nonlinearities than the equations c o n s i d e r e d in the preceding chapter. The s i m p l e s t examples of such equations a r e equations of the f o r m ( - - 1 ) m ~ D~(ID~ulP-2O~u)=h(x), p > l . i c, l=,n

(0.1)

It is c l e a r that to prove the solvability of boundary-value p r o b l e m s for Eqs. (0.1) the minimal e s t i m a t e s the D i r i c h l e t integral - a r e inadequate. The basic additional concept which b e a r s fruit consists in using the fact that the c h a r a c t e r of the nonlinear e x p r e s s i o n s in Eqs. (0.1) is monotone. In view of this we f i r s t consider a c l a s s of o p e r a t o r equations p o s s e s s i n g monotonicity p r o p e r t i e s and establish the basic t h e o r e m s for the solvability of equations with monotone nonlinear o p e r a t o r s . We then c o n s i d e r nonlinear elliptic and parabolic equations. 1.

Basic

Theorems

from

the

Theory

of Monotone

Operators

In this section we establish the basic r e s u l t s of the theory of monotone o p e r a t o r equations - the s o l v a bility t h e o r e m s . We f i r s t c o n s i d e r monotone o p e r a t o r equations and then m o r e g e n e r a l equations with semibounded v a r i a tion. 1~ Solvability of Nonlinear Equations with a Monotone Operator. Let X be a s e p a r a b l e complex (or real) reflexive s p a c e , and let X* be the dual space, i.e., the space of continuous, linear funetionals on X. The value of a functional w E X* on an element u E X we denote by <w, u>. We c o n s i d e r the o p e r a t o r (in g e n e r a l , nonlinear) A(u) acting f r o m X to X*. Definition 1.1. The o p e r a t o r A is called monotone if for any u E X and v E X Re < A(u)--A(v), u - - v ) >~0. Definition 1.2. A monotone operator is called strictly monotone if'the equality Re < A (u)-- A (v), u - - v ) = 0 implies that u = v. 488

(LI)

D e f i n i t i o n 1.3. A n o p e r a t o r A : X - - X * i s c a l l e d s e m i c o n t i n u o u s i f i t i s b o u n d e d ( i . e . , i t t a k e s a b o u n d e d s e t i n X i n t o a b o u n d e d s e t in X*) and f o r a n y u E X , v E X

< A ( u - - ~ v ) , w > -+ < A(u), w ) , ~-~ + 0 , where w E X is arbitrary. T H E O R E M 1.1. L e t a m o n o t o n e o p e r a t o r s a t i s f y t h e f o l l o w i n g two c o n d i t i o n s : 1. ( C o e r c i v i t y of A). I f u E X and ]Iu IIx --" ~o, t h e n limRe
=+r

Then the mapping A:X-~X* is a surjective mapping, i.e., for any A( u ) : h

h a s a t l e a s t one s o l u t i o n u E X. P r o o f . L e t t h e s y s t e m v i E X (i = 1, 2 . . . . l i n e a r h u l l of t h e f i r s t k b a s i s e l e m e n t s v 1. . . . .

) c o n s t i t u t e a c o m p l e t e s y s t e m in X. W e d e n o t e by X k the v k. O b v i o u s l y , X k c X

and

~ Xk i s d e n s e in X. k=,

W e f i r s t find a n a p p r o x i m a t e s o l u t i o n u k E X k of the e q u a t i o n A(u) = h w h i c h s a t i s f i e s the r e l a t i o n s < A ( u k ) , v j > = , j = l

.....

k,

or, w h a t i s t h e s a m e , t h e r e l a t i o n ( A (uk), v ) = ( h, v >, vvEXk.

(1.2)

T h e s o l v a b i l i t y of Eq. (1.2) f o l l o w s f r o m the " a c u t e - a n g l e " l e m m a and the c o e r c i v e n e s s c o n d i t i o n . W e s h a l l o b t a i n a n a p r i o r i e s t i m a t e of the a p p r o x i m a t e s o l u t i o n s

II u~ [Ix ~
(1.3)

w h e r e C = C (11h IIX*) is a c o n s t a n t . I n d e e d , s e t t i n g v = u k i n (1.2), w e find t h a t Re < A (uk), uk '> = Re < h, ~z~ > -.< [[ h ]Ix* [] u,~ Jlx, w h e n (1.3) f o l l o w s f r o m t h e c o e r c i v e n e s s c o n d i t i o n f o r t h e o p e r a t o r A. I t f o l l o w s f r o m t h i s e s t i m a t e t h a t s o m e s u b s e q u e n c e u k c o n v e r g e s w e a k l y to u E X. W e s h a l l s h o w t h a t A(u) = h, i . e . , u i s t h e d e s i r e d s o l u t i o n . S i n c e A(u) i s a b o u n d e d o p e r a t o r , t h e n o r m s II A(uk)IIX* a r e bounded by s o m e c o n s t a n t . I t is thus p o s s i b l e to a s s u m e ( a f t e r c h o i c e of a s u b s e q u e n e e ) t h a t A(u k) ~ a E X* w e a k l y in X*. F r o m Eq. (1.2) w e i m m e d i a t e l y o b t a i n (as k ~ .o) a = h. W e now m a k e u s e of t h e m o n o t o n i c i t y c o n d i t i o n Re ( A ( u k ) - - A ( v ) , u k - - v ) >~0,

(1.4)

where v E X is an arbitrary element. N o t i n g (1.2), w e h a v e Re[ < h, u~ > - - < A ( u k ) , v >

< A(v), u ~ - - v > ] > 0 ;

whence as k-- ~ we deduce the inequality Re[]:~O, o r , w h a t i s the s a m e , Re ( h - - A ( v ) , u - - v > >~0. I n t h e l a s t e q u a t i o n w e s e t v = u - ~w, w h e r e w E X i s a r b i t r a r y and ~ ~ + 0 . By t h e s e m i c o n t i n u i t y of t h e o p e r a t o r A w e a r r i v e a t the i n e q u a l i t y

489

Re < h - - A ( u ) , ~ ) > 0 .

(1.5)

Since w is a r b i t r a r y , inequality (1.5) is not c o n t r a d i c t o r y only for h = A(u). Thus, the e l e m e n t u E X found is a solution of the original equation. The p r o o f of the t h e o r e m is complete. We note two useful a s s e r t i o n s . A s s e r t i o n i (the uniqueness t h e o r e m ) , I f the o p e r a t o r A is s t r i c t l y monotone, then the solution of the equation A(u) = h is unique. The a s s e r t i o n is obvious. A s s e r t i o n 2. Suppose that the o p e r a t o r A is semicontinuous and s a t i s f i e s a condition s t r o n g e r than monotonicity: Re < A(u)--A('o), u--v ) :~c(Ilu-~I]x)I[u--c3l]x, ~

(1.6)

w h e r e c(r) -> 0 is a continuous, nondecreasing function which is equal to zero only for r = 0; c(r) -~ + ~ as r - - + ~. Then the mapping A-1 :X*-+ X is continuous. Proof. We have the inequality c ([i u-- v Hx)][u -- v [Ix--<]]A (u)-- A (~)]Ix*"[lu-- v fix, i.e., [[u - v [IX -< e-1 ([[A(u) - A (v)fiX.). This inequality implies the continuity of the inverse operator A-i. We shall establish the existence of A-I on the basis of Theorem I.i. We verify the eoercivity condition. Setting v = 0 in (1.6), we find that Re Re rl u !Ix


II u Ifx

as [[u [Ix ~ + =- All the h y p o t h e s e s of T h e o r e m 1.1 a r e s a t i s f i e d . The p r o o f of the a s s e r t i o n is complete. COROLLARY 1.1. If the o p e r a t o r A : X - - X * is continuous and s a t i s f i e s condition (1.6), then the mapping A:X--+ X*

is a h o m e o m o r p h i s m . 2 ~ Solvability of Nonlinear Equations with Semibounded Variation. In applications the monotonicity c o n dition is frequently v e r y r e s t r i c t i v e . Roughly speaking, monotone o p e r a t o r equations " s e r v i c e ~ p a r t i a l d i f f e r e n t i a l equations c o n s i s t i n g only of a p r i n c i p a l p a r t . C o n s i d e r a t i o n of equations having l o w e r o r d e r t e r m s as well leads to a l o s s of the monotonicity p r o p e r t y (in general). In view of this we shall c o n s i d e r o p e r a t o r equations which s a t i s f y the much w e a k e r condition of semibounded v a r i a t i o n which e n c o m p a s s e s differential equations containing l o w e r o r d e r t e r m s . We c o n s i d e r the o p e r a t o r equation A(u)=h,

(1.7)

w h e r e the o p e r a t o r A(u) s a t i s f i e s the following condition (the condition of semibounded variation): F o r any e l e m e n t s u ~ X, v E X s u c h that tl u IIx -< R, tl v IIX -< R Re < A(u)--A(v), u--v >> --c(R, [[u--vii'), where []u [[' is a norm compact relative to [[u ]IX, and the continuous function e(R, p) -> 0 is such that c(R, }p) / } - - 0 as ~--+0 for anyR-> 0 a n d p -> 0. It is obvious that the condition of semibounded variation is broader than the condition of monotonicity. THEOREM 1.2. Let the operator A : X ~ X* be coercive, semicontinuous, and have semibounded variation. Then for any h EX* Eq. (1.7) is solvable. The proof proceeds just as the proof of Theorem 1.1. Moreover, up to inequality (1.4) it is word-forword the same. To prove the equation A(u) = h we now make use, in place of the monotonicitycondition, of the condition of semibounded variation

490

Re < A(uk)--A(v), u k - - v :> >1 --c(R, liuk--vlI'),

(1.8)

w h e r e II Ukl[X - C [C > 0 i s t h e c o n s t a n t i n t h e r i g h t s i d e of the a p r i o r i e s t i m a t e (1.3)]. N o t i n g (1.2), w e h a v e R e [ < h, uk ) - - < A(uk),~ > - - ( A ( , ) , u k - - v > ] > - - c (R, [[uk--~lt'). C o n s i d e r i n g t h e c o m p a c t n e s s o f the n o r m II u ]l', w e o b t a i n f r o m t h i s a s k Re[ --

-- l>--c(R,

oo

ilu--vll'),

or, what is the same, Re < h - - A (*), u - - v ) > - - c (R, I[ u - - v [['). I n t h e l a s t e q u a t i o n w e s e t v = u - }w, w h e r e w ~ X l]w [Ix <- R , ~ ~ + 0. By t h e s e m i c o n t i n u i t y of t h e o p e r a t o r A and p r o p e r t i e s of the f u n c t i o n c ( R , p) w e a r r i v e a t t h e i n e q u a l i t y Re < h - - A ( u ) , w > > 0 .

(1.9)

S i n c e R >- C (and h e n c e a l s o w ~ X) i s a r b i t r a r y , i n e q u a l i t y (1.9) i s not c o n t r a d i c t o r y o n l y f o r A(u) = h. T h u s , u E X i s a s o l u t i o n of the o r i g i n a l e q u a t i o n . T h e p r o o f of t h e t h e o r e m is c o m p l e t e . 3 ~ S t r o n g C o n v e r g e n c e o f the G a l e r k i n A p p r o x i m a t i o n s . I n t h e p r o o f of t h e t h e o r e m s o f P a r t s 1 ~ and 2 ~ w e m a d e u s e o f t h e w e a k c o n v e r g e n c e of s e q u e n t i a l a p p r o x i m a t i o n s u k to t h e s o l u t i o n u. W e s h a l l show t h a t i f c o n d i t i o n (1.6) i s s a t i s f i e d ,

Re < A (u)--A (v), u - - v > >1c (II u-v[Ix)'il u - v [Ix ,

(1.6)

t h e n s t r o n g c o n v e r g e n c e u k - - u a c t u a l l y t a k e s p l a c e i n t h e s p a c e X. In t h e c a s e o f a c o n t i n u o u s o p e r a t o r A(u) t h i s s i t u a t i o n c o n s i d e r a b l y s i m p l i f i e s t h e theory,, s i n c e i n t h i s case obviously

A(uh)-+A(u) in t h e s p a c e X*. W e s h a l l e s t a b l i s h the f o l l o w i n g t h e o r e m . T H E O R E M 1.3. L e t c o n d i t i o n (1.6) be s a t i s f i e d [this c o n d i t i o n is n a t u r a l l y c a l l e d the c o n d i t i o n of d e f i n i t e n e s s of t h e v a r i a t i o n of t h e o p e r a t o r A(u)]. T h e n the G a l e r k i n a p p r o x i m a t i o n s u k (k = 1, 2, . . . ) c o n v e r g e s t r o n g l y i n X. Proof.

The a p p r o x i m a t i o n s Uk s a t i s f y t h e r e l a t i o n < A (u,), ~ > = < h, v >,

(1.2)

where v EXk is arbitrary. In particular, for v = uk we have

< A(uk), Uk > = < h, uk > 9

(1.10)

L e t u be t h e w e a k l i m i t e l e m e n t o f t h e s e q u e n c e u k . BY c o n d i t i o n (1.6) w e h a v e

c (![ u - - uki]x)" [! u - - ukl[x ~ Re < A ( u ) - - A (uk), a - - uk >, w h e n c e , u s i n g (1.10), w e o b t a i n the i n e q u a l i t y c ([[ u--uk[[x)" [[ u--uk]Jx ~
) - - R e ( A (uk), u > + Re ( h, uk ) .

W e l e t k - - + oo n o t i n g t h a t u k --- u w e a k l y i n X , w h i l e A(u k) --- a w e a k l y in X*. F r o m t h e l a s t i n e q u a l i t y w e o b t a i n l i m c (l] u--us[Ix)][ u--u~[[x= Re < - - a + h , u > =0.

k*20

T h i s m e a n s t h a t Hu - Ukllx - - 0, w h i c h we w e r e r e q u i r e d to p r o v e . T h e p r o o f of the t h e o r e m i s c o m p l e t e . R e m a r k 1.1. D u r i n g the p r o o f w e had to r e p e a t e d l y c h o o s e s u b s e q u e n c e s of the m a i n s e q u e n c e u k. Now (a p o s t e r i o r i ) i t i s c l e a r t h a t i f t h e u n i q u e n e s s c o n d i t i o n s a r e s a t i s f i e d t h e e n t i r e s e q u e n c e u k c o n v e r g e s t o the s o l u t i o n u 6 X. I n d e e d , o t h e r w i s e i t w o u l d follow f r o m t h e a p r i o r i e s t i m a t e (1.3) t h a t t h e s e q u e n c e u k had a t l e a s t two d i s t i n c t w e a k l i m i t p o i n t s . E a c h of t h e s e , by w h a t h a s b e e n p r o v e d , m u s t be a s o l u t i o n of the e q u a t i o n A(u) = h, w h i c h i s i m p o s s i b l e by t h e u n i q u e n e s s a s s u m p t i o n . T h i s s i t u a t i o n is e x t r e m e l y i m p o r t a n t f o r numerical computations.

491

2. Quasilinear

Strongly

Elliptic

Equations

In this section we c o n s i d e r q u a s i l i n e a r elliptic equations having the "ideal" p r o p e r t y of sSrong positivity of the v a r i a t i o n - s t r o n g l y elliptic equations. On the basis of the r e s u l t s of Sec. 1, we establish the c o r r e c t ness of the D i r i c h l e t problem. An algebraic c r i t e r i o n of strong ellipticity is given. 1% H o m e o m o r p h i s m T h e o r e m . Let G be a bounded domain of n-dimensional Euclidean space with bounda r y F. In the domain G we c o n s i d e r the f i r s t b o u n d a r y - v a l u e p r o b l e m for the q u a s i l i n e a r elliptic equation of order 2 m:

L (x, D) (tt)-~ ~.a (-- 1)l~lD~A,z(x, Dvtt)=h (x), [ 7[~< m,

(2.1)

l~zl-<:m

D~ttlr=f~(x'), D~I...

x'Er,

[~[~m--1.

(2.2)

H e r e ~ = (~l, 9 9 9 ~n) (as usual) is an integral differentiation multiindex, I~t = ~1 + , 9 9 + ~n, DC~ = Dn~n, D i = 0/Oxi; the s a m e holds for 3' and w.

The functions Ac~(x, ~3"), w h e r e ~3' = (~3`~. . . . indices ~ / f o r t3`t --- m.

, ~,/n ), a r e in general nonlinear and depend on all possible

We a s s u m e that the following conditions a r e satisfied: I. The functions Ac~(x, ~3`) a r e defined for x E G and any ~-/and a r e continuous i n x and ~T. M o r e o v e r , \[yl~<m

w h e r e p > I and K > 0 a r e constants. II. The definiteness condition (strong ellipticity). F o r any functions u(x) E w~n(G) and v(x) ~ w~n(G) such ~ (G) the following inequality holds: that u (x) - v{x) E Wp Re ( L (x, D) (u)-- L (x, D) (v), tt -- v ) a -~ ~ Re ( As(x, Dvu)-- A~(x, Dvv), D~(tt-- v) ) o >1altl u - - v I1s lal,~m

w h e r e a s > 0 is a constant. [Here and below IIu tlm,p denotes the n o r m in the space W~a(G).] III. Regarding the boundary conditions (2.2) we a s s u m e that they admit an extension to the i n t e r i o r of the domain G f r o m the space W ~ ( G ) , i.e., t h e r e exists a function f(x) E W~n(G) sueh that DWf(x)lr = fw(xq, x' E F. Finally, we a s s u m e that h(x) E Wpm(G), p' = p / ( p - 1). 0

We denote by W~n(f) the set of functions of the f o r m u(x) = f(x) + z(x), w h e r e z(x) EW~n(G); the m e t r i c in w~n(f) is by definition the s a m e as in w~n(G). The following r e s u l t holds which is basic in this seetion. THEOREM 2.1 ( H o m e o m o r p h i s m T h e o r e m ) . Let conditions I-III be satisfied. Then the mapping L (x, D) (it): W~' (f) ~ W~','n CO)

(*)

is a h o m e o m o r p h i s m . H e r e

II u - v I!~C~ < K IIZ (x, D) (tt)-- L (x, D) (v)ll-m, p,, w h e r e K > 0 is a constant and II. II_m,p, is the n o r m in the s p a c e Wpm(G). Proof. The d i r e c t a s s e r t i o n of the t h e o r e m is obvious, since continuity of the o p e r a t o r L(x, D)(u) u n d e r condition I is well known (eL, e.g., K r a s n o s e l ' s k i i [32],Sec. 2). Thus, to complete the p r o o f of the t h e o r e m it suffices to establish the unique solvability of p r o b l e m (2.1), (2.2) and the continuous dependence of the solution u(x) E W~(f) on the r i g h t side h(x) E w~m(G). This will be accomplished on the basis of the general o p e r a t o r t h e o r e m 1.1. F i r s t of at1, however, we prove a l e m m a .

0

LEMMA 2.1. If conditions I and II a r e satisfied, then for any function u(x) E W~n(G), u(x) - f(x) E W~n(G) we have the inequality Re ( L (x, D) (u), tt ) o ~ ' ~ Re ( A~ Cx, Dvtt), D~tt ) q ~- a2 IIu It~, ~,-- K, lc~l~
w h e r e a 2 > 0 and K > 0 a r e constants.

492

(2.3)

Proof. In the c a s e w h e r e A~(x, 0) - 0 for all I ~l - m the a s s e r t i o n of the l e m m a follows immediately f r o m condition II if we set v(x) - 0; h e r e a 2 = al, K = 0. We c o n s i d e r the c a s e w h e r e the functions Ac~ (x, 0i do not vanish identically. F r o m II we then find that

Re'~ a,l j a[l~,p.

(2.4)

lctl
By condition I we have the inequality I Ac~ (x, 0)! --< K; hence

I< Ac,(x, O),D~tt >aI~K ( 1, !D~tt[ >~, whence, using Young's inequality, we find that a

,

l

Choosing e > 0 sufficiently s m a l l , f r o m inequality (2.4) we immediately obtain the inequality of the l e m ma. The proof of the l e m m a is complete. We now turn to the solvability of p r o b l e m (2.1), (2.2). With an eye to T h e o r e m 1.1, we make the change 0

of variable u(x) = f(x) + z(x), w h e r e z(x) ~w~n(G) is the d e s i r e d function, and we set A (z) ~ L (x, 0) ( f q- z). Obviously, the mapping (.) is equivalent to the mapping 0

A (z): W~ (0)-~ W~,~ (C). It is not hard to see all the hypotheses of T h e o r e m 1.1 a r e satisfied, and the p r o p e r t i e s of semicontinuity and s t r i c t monotonicity a r e obvious. Indeed, by condition I the o p e r a t o r A(z) is continuous and. hence s e m i c o n tinuous. Condition II i m m e d i a t e l y implies the definiteness of the variation of the o p e r a t o r A(z): Re < A ( z ) A (w), z -- w ) ~= Re ( L (x, D) ( f + z) -- Z (x, D) 0 e+ w), ( f + z) -- ( f + w) > a > ch [IZ-- w lIs p and hence its s t r i c t monotonieity. Thus, only the c o e r c i v e p r o p e r t y of the o p e r a t o r A(z) r e q u i r e s verification, and this is obviously i m plied by the following a s s e r t i o n . 0 A s s e r t i o n 1. F o r any function z(x) E w~n(G) we have the inequality (2.5) w h e r e a 3 > 0 and K > 0 a r e constants. P r o o L Indeed, Re < A (z), z ) ~ Re < L (x, D) ( f q- z), f q-z } ~ " R e ( L (x, D) ( f q- z), f > a, whence by L e m m a 2.1

Re < A (z), z > > a~ll f + z l!~ ~--K --Re< L (x, D) (f + z), f > G.

(2.6)

To e s t i m a t e the t e r m Re
[Gl~
-----K ~

.~ Iov(y+z)[~'-'+l, lD~'Yl >o < "~ 17l
~+K ~

ltzl,l~[m

a.

l~]<m

Since f(x) 6 W~n(G), again applying Young's inequality, we deduce that

I Gl~<~llf+zllS,p+c(~)(!lfll~.,,+l), which together with inequality (2.6) leads to the d e s i r e d inequality (2.5) if e > 0 is sufficiently small. The a s s e r t i o n and hence the e o e r e i v i t y of A(z) has thus been established. 0

Applying now T h e o r e m L 1 , we see that the mapping A(z) :w~n(G) ~ W p m ( G ) is surjective; t h e r e f o r e , for any function h(x) 6 Wpm(G) the original p r o b l e m is uniquely solvable in the "space" w~n(f). 493

It r e m a i n s to show that the i n v e r s e o p e r a t o r thus found is continuous. This a s s e r t i o n follows immediately f r o m condition H and A s s e r t i o n 2 of the preceding section. H e r e the modulus of continuity of the i n v e r s e o p e r a t o r is c h a r a c t e r i z e d by the inequality

II.LT' (/0.~ L'~,(D 11s --< K II h - g .11-=.r w h e r e K > 0 is a constant. This is what was r e q u i r e d to prove. The p r o o f of the t h e o r e m is complete. R e m a r k 2.1. We o b s e r v e that under the conditions of the t h e o r e m just proved the sequence of Galerldn approximations Uk(X) (and not m e r e l y s o m e subsequence of it) c o n v e r g e s to the solution u(x) strongly in w~n(G) (see the concluding r e m a r k 1.1 of Sec. D: 2 ~ An A l g e b r a i c C r i t e r i o n for Strong Ellipticity. In the p r e c e d i n g s e c t i o n the condition of strong ellipticity of the o p e r a t o r L(x, D)(u) was r e p r e s e n t e d in functional form. We shall now give sufficient algebraic conditions under which we have the inequality

Re ( L(x, D)(u)--L(x, D)(v), u--v ) o >aoll u-vllg.v

o.>o.

(2.7)

The following t h e o r e m holds. THEOREM 2.2. Suppose that p >- 2 and the functions Ac~ (x, ~T) a r e differentiable with r e s p e c t to ~3'" Suppose f u r t h e r that for any $ and ~1 t h e r e is the algebraic inequality

A~,(x,r

Re ~

tr

Icq ,ll~l~m

-~, a~>o,

I~l=m

w h e r e Aa~(x, ~T) = 8Acz(x, ~ T ) / ~ f l . Then the o p e r a t o r L(x, D)(u) is strongly elliptic, i.e., inequality (2.7) is satisfied. Proof. We have 1

Re ~

< A~ (x, D~u) -- A~ (x, DVv), D = (u -- v) ) o= Re __~

I~l<:m

< f Aal~(x,

lal ,II~t<:m

DVv+ : (DVu-- DVv)) d'~Da (u-- v),

0

< f I D~z, + ~ (DVu.-- DYe.,) I .-2d~D ~ ( u - - r

D ~ (u-- v) ) o > % ~ lcq=m

(2.8)

0

D ~ ( u - v) ) ~.

E l e m e n t a r y computations show that for any a and b I

I [a T.cb [p-2d~>~a4[b IP-~, o w h e r e a 4 > 0 is a c o n s t a n t . Using this inequality, f r o m (2.8) we find that Re ~ I~l~:"*

< A~(x,

OVu)--A~(x, DVv),D~(u--v) ) o~as ~,~ ( ]D~(u--v)[ p-'' [O~(u--~v)[ ) o>ao]lu:'-~[[~.p 9 lal=m

This inequality obviously coincides with inequality (2.7). The proof of the t h e o r e m is complete. Example. L e t

L(x,D)(u)~ ~ (--1)mOC'([O~uin-'D~u)r Icq~gra

w h e r e p > 1 is a fixed number. All the conditions of the p r e s e n t s e c t i o n a r e satisfied f o r the o p e r a t o r L(x,D)(u). 3 ~ P e r i o d i c Solutions. In the same way as the D i r i c h l e t p r o b l e m for the equation

L(x,

D)(tt)~ ~

(--l)tC'tD~Av,(x,D~tt)=h(x), lT[~
(2.9)

l**l~m

it is possible to study the p r o b l e m of finding 27r-periodic solutions. Namely, i t is r e q u i r e d to find a solution u(x) of Eq. (2.9) in the Sobolev space W~a(Tn), w h e r e T n is the n-dimensional torus (T n = S 1 x . . . x S ~, S 1 is the unit c i r c l e ) .

494

We a s s u m e that conditions I and II of part 1~ are satisfied [of course, the condition of definiteness of the variation of the operator L(x, D) (u) is now postulated for functions u(x), v(x) of the space W~n(Tn)]. THEOREM 2.3. Under conditions I and II the mapping L (x, D) (u): W~ (T")-+ W~;~ (r") is a homeomorphism. The proof of this t h e o r e m is exactly the same as that of T h e o r e m 2. I. 4 ~ Clarkson~s Inequalities in the Space Wpm(G). As an application of the preceding r e s u l t s we establish that Clarkson's inequalities a r e valid in the spaces Wpm(G). As is known, Clarkson's inequalities imply the uniform convexity of the space Wpm(G). The F i r s t Clarkson Inequality. Let p~ _> 2. Then for any h~ ~ W~tn(G) and h 2 ~ WpPa(G)

0

0 Proof. It follows f r o m the example presented in part 2 ~ that there exist functions u~ ~ W~(G) and u 2 w~n(G) such that

~

i~'~---- ~] (--I)1~tD ~ (ID~tt, 1~D~u,) h~- ~

( - 1)~ID~ (D~u0,

I~.l,~m

l~l<m

(--1)laIDC~(ID~u ~[~"~D~u~)-

~ (-- I)I~'tD~(D~u~).

Here it is e a s y to see that

lI,~, 1!_,,,,,,, = II u, I1~_~, ~ = t, 2.

(2.10)

We have further [ ha _+.h2 _ ~ p, = s u p "

( ha + h:, v ) ]

I~ lJm,p<:l 1

livllm,p<

\]~l<m 0 !

I]~llm,p<:l

!

\tt~l<m e

{we have h e r e f i r s t used the H61der inequality for integrals and then the HSlder inequality for sums). F r o m (2.11) we obtain

II " - - 2 - - - 1 _ ~ , r

11_..,. o o(11

2

l: + t llO, p'

2

L.)pV

Applyiang to the right side of the inequality obtained the classical C1arkson inequality II(u + v ) / 2 II0,pT + II( u v)/2 IiP,pv -< 1/2(11 u II0P,Tp, + IIv UP,p,) in the space Lp,, we find that

+ I1r (o~u=)llgY p,) =- 21--~ (!10,~u, llg. ,,,+ tl O~ue Itg,p) =-- ~ (t1 u~ i!,~. ,,,+ il u.~IlL ,,){~t<m Recalling formula (2.10), we obtain the d e s i r e d inequality. The Second Clarkson Inequality. Let 1 -< p' --< 2. Then for any h~ E wpm(G) and h 2 E wpm(G)

The proof is s i m i l a r to the preceding proof using the second Clarkson inequality in the space Lp,.

495

3. Quasilinear

Elliptic

Equations

with

Semibounded

Variation

In this section we continue the study of the D i r i c h l e t p r o b l e m for quasilinear elliptic equations having d i v e r g e n c e form. On the basis of T h e o r e m 1.2, we establish the solvability of the p r o b l e m indicated under v e r y broad conditions (the conditions of c o e r c i v i t y and semiboundedness of the variation), w e f u r t h e r give a c r i t e r i o n for the Semiboundedness of the variation and also examples and c o u n t e r e x a m p l e s . 1~ Solvability of the D i r i c h l e t ]Problem. In a domain G with boundary F we again c o n s i d e r the p r o b l e m

,I. (x, D)(u) =--~ (-- 1)'~D=A~ (x, Ova)= h (x), [ 7 [ ~
(3.1)

D ~ l r = f O ( x ' ), x ' O ~, [o~l~oz-- 1.

(3.2)

We a s s u m e that the following conditions a r e satisfied: I. The functions Ao~{x, ~ ) a r e defined for x e G and all 4~/, a r e continuous in x and 43~, and satisfy the inequality

[A,~(x,~v)l~<X( ~

t~,lP-'q-1), 1,14m,

w h e r e p > 1 and K > 0 a r e constants. II. C o e r e i v i t y condition. F o r any function u(x) E W~n(f) we have Re ( L(x, D) (u), u >G - a0 IIu IlP p - K, a0 >0, K >0. III. Condition of semiboundedness of the variation of L(x, D)(u). F o r any functions u(x) e w~n(f) and v(x) W~a(f} in the ball I1u llm, p -< R ~ > 0) we have the inequality Re Q> --c (R, [! u--v I[.,-,, v), w h e r e e(R, p) -> O is a continuous function of R and p such that c ~ , 40)/4 - - 0 (4 - - +O) for any R and p. We note, in p a r t i c u l a r , that a monotone o p e r a t o r L(x, D)(u) obviously satisfies condition III. THEOREM 3.1. If conditions I-III a r e satisfied, then for any right side h(x) ~ W p m ( G ) p r o b l e m (3.1), (3.2) has at l e a s t one solution u(x) E W~a(f). The proof of the a s s e r t i o n follows f r o m the o p e r a t o r t h e o r e m 1,2, Indeed, as before we set u(x) = f(x) + z(x) and

A (z) = L (x, D)(f+ z), w h e r e f(x) E W~n(G) is an extension of the boundary conditions (3.2). We have 0

A (z) : W~ (Q)-+W-~, m (O).

(3.3)

It is not hard to s e e that conditions I-III e n s u r e that the conditions of T h e o r e m 1.2 a r e satisfied: s e m i continuity, c o e r c i v i t y (cL Sec. 2), and semiboundedness of the v a r i a t i o n of the o p e r a t o r A(z). The mapping (3.3) is t h e r e f o r e surjective. This means that the p r o b l e m (3.1), (3.2) is solvable for any functions h(x) E Wpm(G). The proof of the t h e o r e m is complete. 2 ~ An Algebraic C r i t e r i o n of Semiboundednes s. We consider an o p e r a t o r of the f o r m

L (x, D)(u)=--~ (--1)mO~ ( A~ (x, O~u) + B~ (x, Dvu)) + ~_~(--1)I~IDST6 (x, D~u), [[31=m, 171~<m, [~[..<m, l~[=ra

16t,r

w h e r e the functions An(x, 4/3), Ba( x, ~y), Ts(x, 4z) a r e continuous in all arguments. THEOREM 3.2. Let t h e following conditions be satisfied:

a) I A~ (x, E~)I ~ K G ~t__~I E~Ir' + I ), p >~2; b) the o p e r a t o r Lo(x, D) (u) -----(--1) m ' ~ D~A~( x, Dt~u) is strongly elliptie; I~l=m

c) the functions B~ (x, ~ ) ,

496

Ts(x, 4~-) a r e differentiable in 4qr 4r, b.nd

I OB~(x,~v)

0~,t or6(x, ~)

]~] B~zv(x' ~v)'~
Then the o p e r a t o r L(x, D)(u) s a t i s f i e s the conditions of T h e o r e m 3.1. Proof. Condition I obviously coincides with condition a). We shall v e r i f y condition II. Because of the s t r o n g ellipticity of the o p e r a t o r L0(x, D)(u) and L e m m a 2.1 we have the inequality Re ( Lo (x, D) (u), u ) o ~ Re

~.~ ( A~ (x, D~u),D~tt ) o > al []u ][~,p-- K,

(3.4)*

lal=m

w h e r e a 1 > 0, K >0 a r e constants. F r o m this for the o p e r a t o r L(x, D) (u) we obtain the inequality Re ( L (x, D) (u), tz ) o :> at I1u [[Pm,p--K

-.}-Re ~.~ ( B~ (x, DVu), D~tt ) o + Re ~_j ( T~ (x, D~u), D6u ) G. l~lffim

lS]<:m--1

By condition c) e a c h t e r m under the s u m m a t i o n sign has d e g r e e of nonlinearity in DYu no higher than q + 1 < p. F r o m this, on applying Young's inequality with exponent p / ( q + 1), we find that Re (

L(x, D)(a).u ) o >ao[] ttHg,p--K,

w h e r e a 0T > 0, K >0. This is what we w e r e r e q u i r e d to show. To c h e c k condition III we c o n s i d e r the d i f f e r e n c e (L(x, D)(u) - L(x, D)(v), u - v) G. By condition b) Re ( L (x, D) (u)-- L (x, D) (v), u - - v ) o > a0 [Iu - - v [[~.p + Re

[~

k I~I=m

+ Z

{ B~ (x, DVu)-- B~ (x, OVv), D ~ (u-- v) ) G

( Y~ (x, D~tt)-- T6 (x, D~v), D ~ (u-- v) ) o] ~- ao [I u - - v I[~,. + Re [/'~ + I21.

(3.4)*

J

I~,i<:m- I

Further, 1

I~l~m,l~'l<:rn 0

leq=m, I Y I<m

By condition c) the functions baT(u, v) have growth in u and v not exceeding p - 2; t h e r e f o r e , the functions bay(u, v) E Lp/(p_2)(G) provided that u(x) E w~n(G) and v(x) ~ W~n(G). M o r e o v e r , the bay(U , v) f o r m a bounded s e t in Lp/(p_2) (G) provided that I[u [[m,p -< R, IIv Hm, p -< R. Using now the Young and H~lder inequalities with suitable exponents, we find that P

12 is e s t i m a t e d in the s a m e way. In s u m m a r y , for sufficiently s m a l l e > 0 we obtain f r o m (3.4) the inequality Re (

Z(x, D)(tt)--t (x, D)(v), u--,v ) o >,gllu'~'ll~,p-~@,

P

-~)1[u-vllF~-Lp, ,g>o,

which obviously e n s u r e s the validity of condition III. The p r o o f of the t h e o r e m is complete. 3 ~ E x a m p l e s and C o u n t e r e x a m p l e s . We c o n s i d e r the differential o p e r a t o r

L(x'D)(tt)=( -1)~' X D~([ D~tt[~D~u+]D~-~ttlqD~u) + ~,~ (~-l)'~tOta( 1~ a~o(x)]O~u!P~, llIl4m

l~l=m

p>2,

q
\lol<m-1

]

~.~ p,o~
All the conditions of T h e o r e m 3.2 a r e s a t i s f i e d for the o p e r a t o r L(x, D) (u). R e m a r k 3.1. We p r e s e n t an e x a m p l e which shows that under the hypotheses of T h e o r e m 3.2 the solution of p r o b l e m (3.1), (3.2) m a y be nonunique even in an a r b i t r a r i l y s m a l l domain (in c o n t r a s t to the l i n e a r case). *Two Eqs. (3.4)

denoted in Russian o r i g i n a l - P u b l i s h e r .

497

We c o n s i d e r a f i r s t b o u n d a r y - v a l u e p r o b l e m of the f o r m

a It-----0.

(3.6)

We s e t w{x) = r S s i n ( l / r ) , r 2 = x~ + . . . + X2n. Then p r o b l e m (3.5), (3.6) obviously has in any d i s k i r I (k~) - t (k = 1, 2 , . . . ) at l e a s t two solutions: u(x) - 0 and u{x) = w(xL R e m a r k 3.2. We p r e s e n t an e x a m p l e which shows that for a n a r b i t r a r i l y s m o o t h r i g h t side h{x) the s o l u tion u(x) of p r o b l e m (2.1), (2.2) [and a l s o of p r o b l e m (3.1), (3.2)] is not smooth. Indeed, let h(x) ~ C~(G) and be o f constant sign, i . e . , e i t h e r h(x) > 0 in G or h{x) < 0 in G. Then the p r o b l e m r$

L(x,D)(u)---- _ ~

~ (l~176 ~176 =h(x), ~ XZ,

ulr=O@>2)

(3.7)

cannot have a twice d i f f e r e n t i a b l e solution. Indeed, if u(x) 6 C2(G), ul r = 0, then inside the d o m a i n G t h e r e is at at l e a s t one e x t r e m a l point x 0. At this point, obviously, L (x, D) (u) = 0, which c o n t r a d i c t s (3.7). R e m a r k 3.3. The p r e c e d i n g r e s u l t s a r e valid f o r quasi=linear elliptic equations with d e g r e e s of n o n l i n e a r i t y which a r e d i f f e r e n t f o r d i f f e r e n t d e r i v a t i v e s . The o r d e r s of the d e r i v a t i v e s m a y a l s o be different. Since no s p e c i a l f e a t u r e s a r i s e in t h i s c a s e , we r e s t r i c t o u r s e l v e s to an e x a m p l e . The o p e r a t o r

L(x,

o~

m a p s WIn(G) onto

D)(u)=_~.~ (--I)I=WD*'(ID~'uIP~-~D~ p ~ > l , la~<m

W

(G) h o m e o m o r p h i c a l l y , w h e r e ~Vm(G) is the c l o s u r e of c o m p a c t l y s u p p o r t e d functions on P the d o m a i n G in the n o r m

Ifutr~.-; = ~ 9

IID%IE,~ -

lal<#z

CHAPTER MONOTONICITY

METHOD

QUASILINEAR

III

IN THE

PARABOLIC

EVOLUTION

CASE.

EQUATIONS

Introduction In this c h a p t e r we develop the r e s u l t s obtained in Chap. II w h i c h a r e a p p l i c a b l e to nonlinear parabolic equations both in a b s t r a c t o p e r a t o r f o r m and in the f o r m of p a r t i a l d i f f e r e n t i a l equations. The s i m p l e s t e x a m p l e s of the l a t t e r equations a r e q u a s i l i n e a r equations of the f o r m

~Oa+ ( - - t )

,n

Z D ~ (]D~ulP-ZD~u)=h(t, x) l~l=m

(0.1)

or essentially nonlinear equations, e.g., Ou

-2

--

(t, x ) ,

p>l.

(0.2)

To p r o v e the solvability of b o u n d a r y - v a l u e p r o b l e m s f o r Eqs. (0.1), (0.2) we use the method of monotonicity. 1. N o n l i n e a r

Parabolic

Equations

with

a Monotone

Operator

In this s e c t i o n we e s t a b l i s h the unique solvability of the Cauchy p r o b l e m for the d i f f e r e n t i a l - o p e r a t o r equation + A (0 (u)----h (t), w h e r e the o p e r a t o r A(t)(u) is monotone.

498

The method of monotonicity for p a r a b o l i c p r o b l e m s is applied for the proof. We note, in p a r t i c u l a r , L e m m a 1.1 on the solvability of a nonlinear s y s t e m of o r d i n a r y differential equations which in subsequent c o n s i d e r a t i o n s plays a r o l e analogous to the r o l e of the " a c u t e - a n g l e " l e m m a (cL Chap. I, Sec. 1). L e t X be a s e p a r a b l e r e f l e x i v e Banach s p a c e , and let X* be its dual space; we denote the duality of the p a i r (X*, X) by
XcHcX*, w h e r e this imbedding is dense and is valid together with duality, i.e., if u ~ H, v ~X, then (u, v) =
A(t)(tt):L,(O, Ti" X)-+L,,(O, T; X*),

p>l,

p'=p/(p--l).

The duality of Lp,(0, T; X*) and Lp(0, T; X) we denote by [u, v]. 1~. E x i s t e n c e and Uniqueness T h e o r e m . We c o n s i d e r the p r o b l e m of finding solutions of the parabolic equation du

d--F-}- A (t) (tt) = h (t),

/6[0, r],

(1.1)

tt (0) = 0 ,

(1.2)

w h e r e h(t) is an a r b i t r a r y e l e m e n t of the s p a c e Lp,(0, T; X*). A solution is sought in the s p a c e

W = {tt(t):u(t)6Lp(O, T; X), -d76Lp, au (0, T; X*),u(O)=O}. R e m a r k 1.1. We note that if u(t) ~W, then u(t) ~ C[0, T; H], since du

This m e a n s that the mapping W9 u(t) - - u ( t ) I t = t, E H is continuous in t' (in p a r t i c u l a r , at t' = 0). We a s s u m e that the following conditions a r e satisfied. I. C o e r c i v i t y . F o r any function u(t) ~ Lp(0, T; X) the following r e l a t i o n is valid u n i f o r m l y with r e s p e c t to t ~ [0, T]: t

S ( a (x) (u), u ) d~

Re 0

11u II%[0.t;xi

* +

[I a

+

II. The o p e r a t o r

A (t)(tt) :Lp (0, T; X)-+Lp, (0, T; X*) is bounded [i.e., it takes a bounded s e t of Lp(O, T; X) into a bounded set of Lp,(O, T; X*) and is semicontinuous. We r e c a l l tbat semicontinnity of the o p e r a t o r A(t)(u) means that for any functions u(t), v(t), w(t) of Lp(0, T; X) we have the r e l a t i o n lira [A (t) (tt--~v), w] = [A (t) (u), w]. ~+0

III. The o p e r a t o r A(t)(u) is a monotone o p e r a t o r , i.e., Re [A (t) (u) - - A (t) (v), it-- v] > 0 f o r any function u(t)ELp(0, T; X) and v(t)s

T; X).

THEOREM 1.1. If conditions I - I I I a r e s a t i s f i e d , then for any r i g h t side t h e r e exists a unique function u(t) which is a solution of p r o b l e m (1.1), (1.2). In other w o r d s , under conditions I - I I I the mapping

L (tt)-~ ~ q- A (t) (u) : W

Lp, (0, T;

X*)

is s u r j e c t i v e . 499

The p r o o f will be c a r r i e d out by the method of F a e d o - G a l e r k l n - H o p L Let vl, v2, . . . be a complete s y s t e m in X. An a p p r o x i m a t e solution is sought in the f o r m k

c, (0 ~ ,

u~ {0 = ~ ~=I

w h e r e the unknown functions cu(t) a r e d e t e r m i n e d f r o m the following s y s t e m of o r d i n a r y differential equations:

(~.kat(0 ' v,

) + ( A (t) (uk), vv ) = ( h (t), v, ) ,

e~(0)=0,

v = l . . . . . k,

(1.3)

te[0, r].

(1.4)

Condition I and L e m m a 1.1 (the analogue of the " a c u t e - a n g l e " l e m m a which is proved in part 2 ~ imply the solvability of the s y s t e m (1.3), (1.4) on the s e g m e n t [0, T]; here there is the a p r i o r i estimate T

vraimax l[ u~ (t) ll~ + I ][uk (t) liPdt ..
(1.5)

o

w h e r e K > 0 is a constant not depending on uk(t). Indeed, to d e r i v e e s t i m a t e (1.5) we multiply Eq. (1.3) by cv(t) and s u m on u f r o m 1 to k. For any t ~ [0, T] we obtain duk

{ --~-, uk ) + ( A (t) (uk), uk ~ = { h (t), uk ). By integrating and using condition I, we find f r o m this that t

-~ (u~ (0, u~ ( 0 ) . + Co(11u IIL~(o,,;x))-- I d~ 0 1

~
w h e r e c0(P)/p

~

oo

[lh(,)Jt.~Ld~

1

.

[[ uk (,) ll,~ d ,

,

(1.6)

0

as p --- 0%

Since t E [0, T] is a r b i t r a r y , the d e s i r e d inequality (1.5) follows i m m e d i a t e l y f r o m (1.6), which we were required to show. We continue the proof of the theorem. F r o m the e s t i m a t e (1.5) and the weak c o m p a c t n e s s of the sphere in Lp(0, T; X) it follows that there exists a function u(t) e Lp(0, T; X) and a subsequence uk(t) (we r e t a i n the f o r m e r notation) such that uk(t) --* u(t) weakly in Lp(0, T; X). The element found is the d e s i r e d solution of p r o b l e m (1.1), {1.2). For the p r o o f we r e m a r k that by the boundedness of the mapping A(t)(u) : Lp(0, T; X) ~ Lp, (0, T; X*) the sequence A(t)(u k) may be a s s u m e d to conv e r g e weakly in Lp,(0, T; X*) to s o m e element a(t) E Lp,(0, T; X*). F u r t h e r , on multiplying r e l a t i o n (1.3) by a s m o o t h function ~(t) which vanishes at t = T and integrating by pax ts, we deduce that (' denotes differentiation with r e s p e c t to t) - - [u,, ~" (0 ~'~]+ [ A ( 0 (u,), ~ (0 ~'~]= [h, 9 (0 ~ ] ,

~ = 1.....

k.

F r o m this as k "-* oo we obtain

- [u, ~" (0 v~]+ [a (0, ~ (0 v~] = [h, ~ (0 v~], ~ = 1, 2 . . . . . . Since the s y s t e m vl, v 2. . . .

(1.7)

is complete in X, (1.7) is equivalent to the identity

- [u, .,i + [o c0 o1-- [. v i

(1.8)

w h e r e v(t) ~ Lp(0, T; X) N C1(0, T; H) is a r b i t r a r y [here v(T) = 0]. The equality (1.8) means that d u ( t ) / d t E Lp,(0, T; X*), since a(t) and h(t) a r e such functions. M o r e o v e r , since the value of v(0) is a r b i t r a r y , it follows that u(0) = 0. R e m a r k 1.2. We o b s e r v e that the estimate (1.5) implies that for t,= T s o m e subsequence uk(T) - - a weakly in H. In exactly the s a m e way as we proved that u(0) = 0 i t can be shown that u(T) = a . Thus, uk(T)-- u(T) weakly in H. We shall need this r e m a r k in part 3~

500

We have thus found that u(t) E W, and, m o r e o v e r ,

(1.9)

+ a (t) = k (t).

i t r e m a i n s to p r o v e (and this is the m a i n thing) that a(t) - A(t)(u). To this end we make u s e of the condition of monotonicity III. L e t N > 0 be an a r b i t r a r y i n t e g e r . We s e t

w ~ = vCt, x ) : ~ ( t ,

x)=~

c~(t)v,, e,(t)~C~ (o, r),

o~(O)=O.

A c c o r d i n g to condition III, we have the inequality

dvdt,a,--v]+[A(t)(u~)--A(t)(v),

Re i['Cduk~

ttt~-- v]} > O,

(1. !0)

w h e r e v(t) ~ W N is an a r b i t r a r y function. We o b s e r v e that f r o m r e l a t i o n s (1.3), w h i c h d e t e r m i n e the a p p r o x i m a t e solution uk(t), it follows that -~, for any function v(t) ~ WN with N -< k. In p a r t i c u l a r ,

[~

dt

"

u~ -- ~] +

tA(0(~)'

~o--v] = [h, u~-- v],

(1.8)k

s i n c e uk(t) - v(t) E W k. We fix N > 0. Making u s e of (1.8)k, we obtain f r o m inequality (1.10) for k -> N

'~~

,,,,-.,.,I- [~. ,k-,-,,]-IA (,) (~). ,i-.,:,l} >~o.

Since this inequality contains no n o n l i n e a r i t y in uk(t), as k -~ r we a r r i v e a t the inequality

~olt,,.~'~1

_

dv [~.,~-.o]-r~(,)(o).

u--v]

} >0,

o r , r e c a l l i n g the r e l a t i o n (1.9), a t the inequality-

Rol[~"" Since the union

"",,,. ,,-.,.,] + [o(t): ~ (t~(~),,-oi} > o.

~ tV~ is d e n s e in Lp(0, T; X), on p a s s i n g to the c l o s u r e in the l a s t inequality on v(t), we s e e

/V~I

that it is valid for any function v(t) ~ Lp(0, T; X). In p a r t i c u l a r , setting v(t) = u(t) - ~w(t), ~ > 0, w(t) E Lp(0, T; X), we obtain

Re {~"[ %

o] +~ Io(,)- ~ (,)(,,- ~o), ,,,,,}> O.

Dividing both s i d e s of the inequality by ~ and letting ~ - - + 0, we deduce that Re

[a(t)--A(t)

(u), w] > 0

(1.11)

[here we h a v e used the semicontinuity p r o p e r t y of the o p e r a t o r A(t)(u)]. Since the function w(t) E Lp(0, T; X) is a r b i t r a r y , inequality (1.11) is not c o n t r a d i c t o r y only for a(t) = A(t) (u), which we w e r e r e q u i r e d to show. Combining the equation obtained with (1.9), we find that du

dt F A(t) (tt)=h(t),

i.e., u(t) is the d e s i r e d solution. The p r o o f of e x i s t e n c e is complete. We shall show that the solution found is unique. Indeed, s u p p o s e ul(t) and u2(t) a r e two solutions of p r o b l e m (1.1), (1.2) f r o m the s p a c e W. F o r the d i f f e r e n c e ul(t) - u2(t) we have

501

t

I<

d_~, au,' u , _ ~ >-d ~ + at -~..

0

I < A(~)(u,)--A(~)(~),

~--~)&:=O.

0

Using the mouotonicity of the o p e r a t o r A(t)(u), we find f r o m this that t d tt ~

d tt 2

Re.~< d'~...... at ' u i - - ~ > d ~ 4 0 , 0

o r , what is the s a m e (we r e c a l l that X c . H c . X*)

•2 I[u, (t)- ~, (t)il~,~ c ([l u - - v II,,x)l[ u - - v lip.x,

(1.12)

w h e r e IIu - v Ilp,X is the n o r m of u - v in Lp(O, T; X); c(p) >- 0 is a n o n d e e r e a s i n g continuous function, which is equal to z e r o only at p = O. Then the mapping

au }-A(t)(u): W-+Lp,(O, T; X*) dt is a h o m e o m o r p h i s m . Indeed, the d i r e c t continuity is obvious. In the o t h e r d i r e c t i o n , taking account of (1.12), just as in the p r o o f of uniqueness we obtain

2 IIu ( r ) - - ~ (r) I1~+ c(ll u - - ~ II~.x) I1u - - ~ tIp,x --[[ A (t) (u)-- A (t) (v)lip', ~*" [[ u - - ~ []p.x. F r o m this it follows that

I[u - v ll~,x ~< c-* (ll A (0 ( * 0 - A (0 (~*)lb',aO, which i m p l i e s the continuous dependence of the solution in Lp(0, T; X). Since by hypothesis the o p e r a t o r A(t)(u) is continuous f r o m Lp(0, T; X) to Lp,(0, T; X*), f r o m Eq. (1.1) [for u(t) and v(t)] we i m m e d i a t e l y find that Hd u / d t - dv/dtt[p,,X* --~ 0 if IIA(t) (u) - A(t) (v) lip, ,X* - - 0. The inv e r s e mapping is thus a l s o continuous. R e m a r k 1.3. As in the elliptic c a s e , the following point should be noted. During the p r o o f of the t h e o r e m we had to r e p e a t e d l y choose a subsequence of the b a s i c sequence of a p p r o x i m a t i o n s uk(t). Now (after the p r o o f o f uniqueness) it may be a s s e r t e d that the e n t i r e sequence c o n v e r g e s to the solution u(t). 2 ~ The L e m m a on the E x i s t e n c e of a Solution to the A p p r o x i m a t e Equations: We shall now e s t a b l i s h the solval~iiity of the s y s t e m of o r d i n a r y d i f f e r e n t i a l equations (1.3), (1.4) which d e t e r m i n e the a p p r o x i m a t e s o l u tions of p r o b l e m (1.1), (1.2). F i r s t , however, we shall p r o v e a m o r e g e n e r a l a s s e r t i o n . LEMMA 1.1 (the evolution analogue of the " a c u t e - a n g l e " l e m m a ) . Let c(t) = (cl(t) . . . .

, ck(t)), 0 -< t -< T,

and s u p p o s e that the v e c t o r - v a l u e d function, f(t, c) is defined and continuous for all t and c. Suppose f u r t h e r that f o r any c(t) and any t ~ [0, T] Relf(t, e)c(t)dt>--

a~ l e(t)l~dt.+g(t) ,

0

w h e r e a 0 > 0 and K(t) -> 0 is continuous on the s e g m e n t [0, TI. Then the s y s t e m of nonlinear differential e q u a tions 502

-->

~ ~ adet ~-f(t, =0,

c(0)=0,

(1.13)

has at l e a s t one solution defined for all t E [0, T]. 0

~

--~

Proof. We c o n s i d e r the s p a c e Hi(0, T) of v e c t o r - v a l u e d functions c(t), c(0) = 0, with n o r m defined by

0 O1

It is known that H (0, T) C C(0, T) and this imbedding is c o m p a c t . The s y s t e m of equations (1.13) is obviously equivalent to the s y s t e m of i n t e g r a l equations t

A (c~ - - c ( t ) + 17(~' ~ d~ --_--~(t)+ V (c ~ ) : 0.

(1.14)

0 -..

0 1

Since f(t, c) is continuous in its a r g u m e n t s , the o p e r a t o r V(c) is c o m p l e t e l y continuous in H (0, T). T h e r e 0

f o r e , the d e g r e e of r o t a t i o n of the o p e r a t o r A(c) on the s p h e r e II c II = R in Hi(0, T) is defined. We shall show that for R >> 1 the d e g r e e of r o t a t i o n is equal to one; this implies the solvability of (1.14) and hence of the original s y s t e m (1.13). To this end we c o n s i d e r the family of o p e r a t o r s

A~(c-')--;+~V(c-'),

0.<~.< 1.

It is obvious t_,hat A0(~ - c is the identity o p e r a t o r , and Al(c -~) is the initial o p e r a t o r A(c). T h e r e f o r e , if we show that A)~(c) ~ 0 on a s p h e r e of sufficiently l a r g e r a d i u s , then the d e g r e e of r o t a t i o n of Ak(c) on this s p h e r e is equal to one for all ~ ~ [0, 1]. We s u p p o s e o t h e r w i s e , i.e., AX(c) = 0, II c II = R. Differentiating the equation A;~(c) -~ 0, we obtain

d-7
+~y(t,~')=o,

~(o)=o.

(1.15)

Multiplying (1.15) by c(t) and integrating, we obtain 1

t ~--~

-->

:-~

-~[ c 12(t)+ ~ .~f(t' c)cd~=O. 0

F r o m this and the hypothesis of the l e m m a we conclude that

I~l ~(0 4 2a0x ! I~l' (*) d, + 2XK (0--< 2 a~ o I c !' (9 d, + K (0 ' Using a c l a s s i c a l l e m m a , we find that [-~12(t) ..< 2/((t) exp {2aot}, i.e., all p o s s i b l e solutions of Eq. (1.14) a r e u n i f o r m l y bounded. After this f r o m the s y s t e m (1.14) we find also that the n o r m s T

-->

IJcll~=.~j~j 0

T

dt=~[f (t, c)12dt 0

a r e u n i f o r m l y bounded, i.e., [I c II -< K. But this is i m p o s s i b l e i f R > K. The p r o o f of the l e m m a is complete. This l e m m a e a s i l y i m p l i e s the solvability of the s y s t e m (1.3), (1.4). Indeed, we may a s s u m e with no loss of g e n e r a l i t y that the s y s t e m vl, v 2. . . . is o r t h o n o r m a l in H. The s y s t e m (1.3), (1.4) then a s s u m e s the f o r m (1.14); h e r e conditions I and II of T h e o r e m 1.1 e n s u r e that the hypotheses of L e m m a 1.1 a r e satisfied. This m e a n s that the s y s t e m (1.3), (1.4) is solvable. The t h e o r e m has now been c o m p l e t e l y proved. 3 ~. Strong C o n v e r g e n c e of the A p p r o x i m a t e Solutions. The b a s i c e s t i m a t e (1.5) i m p l i e s that the sequence of a p p r o x i m a t e solutions uk(t) c o n v e r g e s to the solution u(t) weakly in Lp(0, T; X). We shall now show that in t h e c a s e w h e r e the o p e r a t o r A(t)(u) has definite v a r i a t i o n the c o n v e r g e n c e is actually strong. 9We have the following t h e o r e m , which is the analogue of T h e o r e m 1.3 of Chap. IV.

503

THEOREM 1.3. Suppose that the o p e r a t o r A(t)(u) satisfies condition (1.12), i.e., Re IA (t) (u)-- A (t) (v), u - - v] > c (11u " v IIp.D IIu - v [Ip,x.

(1.12)

Then the approximate solutions uk(t) c o n v e r g e to u(t) strongly in the space Lp(0, T; X). Proof. F r o m (1.12) we have the inequality 1 Tllu(T)-uk(T)lI~+c(l[u-ukIlp,x)l]u-ttkllp,x "-<[~t---dT'duku--ttaJ4-[A(t)(tt)--A(t)(uk),tt--un].

(1.16)

Recalling the definition of uk(t), from this we find that

-

-

(ttk (r), u (r)) 4- [A (t)(u), tt--u, l --[A (t)(ttk), it] + [It, ttk]

---[~, u]-(u~ (r), u (T))+iA (0(u), u--u~l --[A (0(u~), ul + [h, u~l.

(1.17)

We now o b s e r v e that as k ~ + ~o the following limit relations hold: 1) (uk(T), u(T)) --- (u(T), u(T)), since Uk(T) - - u ( T ) weakly in H and u(T) E H (el. R e m a r k s 1.1 and 1.2); 2) [A(t)(u), u - Uk] ---0, since A(t)(u) 6 Lp, (0, T; X*) and uk(t) ~ u(t) weakly in Lp(0, T; X); 3) [A(t)(uk), u] - - [ a ( t ) , u], since A(t)(u k) - - a ( t ) [by choice of the sequence uk(t) ] weakly in Lp,(0, T; X*); 4) [h, Uk] - - [h, u]. Taking account of these r e l a t i o n s , f r o m (1.17) we find that 'ica_co --m (ll it--ttk lip, x)t' i t - i t , IIp.x ~<[ ~ , t t ] - ( u (T), tt ( T ) ) - [a (0, u] 4- [h, u] ~ --tu, ~( t7) ], d_ u[a

u] 4- [h, i t ] = 0

(1.18)

by virtue of the r e l a t i o n (1.9). F r o m (1.18) it follows i m m e d i a t e l y that II u - ukllp, X ~ 0,which we w e r e r e q u i r e d to show. The proof of the t h e o r e m is complete. R e m a r k 1.4. It is c l e a r f r o m the p r o o f that under the s t r o n g e r condition Re c01 u - vtlx) IIu - viiX c o n v e r g e n c e (strong) takes place in Lp(0, T; X) fq L~(0, T; H). 2.

Nonlinear

Parabolic

Equations

with

Semibounded

Variation

In this section we p r o v e a t h e o r e m on the solvability of the Cauehy p r o b l e m for the equation d u / d t + A(t) (u) = h(t) when the o p e r a t o r A(t)(u) satisfies a b r o a d e r condition than the condition of monotonicity - the condition of semiboundedness of the variation. 1~ Solvability of the Cauchy P r o b l e m . We c o n s i d e r the Cauchy p r o b l e m

~-t+A(O(u)=h(O, u(0)=0.

(2.1)

te[0, r l ,

(2.2)

As before, the solution of p r o b l e m (2.1), (2.2) is sought in the space

W={tt (t):u(t)~Lv(O, T; X), with

norm

du -d-IELu, (0, T; X*),

u (0) = 0t,J

defined by

We a s s u m e that the o p e r a t o r A(t)(u) satisfies the following conditions: I. C o e r e i v i t y condition. If IIu(t)IIp,X ~ ~, then Re [A (t) (u), u] ~ q_ c~ ; IIu [Ip,x II. The o p e r a t o r A(t)(u) : Lp(0, T; X) --.-Lp, (0, T; X*) is bounded and semieontinuous;

504

III. The c o n d i t i o n of s e m i b o u n d e d n e s s of the v a r i a t i o n . Lp(0, T; X) in the ball II u lip, X -< R

F o r any functions u(t) E Lp(0, T; X) and v(t) E

Re [A (t) (tt) - - A (t) (v), tt-- v] > - - c (R, IItt - - vl['), w h e r e IIu I1' is a n o r m c o m p a c t with r e s p e c t to the n o r m II u ItW, and c(R, p) is a continuous function s a t i s f y i n g the condition c(R, ~ p ) / ~ --- 0 (~ - - +0) f o r any R and p. U n d e r t h e s e conditions the following r e s u l t holds. T H E O R E M 2.1. If conditions I - I I I a r e s a t i s f i e d , t h e n f o r any function h(t) E Lp,(0, T; X*) t h e r e e x i s t s at l e a s t one s o l u t i o n u(t) e W of p r o b l e m (2.1), (2.2). In o t h e r w o r d s , the m a p p i n g du

d-g § A (t) (u): W-,,-Lp, (0, T; X*) is s u r j e c t i v e . We p r e s e n t the p r o o f of this t h e o r e m in two s t a g e s : w e f i r s t s o l v e a s t a t i o n a r y p r o b l e m with p a r a m e t e r e > 0 and we then let e - - 0.

Stage 1 (solution of an auxiliary problem with "viscosity"). On the segment tionary problem for the differential-operator equation of second order: d*u e

du e

--~-j~-+~+A(t)(u~)=h(t), u~ ( 0 ) = 0 ,

~>0,

[0, T] we consider a sta-

tet0, V l,

(2.1)e

d"~(r) = 0 .

at

(2"2)e

H e r e by a s o l u t i o n of p r o b l e m (2.1)e, (2.2)e we m e a n a function u~ (t) E ~r = {u(t) : u(t) E Lp(0, T ; X ) , (du/dt) EL2(0, T; H)} which a s s u m e s the b o u n d a r y v a l u e s (2.2)s [the s e n s e in which d u e ( T ) / d r = 0 will b e c o m e c l e a r below] and s a t i s f i e s the identity

rd., do ] [due ]

~ I-a-' arj§

~ + [A (t) (u,), vl=[a, ~1

(2.3)

for any function v(t) E W s u c h t h a t v(0) = 0. T H E O R E M (Auxiliary). If conditions I - I I I a r e s a t i s f i e d p r o b l e m (2.1)e, (2.2) e has at l e a s t one s o l u t i o n ue(t) E W, and t h e r e is the e s t i m a t e du e 2

(2.4)

~ 2,n+ll uellp,x ~
where K

=

K(h) does not depend on e i t h e r e > 0 o r on ue(t).

P r o o f . L e t v~(t), v2(t), . . . . vk(t), . . . be a c o m p l e t e s y s t e m in W with vj(0) = 0, j = 1, 2 . . . . . s e e k an a p p r o x i m a t e s o l u t i o n in the f o r m

We

k 1=0

w h e r e the unknown c o n s t a n t s a r e found f r o m the s y s t e m of equations

V%o rdv , l-aT-' dt J + I--tiT-' vj] + [A (t) (a,~), vii = [h, v~],

(2.5)

j = 1,...,k. By virtue of the coercivity condition I for the approximate

solutions there is the obvious a priori estimate

d//sk 2

- ~ - 2,. + Ila~jtp,x--< K,

(2.6)

w h e r e K > 0 is a c o n s t a n t [depending on h(t)]. F r o m the e s t i m a t e (2.6) it follows that s o m e s u b s e q u e n e e Uek(t ) c o n v e r g e s w e a k l y in Lp(0, T; X) to a function ue(t) , and d u e / d t d u / d t w e a k l y in L2(O, T; H). Thus ue(t ) EW, and, m o r e o v e r , ue(0) = 0. We shall show that ue(t) is the d e s i r e d s o l u t i o n of p r o b l e m (2.1)~, (2.2)~. T o this e n d w e f i r s t e s t a b l i s h that the function ue(t) s a t i s f i e s the identity (2.3). P a s s i n g to the l i m i t in E q s . (2.5), we c o n f i r m the v a l i d i t y of the identity

505

[due dv

L'Ti-' ~7] + Fa%

L~-' ~J7 +[a~ (t), ~,l--V,, ~1, v (t)eW,

(2.7)

~ (o) = O,

w h e r e as(t) is the w e a k l i m i t of the s e q u e n c e A(t) (Usk) (k -* .o) in Lp(0, T; X). We shall show that as(t) - A(t)(us). F o r this we m a k e u s e of the condition III of s e m i b o u n d e d n e s s of the v a r i a t i o n . We h a v e

Re

{['%~, ~ ,z~,at,~"*~t

:d~J+L~ao] ra.~k aOat,u~'*]+[A(t)(u*~)--A(t)(~)'u*~--vl}>--c(R'llu~,~--W),.

(2.8)

w h e r e I1" I1' is a n o r m c o m p a c t r e l a t i v e to the n o r m in W (and hence W), IIUskllp, X -< R, II v IIp,X --- R, R -> K, and K is the c o n s t a n t in i n e q u a l i t y (2.6). R e c a l l i n g the equations f o r the a p p r o x i m a t e s o l u t i o n s (2.5), we have

[due, dv l

fdv dUe~ do q at at

Re { [h, u,k[--~l_--jT-, - a - / j - - e l ~ ,

[ a*'e, v]--I-a-/-, av --L--E-,

.J

u~, --v] + [ A (t) " (u,,), v] " } > . c (R,

ue,--vJ--IA(t)(v),

Ilue~-vll').

(2.9)

By c h o i c e of the s e q u e n c e Uek(t) and the function v(t) E W the following l i m i t (k --~ ~) r e l a t i o n s hold: 1) [h, us~]-+ [h, us],

rduek au ]

3)

~'

at

[ d% at

7], a t ' at

~

at'

4) L-m-' VJ~l-Z?' 6) [A (t)(v), u ~ - - v l ~ I A ( t ) ( v ) ,

u~-z,],

7) [A (t) (usk), v] -+ Ia~ (t), v]. M o r e o v e r , s i n c e the n o r m II- II' is c o m p a c t r e l a t i v e to the n o r m in W, it m a y be a s s u m e d (by c h o o s i n g a s u b s e q u e n c e if n e c e s s a r y ) that II U e k - vii' ~11 u e - vii' and h e n c e 8) c(R, I l u e k - vii') - - c ( R , I l u e - vii'). A f t e r t h i s , on letting k -

~o in (2.9), w e a r r i v e at the i n e q u a l i t y

{

Re [h, u~]--~

r,'u,, L d t ' -,,,,] ~ -- ~

dr' dt

dt J

L

at ' v

] (2.10)

o r , taking a c c o u n t of the identity (2.7), at the i n e q u a l i t y Re ~ at

at" dt

at +

at

at,u~--v

+[a~(t)--A(O@),u~--vl

>--c(R,l[u~.vH"

).

(2.11)

We now s e t v(t) - ue(t) - ~w(t), w h e r e ~ > 0, and w(t) ~ W is an a r b i t r a r y function. F r o m (2.11) we then find the inequality ' c(R,

~t1~11%

w h e n c e as } - - 0 we find [using the semiconLinuity of the o p e r a t o r A(t) (u) and the p r o p e r t i e s of the function c(R, p)] t h a t Re [ae (t)-- A (t) (us), zoI > O. Since the function w ~ W is a r b i t r a r y ,

w h i c h we w e r e r e q u i r e d to show.

506

this m e a n s that a~ ( t ) ~ A (t) (ue),

It r e m a i n s to o b s e r v e that the identity (2.3) and the e s t i m a t e (2.4) [it is c l e a r that (2.4) is obtained f r o m (2.6) as k - ~] imply that d2%

(2.12)

- - d ~ : g~ (t) + g2~ (t), w h e r e gle(t) -= h(t) - A ( t ) ( u e) ~ Lp,(0, T; X*), g2e(t) - - ( d u J d t ) E L 2 ( 0 ,

T; H).

Thus, (d2ue/dt 2) E Lp,(0, T; X*) + L2(0, T; H), so that the t r a c e ( d u e / d 0 ( T ) is meaningful. Since the values of v(T) a r e a r b i t r a r y , (2.3) implies that (due/d0(T) = 0. It has thus been established that us(t) is the d e s i r e d solution of p r o b l e m (2.1)e, (2.2) e. The proof of the t h e o r e m is complete. Stage 2 (passage to the limit as e - - 0). We now let s - - 0 and show in the p r o c e s s that the family of solutions found is weakly compact in W. We o b s e r v e , f i r s t of all, that by virtue of the a p r i o r i estimate (2.4) it may be a s s u m e d that us(t) --u(t) weakly in Lp(0, T; X), w h e r e u(t) ~ Lp(0, T; X) is some function. Secondly, turning to (2.12), we find that the derivative du e / d t satisfies the relations

~d2% + ~ du e =gt~ (t),

(2.13)

due (T)~ O,

(2.14)

dt

w h e r e gle(t) runs through a bounded set in Lp,(0, T; X*) as e ~ 0 .

Solving (2.13), (2.14), we find that

T--t du e . .~

1 0

1 oo

Since -/~., exp t~ - - ~~J d~q=-=1, it follows f r o m a well-known inequality for the convolution* that (due/dr) e Lp~(0, T; 0

X*), and they r u n through a bounded set t h e r e , i.e., du e

~
w h e r e K > 0 is a constant. Thus, it is possibIe to a s s u m e that not only

u8 (0-~u(t)w*akly ~ Lp(0, T; x), but also du e (t) da (t) dt + ~ weakly in g p , (0, T ; X * ) ,

i.e.,

ue (t)-+ u (t) w~akly in W. We note that h e r e u(0) = 0. We shall show that the function u(t) found is the d e s i r e d solution of the original p r o b l e m (2.1), (2.2). Since as e ~ 0 dv

dv

[cf. the e s t i m a t e (2.4)], f r o m the r e l a t i o n (2.3) we find that the limit function u(t) satisfies the identity (2.15) w h e r e a(t) E Lp,(0, T; X*) is the weak limit of A(t)(u e) in Lp,(0, T; X*). TWe have in mind the inequality II f,g lip - [ I flll*ll g lip, p > 1.

507

Since r~r is d e n s e in Lp(0, T; X*), f r o m (2.15) we deduce that [ - ~ , v] + [a(t), v] = lb, v],

(2.16)

w h e r e v(t) E Lp(0, T; X) is an a r b i t r a r y function. It r e m a i n s to show that a(t) - A(t)(u). To this end we again make use of the monotonicity condition III. We have for any function v(t) E W, IIv IIp,x ~ R the inequality

Re~3[~"

,t-T' at

at + a t

at,u,--v

-(-IA(t)(u~)--A(t)(v),n.--v]

>-c(R,

Ilu.-+,l['),

(2.17)

w h e r e we r e c a l l that the n o r m I1" I1' is c o m p a c t r e l a t i v e to the n o r m II- Ih Using the identity (2.3), f r o m (2.17) we a r r i v e at the inequality (it

Letting e --* 0, we find that

Re {[h, #]--[ ~-~, u--v]--[A (t)(v), u--v]} > --c (R, IIu--v I1"), o r , using r e l a t i o n (2.15), that

,-.+i} >-c(,+.tI,,-",,ll').

<+..i+>

The l a s t inequality was obtained for v(t) Gr~, but it is obviously a l s o true for any function v(t) E W, since ~r c W is dense. Setting v(t) = u(t) - ~ w ( t ) in (2.18) w h e r e ~ > 0, w(t) EW, we obtain the inequality 9

,

eel} > -y c (R, ell * 113, whence for + + + 0 we find that Re [a (t)-- A (t) (u), ~o] > 0,

weW

(2.19)

[here we have again u s e d the semicontinuity of the o p e r a t o r A(a)(u) and the p r o p e r t i e s of the function e(R, p)]. F r o m (2.19) we find that a(t) = A(t)(u), which we w e r e r e q u i r e d to prove. 2 ~ Strengthening of the R e s u l t s of Sec. 1. We focus attention on the fact that the c o e r c i v e n e s s condition I in T h e o r e m 2.1 is l e s s r e s t r i c t i v e than the c o r r e s p o n d i n g c o e r c i v e n e s s condition in T h e o r e m 1.1 of Sec. 1. This situation m a k e s it possible to s t r e n g t h e n c e r t a i n r e s u l t s of Sec. 1. Namely, we have the following result. THEOREM 2.2 (a strengthening of T h e o r e m 1.1). Suppose that the following conditions a r e satisfied: I. The o p e r a t o r A(t) (u) is c o e r c i v e , i.e., for IIu IIp,X - - .o Re [A (t) (a), u] --> _~. oo ; [I u Ilp.X

(2.20)

II. the o p e r a t o r A(t)(u) :Lp(0, T; X) + Lp,(0, T; X*) is bounded and semieontinuous; Ill. the o p e r a t o r A(t)(u) i s a monotone o p e r a t o r , i.e., Re [A (t) (u) - - A (t) (v), tt

~l > 0,

w h e r e u(t) E Lp(0, T; X), v(t) e Lp(0, T; X) a r e a r b i t r a r y functions. Then for any function h(t) E Lp,(0, T; X*) the Cauehy p r o b l e m dtt

at ~ - A ( t ) ( t t ) = h ( t ) ,

u(0)=0,

has at l e a s t one solution u(t) ~ W [we r e c a l l that W = { u(t) :u(t) e Lp(0, Ti X), (du/dr) ~ Lp, (0, T; X*)}. THEOREM 2.3 Ca strengthening of T h e o r e m 1.2). Suppose that the following conditions a r e satisfied:

508

I.

The c o e r c i v e n e s s condition (2.20);

II.

the o p e r a t o r A(t)(u) :Lp(0, T; X) -- Lp,(0, T; X*) is continuous;

III. the o p e r a t o r A(t)(u) hasdefinite variation, i.e.,

Re [A (0 (u)-- A (0 (v). u--v] > c (II~!--v [I,. x)II u - - ~ [Ip x. w h e r e u(t) ~ Lp(0, T; X), v(t) E Lp(0, T; X) a r e a r b i t r a r y functions, and c(p) > 0 is a nondecreasing, continuous function. Then the mapping d~

dt {-A(t)(u): W ~Lp,(O, T; X*)

is a h o m e o m o r p h i s m . The proof of both t h e o r e m s follows i m m e d i a t e l y by c o m p a r i n g the proofs of T h e o r e m s 1.1 and 2.1. We r e m a r k that these t h e o r e m s (as also T h e o r e m 2.1) a r e v e r y widely used in applications. 3.

First

Boundary-Value

Parabolic

Differential

Problem

for

Quasilinear

Equations

In this section applications a r e given of the general t h e o r e m s proved in the preceding sections to quasil i n e a r partial differential equations of any o r d e r and of parabolic type. 1~ P a r a b o l i c Equations with a Strongly Elliptic Spatial P a r t . We study a q u a s i l i n e a r parabolic equation of of the type ~ t + L (t, x, D) (u) = ~ + Z ( - l)[ ~ID~A~ (t, x, Dvu)= h (t, x), I ~;I ~<m. Icq..<m

(3.1)

In the cylinder Q = [0, T] • G with l a t e r a l s u r f a c e S it is r e q u i r e d to find a solution of Eq. (3.i) with the following initial and boundary conditions: u(0, x ) : ? ( x ) , D~

s~S, I ~ [ ~ m - - 1 .

(3.2) 0

B e f o r e proceeding to formulate the conditions of our problem, we introduce the space W needed below. We a g r e e throughout this section to denote by It u IIm,p(ll u l]_m,p,) the n o r m in the space Lp(0, T; W~n(G)) [ r e spectively, Lp,(0, T; Wpm(G))]. 0

Definition 3.1. The space W is the c l o s u r e in the n o r m

Hurl

--II.rim.+]l 0o

of smooth functions u(t, x) satisfying the conditions u(0, x) = 0, DC~uls = 0, l a)i _< m Definition 3.2. If f(t, x) has finite n o r m s

If film, p

1.

and Jl 0f/0t Lm,pY , then the set of functions of the f o r m

0

u(t, x) = f(t, x) + z(t, x), w h e r e ~.(t, x) E W, f o r m s the space W(f). We r e t u r n to the original p r o b l e m (3.1), (3.2). As sumptions. I. The functions Ac~ (t, x, (~/) a r e defined for (t, x) E Q and ali (,/, are continuous in t, x, and (~,, and satisfy the inequality

Z t

\[~[<:m

9

II. The condition of strong ellipticity of L(t, x, D)(u). F o r any functions u(t, x) e Lp(0, T; W~l(G)) and v(t, x) E Lp(0, T; w~n(G)) such that u(t, x) - v(t, x) e Lp(0, T; ~g~n(G)) t h e r e is the inequality

Re [L (t, x, D ) ( u ) ' L ( t , x, D)(v), u--v]-----Re ~

[As(t, x, DVu)--A=(t, x, DVv), D~(u--V)] >fiao

a,

luI--.<m

w h e r e a o > O.

509

HI. R e g a r d i n g the initial and b o u n d a r y conditions (3.2), we a s s u m e that they a d m i t c o n t i n u a t i o n to Q f r o m the s p a c e W(f), i.e., t h e r e e x i s t s a function f(t, x) having finite n o r m s i] film, p and I] 0 f / 0 t U_m, p, s u c h that f(0, x) = go(x), DWfis = fed (x), I a~ J _< m - 1. T H E O R E M 3.1. If c o n d i t i o n s I - I I I a r e s a t i s f i e d , then the m a p p i n g

Ouot_+_ LCt, x, D) (it): W (f) -+ Lp, (0, T; W-~m(G))

(*)

is a h o m e o m o r p h i s m . 0 P r o o f . We s e t u(t, x) = f(t, x) + z(t, x), w h e r e f(t, x) is a c o n t i n u a t i o n of conditions (3.2) and z(t, x) EW is 9the d e s i r e d function. T h e n p r o b l e m (3.1), (3.2) r e d u c e s to the p r o b l e m az

~T-+L (t, x, D)(j' + z)~g(t,x),

D~

z(0, x)=0,

i3.3)

I0~l~<m-1,

(3.4)

w h e r e git, x) - hit, x) - ( a / 0 t ) f ( t , x) E Lp,i0, T; W p ~ i G ) ) . The h o m e o m o r p h i s m

(*) is t h e r e f o r e e q u i v a l e n t to the h o m e o m o r p h i s m 0

-~-t-LOZ, (t, x, D)(y+z):W-+Lp, (0, T; w-~,~(a)). To p r o v e the v a l i d i t y of (**) w e m a k e u s e of the o p e r a t o r h o m e o m o r p h i s m

(**)

t h e o r e m 2.2.

With this t h e o r e m in m i n d , we s e t ( a s s u m i n g for the t i m e being that p ->

2)

0

x - ve~ (o), H = t~ (O), X*-----W~" (O), A (t) (z)-~ L (t, x, D) ( f + z),

z (t)-------z (t, x).

O b v i o u s l y , X c H c X*, and t h e s e i m b e d d i n g s a r e d e n s e and p r e s e r v e duality. Further,

by v i r t u e of condition I t h e o p e r a t o r 0

A(t)(z):W-+Lp, (0, T; X*) is a continuous o p e r a t o r ,

s o that the d i r e c t continuity of the m a p p i n g (**) is obvious.

In the s a m e w a y it is obvious t h a t the condition of s t r o n g e l l i p t i c i t y of the o p e r a t o r L(t, x, D) (u) i m p l i e s the d e f i n i t e n e s s of the v a r i a t i o n of the o p e r a t o r A(t)z: Re

IA (t)(z)--A (t) (•), z - w] > a011z--~ll~.x,

w h e r e z(t) and w(t) a r e a r b i t r a r y functions in Lp(0, T; X). It r e m a i n s to c h e c k the c o e r c i v i t y of the o p e r a t o r A(t)z. This is c a r r i e d out in a m a n n e r s i m i l a r to the s t a t i o n a r y e a s e (See. 2, Chap. II). N a m e l y , w e f i r s t p r o v e a l e m m a . L E M M A 3.1. If conditions I - I I I a r e s a t i s f i e d , for any function u(t, x) e Lp(0, T; w~n(G)), u(t, x) - fit, x) 0

Lp(0, T; W~I(G)) t h e r e is the i n e q u a l i t y Re [L (t, x, D) (t0, u] -----Re

~

IA~ (t, x, DVu),O~u]> at IIu II~,p- K,

IcqKm

w h e r e a 1 > 0 and K > 0 a r e c o n s t a n t s . P r o o f . If f o r ~y = 0 all the functions A s (t, x, 0) = 0, then the inequality of the l e m m a follows f r o m c o n d i tion II if w e s e t v(t, x) - 0; h e r e al = a0, K = 0. S u p p o s e now t h a t t h e r e a r e functions Ao~ (t, x, 0) ~ 0. F r o m condition II w e then obtain Re ~

[As(t, x, OVu)--A~(t, x, 0), D~ul>aollull~.p.

9 Ic~l~m

By condition I we h a v e the inequality I Ao~ it, x, 0) I - K; h e n c e I:[A~ (t, w h e n c e we find that f o r any e > 0 510

x,O), O~u] l..
(3.5)

I

x, 0), D<< ll

il.ii ,. +

mes 0).

Choosing e > 0 sufficiently s m a l l , we obtain f r o m (3.5) the inequality of the l e m m a . The proof of the l e m m a is complete. On the b a s i s of the p r e c e d i n g l e m m a , it is not hard to e s t a b l i s h the c o e r c i v i t y of the o p e r a t o r A(t)z. 0 A s s e r t i o n 1. F o r any function z(t, x) t Lp(0, T; W~n(G)) t h e r e is the inequality

(3.6)

Re [A (t) (z), zl > c~ IIz I!~,p- K, w h e r e a 2 > 0 and K > 0 a r e constants. Proof. We have

Re [A (t) (z), z]-- Re [L (t, x, D) (f + z), f + z]-- Re [L (t, x, D) (.f + z), f], whence by L e m m a 3.1 Re [A (t) (z), z] > al !1f'q- z [[~,p - - K - - Re L (t, x, D) ( f + z), f].

(3.7)

We e s t i m a t e the e x p r e s s i o n [Lit, x, D)(f + z), t], r e c a l l i n g condition I on the growth of the functions A a (t, x, DYu). According to this condition, we have

IIL(t,x,D)(f+z),fll-.<X ~ [ ~ l~zl<:m

--K

~

l~'l<:m

.1

[ID'
lal ,IVl<:m

I~i,<:m

Since f(t, x) E Lp(0, T; w~n(G)), applying YoungTs inequality, we deduce that l[L (t, x, D) ( f + z), Sl I ~< e Ill + z IIt,.+ c (~) (IIi I[~,p-I"

1),

which together with inequality (3.7) and for sufficiently s m a l l e > 0 leads to the d e s i r e d inequality (3.6). The a s s e r t i o n has thus been proved. It r e m a i n s to note that since p > 1 inequality (3.6) i m m e d i a t e l y yields the c o e r c i v e n e s s of the o p e r a t o r ACt) (z), which we w e r e r e q u i r e d to show. All the hypotheses of T h e o r e m 2.2 a r e thus s a t i s f i e d , and hence the h o m e o m o r p h i s m (**), and thus a l s o (*), is established. L e t us not forget, however, that we have so f a r c o n s i d e r e d only the c a s e p-> 2. The c a s e p < 2 r e q u i r e s m i n o r modifications. The situation is that for p < 2 we do not n e c e s s a r i l y have the imbeddings 0

W";(O)cL~(O)cW;,"(O) [this is valid if np > 2 ( n - mp)]. Taking this into account, we m a k e the change of v a r i a b l e u(t, x) = v(t, x)e yt, ~ > 0. P r o b l e m (3.1), (3.2) then goes o v e r into the p r o b l e m

0~_+ e_~,tL(t, x, D) [vevq + "iv-- h (t, x) e-~.t,

(3. 8)

Dovls=fo(s)e-%

(3.9)

v(0, x ) = r ( x ) ,

I~ol~
After this we s e t X=t~/~(G) f-lZ2(G), H=Le(G),

X*=W'~"(G)+L2(O);

the imbeddings X c H c_ X* a r e now s a t i s f i e d automatically. The r e m a i n d e r of the a r g u m e n t is now exactly the s a m e as in the c a s e p -> 2. We r e m a r k only that the p r e s e n c e in (3.8) of the t e r m ~/v(t, x) with 3' > 0 gives the additional t e r m IIv Ill{ in all e s t i m a t e s . T h e proof of T h e o r e m 3.1 is complete. E x a m p l e . We c o n s i d e r the equation

Ou + L (D) (u) =-~t + ~ (-- 1)i~'iD~<(10~'u lP-~O~u)= h (t, x), p > 2. Ic*[-~m

511

All the conditions of T h e o r e m 3.1 a r e satisfied. T h e r e f o r e , the o p e r a t o r au/Ot + L(D)(u) maps W(f) onto Lp,(0, T; WoO(G)) h o m e o m o r p h i c a l l y . 2 ~ P a r a b o l i c Equations with Semibounded Variation. In the cylinder Q = [0, T] • G we consider the f i r s t boundary-value p r o b l e m for the parabolic equation

Ou + L (t, x, D) (u) ~ ~i" + Z (-- 1)t~tD~A~(t, x, Dvu) = h (t, x),

(3.10)

l~l<:m

Ixl~m, u(O, x ) = w ( x ) ,

DOuls=f~(s),

loll<m--1.

(3.11)

As sumptions. I. The functions A s ( t , x, ~T) a r e continuous in all arguments and satisfy the inequalities

[A~(t,x,}Ol~ l . II. The c o e r c i v i t y condition. F o r any function u(t, x) E Lp(0, T; W~n(G)) t h e r e is the inequality

Re [L (t, x, D) (u), u]----Re ~ [A4t, x, D~u), D=u] > a0[[u ][~,p-- K, w h e r e a 0 > 0, K > 0 a r e constants depending, in general, on D~~ IS, [~o[ _< m - 1. IIL Condition of semiboundedness of the v a r i a t i o n of the o p e r a t o r L(t, x, D)(u). For any functions u(t, x) E Lp(0, T; W~n(G)) and v(t, x) ~ Lp(0, T; w~n(G)) in the ball IIu IIm,p < R such that u - v E Lp(0, T; ~V~n(G)) t h e r e is the inequality Re [L (t, x, D) (u) -- L (t, x, D) (v), u - - v] ERe ~

[A.(t, x, D~u)--A~(t, x, Dvv), D~(u-- v)] > --c(R, II u-v

Jlm-l,p),

w h e r e c(R, p) -> 0 is a continuous functions such that c(R, ~p)/~ --" 0 for any R and p as ~ -- +0. Finally, as in P a r t 1 ~ we a s s u m e that the initial and boundary conditions admit continuation to Q with f(t, x) ~W(f). THEOREM 3.2. If conditions I-III a r e satisfied, then for any function h(t, x) E Lp,(0, T; wpm(G)) p r o b l e m (3.10), (3.11) has at l e a s t one solution in the space W(f). The p r o o f will be reduced to applying the o p e r a t o r t h e o r e m 2.1. We a g a i n s e t u(t, x) = f(t, x) + z(t, x), 0

w h e r e z(t, x) ~ W. We then a r r i v e at the p r o b l e m of the solvability of the equation 0_zz+ L (t, x, D) ( f + z) = g (t, x) (g (t, x)----h r ~ ) , ~ f (t, x))

Ot

o

in the space W. With an eye to T h e o r e m 2.1, we set (assuming p -> 2) 0

x=~,(o), H=&(O), x,=w-;.,,(o). A(t)(z)--L(t, x, D)(f +z), 0

w h e r e z(t, x) is c o n s i d e r e d as a function z(t) taking values in the space w~n(G). It is obvious that conditions I-Ill of our t h e o r e m lead to the validity of the conditions of T h e o r e m 2.1. Only the c o m p a c t n e s s of the n o r m II u I1' -II u IIm_l, p in the space W r e q u i r e s verification. We have 0

0

wp~-~ (O)cW~ (o)cw-;, m (a), and these imbeddings a r e compact. T h e r e f o r e , by a well-known c o m p a c t n e s s t h e o r e m (cf., e.g., Lions [38]) the space

o "0 T; w;', I w={~(t):u (t)~L.(o, r; w;), ~du -Lp,t, 512

0

is c o m p a c t l y imbedded in Lp(0, T; W~a-~(G)). This is what is r e q u i r e d . Thus, the a s s e r t i o n of our t h e o r e m follows f r o m T h e o r e m 2.1. The c a s e p < 2 is handled as in P a r t 1 ~ The p r o o f of the t h e o r e m is complete. Examples. 1. We c o n s i d e r the p r o b l e m --0u 0t

+ ~ (-

1),~z>= (I Dc'tt ["-2D=u) = h (t, x), I < p < 2,

(3.12)

I~L
u(O, x)-=~(X), D~

(3.13)

fo(s), l o l < m - 1 .

In c o n t r a s t to the c a s e p -> 2 in the c a s e 1 < p < 2 the o p e r a t o r L (D) (u) ~ Z ( - - t)I~ID~(I D~tt [P~Dr'tO Ic*l~:m

is not a s t r o n g l y elliptic o p e r a t o r . H o w e v e r , it is a s t r i c t l y monotone o p e r a t o r as follows f r o m the inequality Re [L (D) (tt)-- L (D) @),

u-- v] -- ~ [ID~u P-2D~t*- I D ~ ~'-~Dzv, D ~ (tt--~)]

[~m O+, ~

O~, r

/

w h e r e Q+,a = {(t, x) E Q : I D a u - D a v l -< I D a u + D a v l , Q - , a = Q \ Q + , a }

(el. Dubinskii [26]).

Applying T h e o r e m 1.1 on the solvability of monotone p a r a b o l i c equations, we find that p r o b l e m (3.12), (3.13) is uniquely solvable for any h(t, x) E Lp,(0, T; w~m(G)). 2. We c o n s i d e r the equation (cf. See. 3, Chap. II, P a r t 3~

aa Ot '-]-(--1)m Z Da(I D~u I'-2D"a + I D~-'uiqD~u) -}-Z (-- I)I"ID"( I~ c~ ,ID~u I'~) =h (t' x)' lul=m

where p>2,

l~[<m

(3.14)

lo)]<:m--1

q
F o r this equation the conditions of the v e r y g e n e r a l t h e o r e m 2.1 on the solvability of d i f f e r e n t i a l - o p e r a t o r equations with semibounded v a r i a t i o n a r e satisfied. The f i r s t b o u n d a r y - v a l u e p r o b l e m for Eq. (3.14) is t h e r e f o r e always solvable. 4.

Nonlinear

Having

Parabolic

Divergence

Equations

in the

Plane

Not

Form

In the p r e s e n t s e c t i o n we study p a r a b o l i c equations in two independent v a r i a b l e s (t, x), x E (a, b), which, in g e n e r a l , do not have d i v e r g e n c e form. It is found that the f o r m u l a t i o n of the p r o b l e m which is the analogue of the f i r s t b o u n d a r y - v a l u e p r o b l e m depends on the o r d e r of differentiation c a r r i e d out beyond the sign of the nonlinear function. In p a r t i c u l a r , the f i r s t b o u n d a r y - v a l u e p r o b l e m we obtain for d i v e r g e n c e equations or for e s s e n t i a l l y nonlinear equations (i.e., equations nonlinear in the highest o r d e r d e r i v a t i v e s ) . 1~ Some Definitions and Notation. L e t Q = {0 -< t --- T, a _< x -< b} be the r e c t a n g l e with l a t e r a l boundary F, let u(t, x) be a function defined in Q, and let Du = d u / d x denote differentiation with r e s p e c t to x. L e t p -> 1, q -> 1 be a r b i t r a r y n u m b e r s , and let l -> 0, s - 0 be a r b i t r a r y integers. We denote by wS,/tO) the a n i s o t r o p i c Sobolev s p a c e q'P'~"

oiu -L

"~"

9

W$:~(Q)={~(t, x):-~7 e qffd), O-.<]4s;

DJueLp(q),

04]~
0

F u r t h e r , we denote by W~-/(Q) the s u b s p a c e of functions u(t, x) ~ Wq,p(Q) 1,l such that u(T, x) = 0, u[ F = 0 . . . . .

~,:p(Q) we denote the subspace of functions u(t, x) e Wq,p(Q) l,l such that u(0, x) = 0, uIF = 0,,

D/-IulF = 0; by W D / - I u I F = 0. 0

"" 513

~

The n o r m in the s p a c e Lp(0, T; W / ( a , b)) we denote by Ilu Ill,pand the n o r m in the s p a c e Lp,(0, T; W b)) by IIu" II_/,p,.

(a,

L e t l --- 0, k -> 0, r -> 0 be a r b i t r a r y i n t e g e r s with 2r _< l - 1. On the s e g m e n t (a, b) we c o n s i d e r the s e t of s m o o t h functions u(x) s u c h t h a t u ( a ) ~u(b) =0, . . . . Dr-~u(a) = Dr-tu(b) = 0; D2ru(a) = D2ru(b) = 0 . . . . . D l-lu(a) = D l-lu(b) = 0, and we denote by W~p{a, b) t h e i r c l o s u r e in the n o r m b

IIu IIp = f (I u Ip +1 o ' u 19 dx. a

[The s p a c e r~/p(a, b) depends, of c o u r s e , on r , but, since in each p a r t below r r e m a i n s unchanged, for b r e v i t y we do not indicate this dependence.] The s p a c e W of functions u(t, x) such that u(t, x) ~ Lp(0, T; ff/p(a, b)), (0u/0t) E Lp,(0, T; Wpk(a, b)), u(0, x) = 0 (k -> 0) is of s p e c i a l i n t e r e s t to us. The n o r m in W is defined by

F u r t h e r , if f(t, x) has finite n o r m II f IIw, then we denote by W(t) the s e t of functions u(t, x) of the f o r m tt(t, x ) = f (t, x)q-zCi, x), w h e r e z(t, x) ~ W. Finally, we r e c a l l that [u, v] denotes an i n t e g r a l (possibly in the s e n s e of g e n e r a l i z e d functions) o v e r the r e c t a n g l e Q, while denotes an integral o v e r the s e g m e n t [a, b]. 2 ~ P a r a b o l i c Equations with a Definite Spatial O p e r a t o r (an E x i s t e n c e and Uniqueness T h e o r e m ) . Let k - 0, 1 -> 0, m -> 1 be i n t e g e r s with k + l = 2m. We f i r s t c o n s i d e r the c a s e k ~ l. It is obvious that l - k = 2r is an even n u m b e r . In this c a s e in the r e c t a n g l e Q we c o n s i d e r the p r o b l e m ~J (tt) --- ( - - 1)r ~-t -q- D~ A (t, x, D t t t ) = h (t, x),

(4. 1)

tt (0, x) = ~ (x),

(4.2)

ttlr = f 0 (7). . . . . D r - ' t t l v = L - , (7), D2"tt[," = f 2r ('0 . . . . . Ot-'ulr = Yt-, (7).

(4.3)

We shall f o r m u l a t e the a s s u m p t i o n s under which p r o b l e m (4.1)-(4.3) will be studied. I. The function A(t, x, 4) is continuous in all its a r g u m e n t s for (t, x) e Q and any 4, while IA(t, x, 4)1 -< K(I } IP-1 + 1), w h e r e K > 0, p > 1 a r e constants. II. C o e r e i v i t y condition. F o r any function u(t, x) 6 Lp(0, T; W/(a, b)) t h e r e is the inequality t

t

(-- l)kRe i < A(t, x, D~tt), D~u >dr >a01 dr--K, 0

0

where t E [0, T] is arbitrary; a0 > 0, K > 0 are constants. III. Condition of strict monotonicity. For any functions u(t, x) E Lp(0, T; W/(a, b)) and v(t, x) ~ Lp(0, T; W/(a, b)) there is the inequality (-- l)k Re [A (t, x, Dtu) -- A (t, x, Dry), Dt (u , v)] > 0, while the equality holds if and only if u(t, x) - v(t, x). IV. The initial and boundary conditions admit a continuation f(t, x) to Q such that f(t, x) ~ Lp(0, T; W/(a, b)), 0f(t, x)/3t E Lp,(0, T; Wp,k(a, b)). In other words, it is assumed that there exists a function such that f ( 0 , X ) = ~ (x),

f i r = f 0 (7). . . . . D ' - x f I r = L - ,

('l),

D"~"f l r = f 2r (7). . . . , Dt-' f[r = f t-x (7).

THEOREM 4.1. L e t the conditions I - I V be satisfied. Then for any right side h(t, x) e Lp,(0, T; Wp,k(a,b)) t h e r e exists a unique solution of p r o b l e m (4.1)-(4.3) u(t, x) ~ W(f).

514

We c a r r y out the p r o o f by the F a e d o - G a l e r k i n - H o p f method taking account of the monotonicity p r o p e r ties. Conceptually we a r e a l r e a d y v e r y f a m i l i a r with this method, and we shall t h e r e f o r e be brief. ok

L e t vl(x), v2(x), . . . be a c o m p l e t e s y s t e m in the s p a c e W (a, b). We s e e k a p p r o x i m a t e solutions un(t, x) of p r o b l e m (4.1)- {4.3) in the f o r m un(t, x) = f(t, x) + Zn(t, x), w h e r e Zn(t, x) is d e t e r m i n e d f r o m the r e l a t i o n n

D~tzn (t, x)

=Zc~ (t) v~ (x)

(4.4)

[we r e m a r k that conditions (4.2) for the function zn(t, x) give Znl F = 0 , . . . , D r - l z n I F = 0, DZrznlF = 0 . . . . . Dl-lZnlF = 0, and the f i r s t group of conditions m a k e s it p o s s i b l e to d e t e r m i n e the function Zn(t, x) uniquely f r o m the r e l a t i o n (4.4) if the right side in (4.4) is known.] The unknown functions cu(t) a r e d e t e r m i n e d f r o m the following s y s t e m of nonlinear o r d i n a r y differential equations: 0Un ( - - 1y < --57"' v~ ) + ( - - 1)~ < A (t, x, D'u.), D~v. > = < h, v~ ) ,

c.(0)=0,

~----1,2..... n ~

(4.5) (4.6)

The solvability of the s y s t e m (4.5) under condition (4.6) follows f r o m L e m m a 1.1 of See. 1 of the p r e s e n t chapter. The following a p r i o r i e s t i m a t e holds for the solution Un(t , x):

[I Dt~,, Is'-'. [D'u,, [] ~/<"

(4.7)

[I D~zn [P-L ]Dtz~ ]] ~< K,

(4.8)

or the equivalent e s t i m a t e

w h e r e K > 0 is a constant depending on f(t, x) and h(t, x). To d e r i v e the e s t i m a t e s (4.7) and (4.8) we multiply Eq. (4.5) by % ( 0 , s u m on v f r o m 1 to n, and integrate on t f r o m 0 to T. Recalling Eq. (4.4) we obtain

(__ |)• I~_~t , D2r2~nlA1_(__ ])/t [A (t, x, DtUn), Dtz.] = lit, D~"zn]. F u r t h e r , i n a s m u c h as Un(t, x) = f(t, x) + zn(t, x), we find f r o m the l a s t equation that

(-'>' [2,I,

+ ( - " [% ""'.]

+ ( - - 1)' l A (t, x, D' u.), D'u.] + ( -- 1)'+' [A (t, x, D'u.), D t f ] = [h, D2"z.].

(4.9)

The definite t e r m s in (4.9) a r e the i n t e g r a l s ( - 1 ) k [ 0 z n / 0 t , D2rzn] and (-1)k[A(t, x, D/Un ), D/un], since ( - - 1 ) ' [ ~ - - , D " z n ] = ( Drz:, Drz,, ) , = r > O ,

(4.10)

while ( - - 1)~ [A (t, x, Dtu.), Dttt.] > ao [ I Dttt. Ip-', ! Dtu.

II- K

by the c o e r c i v e n e s s condition II. We now e s t i m a t e the r e m a i n d e r t e r m s in r e l a t i o n (4.9). Since

~ L p , (0, T; W~,k (a, b)), h (t, x)~Lr (0, T; tV;,k (a, b)), 0

o~'z.cL.(o, r; W~(a, 0), we have by the Sehwarz inequality

(4.11)

515

where e > 0 is a r b i t r a r y and K(e) = K(e, f, h) is a constant. In o r d e r to e s t i m a t e the t e r m [A(t, x, D/un), Dlf], we make use of condition I on the growth of the function A(t, x, }) in }, according to which

IA (t, x, 4)1 ~
IIA (L x, Dirt.), O'fll.. 0 is a r b i t r a r y . Considering inequalities (4.10)-(4.12), we obtain f r o m Eq. (4.9) the d e s i r e d inequalities (4.7) and (4.8) if e > 0 is chosen sufficiently small. F r o m the estimate (4.8) and the weak compactness of a bounded set in Lp we find that there exists a function z(t, x) E Lp(0, T; W/(a, b)) and a subsequence [we again denote it by Zn(t, x)] such that Zn(t, X) z ( t , X) weakly in Lp(0, T; W/p(a, b)). We shall show that u(t, x) = f(t, x) + z(t, x) is a solution of the original problem. We must prove that (Ou/0t) 6 Lp,(0, T; w - k ( a , b)), u(0, x) = go(x), andthat u(t, x) satisfies Eq. (4.1) in the sense of generalized func0 tions, i.e., for any function v(t, x) ~ Lp(0, T; wk(a, b)) we have the identity ( , 1 ) r [~, V]--I-[A(t, X, DttO, D%l=[h, v].

(.)

To this end we note f i r s t of all that A(t, x, D/un) E LpT(Q), and

l[ A (t, x, Dea.)llo,o, -. 0 does not depend on un(t, x). Therefore, it may be assumed that A(t, x, D/un) ~ a(t, x) weakly in Lpv(Q), where a(t, x) E Lp,(Q) is some function. F r o m relations (4.5) and (4.6) we demonstrate by closure in the usual way the validity of the identity 0 v]+(--l) '+' [z, 0~]+(--1)k[a(t, x), D'v] -----[h, v], ( - - ') r [~f,

(4.13)

0

where v(t, x) ~ Wi,,k (Q) (for the notation see Part1 ~ is an arbitrary function. P ,P It follows immediately from (4.13) that in the sense of generalized functions

=,,-(-,)'

x)

Hence (Su/0t) ~ Lp,(0, T; Wpk(a, b)). 0

F u r t h e r , since the values of v(0, x) of the function v(t, x) ~ W;,Tkp(Q) are a r b i t r a r y , f r o m (4.13) we see that z(0, x) = 0, i.e., u(0, x) = f(0, x) = go(x) by choice of f(t, x). Recalling what has been proved, we find that for any function v(t, x) v(t, x) G Lp(0, T; W (a, b)) there is the identity !) [y/, v j + ( - - ! ) ~ [a(t, x), D~vl=[h, v], 0

(4.14)

0

since Wp,l,k,p(Q) is dense in Lp(0, T; wk(a, b)). After this the existence of a solution will obviously have been proved if it is shown that Dka(t, x) = DkA(t, x, D/u). Assertion. Dka(t, x) = DkA(t, x, D/u). Proof. Let N -> 1 be an integer. We introduce the space N w,~ = {w(t,

x):D~,w= ~ % (0 v~ (x), % (0 ec ~"CO, r),

%(0)=0, w ] r = O ..... D~-lw Iv=O}

516

and w e note that r e l a t i o n s (4.5} and (4.6} i m p l y that

w h e r e w(t, x) ~ W N (N -< n} is an a r b i t r a r y function. We m a k e u s e of the m o n o t o n i c i t y condition III, a c c o r d i n g to w h i c h f o r any function v = f +w, (w 6 WN, N - n) w e h a v e the i n e q u a l i t y

Since u n - v - Zn - w E Wn, r e c a l l i n g (4.15) with w r e p l a c e d by u n - v, we obtain f r o m this the inequality

T h e l a s t inequality c o n t a i n s no n o n l i n e a r i t y in Un; t h e r e f o r e ,

as n -~ ~ we o b t a i n

Re{[h, D2rCu--e)] + ( " 1)r+, [ aOot D2~lu_v~ " "J~ +(-1)k§

Dry), D l ( a - ~ v ) ] } > 0 .

(4.16)

o

Since o b v i o u s l y D2r(u - v) ~ Lp(0, T; w k ( a , b)), a p p e a l i n g to the r e l a t i o n (4.13), f r o m (4.16) w e obtain the inequality

T h i s i n e q u a l i t y w a s obtained for functions v(t, x) = f(t, x) + w (t, x), w ~ W N, but s i n c e 0

~ W~ is d e n s e in N=~

0

Lp(0, T; Wpk(a, b)) it is valid f o r any function v(t, x) = f(t, x) + w(t, x), w h e r e D2rw(t, x) e Lp(0, T; wk(a, b)). In p a r t i c u l a r ,

s e t t i n g v(t, x) = u ( t , x) - ~w(t, x), w(t, x) ~ W (for the notation s e e P a r t 1~ Re {(--l)'~ 2

we obtain

D2~w]+(--l)'~[a(t,x)--A(t,x,D'~),Dtw]}>~O.

We divide both s i d e s of the inequality by ~ and let ~ - - + 0. We then obtain Re {(-- 1)* [a (t, x)-- A (t, x, Dlu), Dl~v]} >i 0. Since the function w(t, x) E W is a r b i t r a r y , this c a n be t r u e only in the e a s e w h e r e Dka(t, x) -- DkA(t, x, D l u), which we w e r e r e q u i r e d to show. T h e a s s e r t i o n has been p r o v e d , and h e n c e the e x i s t e n c e of a solution to p r o b l e m (4.1), (4.2) h a s b e e n e s t a b l i s h e d . U n i q u e n e s s of the Solution. We s h a l l show that the s o l u t i o n found is unique. Indeed, s u p p o s e that ul(t, x) and u2(t , x) a r e two s o l u t i o n s of p r o b l e m (4.1)-(4.3) in the s p a c e W(f). This m e a n s that f o r any function v(t, x) 0

Lp(0, T; w k ( a , b)) t h e r e a r e the i d e n t i t i e s (--I)' [~t', v]+(--1)*[A(t,x,

Dtut),D*~]=[ h, ~)1,

( - - 1 ) ' [-~t' ' v] + ( - - 1 ) k [a (t, x, D'u2), Dkv] = [tt, v], S u b t r a c t i n g the s e c o n d identity f r o m the f i r s t and s e t t i n g v = D 2 r ( u ~ - u2) ~ we obtain ( - - I ) ' [ 0~- ( u , - ~), D 2' (tit--u2)] + ( - - l)k[A (t, x, Dttt~)-- A (t, x' Dtu2), D' (ut--tt2)] = 0 . F r o m t h i s , on i n t e g r a t i n g by p a r t s (in the f i r s t t e r m ) , w e find that y1 < D~(uI--u2), D'(ut--u~) > t=r + ( - - 1)k [A (t, x, D~u~)--A (t, x, D~uo), D ~(u~--u2)] = 0 , i.e., Re {(-- 1)~ [A (t, x, D~u~)-- A (t, x, D~u2), D ~(u~ --us)]} < 0. In view of the s t r i c t m o n o t o n i e i t y of the o p e r a t o r DkA(t, x, D/u) this m e a n s that Dl(ul - u 2) -= 0, i.e., b e c a u s e

517

of the boundary conditions (4.3) u 1(t, x) - u 2(t, x), which we w e r e r e q u i r e d to prove. T h e o r e m 4.1 has thus been c o m p l e t e l y proved. 3 ~ Continuity of the I n v e r s e Operator. By T h e o r e m 1.1 of the preceding part an o p e r a t o r ~-1 (h) has been constructed which is i n v e r s e to the o p e r a t o r ~ (tt), i.e., to the o p e r a t o r of boundary-value problem (4.1)(4.3). Obviously,

~-, (h):L~, (0, r ; wT, ~ (a, b ) ) - ~ ( f ) . We shall d e m o n s t r a t e the continuity of this mapping. We a s s u m e that in place of the monotonicity condition III the s t r o n g e r condition of definiteness of the v a r i a t i o n is satisfied. Namely, IH'. F o r any functions u(t, x) E Lp(0, T; W ~ a , b)) and v(t, x) E Lp(0, T; w l ( a ,

b)) t h e r e is the inequality

Re {(-- 1)k [A (t, x, D*tt)--A (t, x, D~v), D l (u--v)]} > a 0 [I Dt (n--v)],-1, ]D t (u--v)It, w h e r e a 0 > 0 is a constant. THEOREM 4.2. L e t condition III' be satisfied. Then the i n v e r s e o p e r a t o r ~I-~ ('h) is continuous. Proof. We introduce the notation ttl=g1-1 (h~), u2.=!}[-1 (h2). Obviously, us(t , x) and u2(t, x) a r e solutions of p r o b l e m (4.1)-(4.3) with right sides hi(t, x) and h2(t, x), r e s p e c t i v e l y . J u s t as in the uniqueness t h e o r e m , we obtain for u s - u 2 (--1), [ O (tt,--~), D~' (tt,--tt2)] q-.(--1) * IA(t, x, D t t t l ) - - A ( t , x, Dttt2), D' (~,--u~)l= Int--n~, D~' (g,--u~)]

(4.17)

o r what is the s a m e < Dr (tt,--tt~), D' (tt,--tt.) } ,=r-k(--1)* [A (/, x, O t t t l ) - - A ( t , x, Dttt~), D t ( u l - - t t 2 ) ] = [ k l - - h ~ ,

D~' (tt,--tt2)].

(4.18)

Using condition HI' and the Schwarz inequality, f r o m (4.18) we find that a o []0 l (ttx--t/c)[[oP,~,< [Ihi --l/2 [[-k,t,' l[ D~'r (tQ --tt2) N~,p ~- Hh! -- h2 I[-~,p., [[D t (ttx--tt.o) I[0,p. Hence we obtain 1

1

]l Dl (ttl-- tt2) ][0,p~
oo m

(

o,

(4.19) We have

1), [~t (u, -- tt~), v] + ( - - 1)~ [A (t, x, O'tt~) - - A (t, x, Dirt2), D*v] = [hi - - lu.., v],

whence we find that q F 0 " lOT (u,--u,), VJ Dttz,)--A (t, x. Dta,}, Dkv] [+ I [h,--:-,h,, v]I ll_,.~ .-=~ p . ~ ~<sup v l[A(t'x' ti v [l,.p

' 0 i ar r

< It A (t, x, Dtu~) - - A (t, x, Dttt~) II0,p'+ II h, -- & tl-~.p'.

(4.20)

The inequalities (4.19) and (4.20) imply the continuity of the i n v e r s e o p e r a t o r 91-i (/z). Indeed, let hn(t, x) --" h(t, x) in Lp,(0, T; Wpk(a, b)), i.e., IIh - h n I[_k, p, -" 0 for n -" ~. F r o m inequality (4 19) it then follows that HDl(u - Un) It0. - - 0, i e , D/Un - - D/u in Lo(Q) F r o m condition I r e g a r d i n g the growth of A(L, x, ~) in ~ it follows that [A(t, x, D*u)l -< K(]Dtu W- + 1). This means that the c o n v e r g e n c e D / U n - D/u in Lp(Q) implies the c o n v e r g e n c e A(t, x, Dlun) --~ A(t, x, D/u) in Lp, (Q). Turning now to inequality (4.20), we find that

11

-< il

r x, D , = ) - A

D'..)t'o, + ,

0

Thus, I l u - u n l l w ~ 0 , w h i c h w e w e r e r e q u i r e d to prove. The proof of the t h e o r e m is comt R e m a r k to T h e o r e m 4.2. If A(t, x, ~) is differentiable in ~ w i t h I OA(t, x, ~ ) / ~ 1 --- K(I ~ Ip-2 + 1), then t h e r e is the inequality

518

hi(t, X), h2(t, X) a r e a r b i t r a r y functions in Lp,(O, T; Wp~(a, depends boundedly on ha(t, x) and h2(t, x)].

b)) and K > 0 is a constant [which possibly

where

Proof. I f the conditions stated in the r e m a r k a r e s a t i s f i e d , then

IIA (t, x, Dtth)--

A (t,

X,

-

O~

o,p"

(p--1)(p--2)

..
7

(p--1)(p--2)

+[IDtu~[Io-~p ~

p--1

§

whence, using inequality (4.19) and e s t i m a t e (4.7), we find that I

IIA (t, x, Otth)-- A (t, x,

Dlu2)1]0,,, ~
F r o m this inequality and a l s o inequalities (4.19) and (4.20) the d e s i r e d inequality (4.21) follows. This is what we w e r e r e q u i r e d to prove. We p r e s e n t the s i m p l e s t e x a m p l e which i l l u s t r a t e s the t h e o r e m just proved. E x a m p l e . Let (u) ~ ( - - 1)' - ~ - + ( - - I)'D (I DZtt [~-2Dlu) = h (t, x)

(k~
(u) ~-(- 1)" ~---~4 !D2"~tt Ip-2D2mtt p (Q) (with the conditions of the f i r s t b o u n d a r y - v a l u e r e a l i z e s a h o m e o m o r p h i s m of the Sobolev s p a c e s Wp, 1,2m problem) and Lp, (Q). 4 ~ The C a s e k -> l. If k -> l, then in the r e c t a n g l e Q it is r e q u i r e d to find a solution of the p r o b l e m O~ ~ ( u ) = ( - - 1 ) r -~y-[D A (t, x, Dtu)--h (t, x) ( 2 r = k - - l ) ,

(4.22)

tt (0, x) = + (x),

(4.23)

/t Iv = f0 (7),.--, Dt-au Ir=ft-1 (7).

(4.24)

A solution of p r o b l e m (4.22)-(4.24) is sought in the s p a c e Wt(f), whichby definition c o n s i s t s of functions of the f o r m

u (t, x) = f (t, x ) + z (t, x), 0

w h e r e z(t, x) = D2rv(t, x) and v(t, x) (: Lp(0, T; Wpk(a, b), (0v/~)t) E Lp,(0, T; Wpk(a, b)), v(0, x) = 0. H e r e f(t, X) is an extension of the initial and boundary conditions (4.23), (4.24) which has finite n o r m s II f llk,p and

l /0q_k,p,.

THEOREM 4.3. If the conditions I - I I I of T h e o r e m 4.1 a r e s a t i s f i e d , then for any right side h(t, x) E Lp,(0, T; w - k ( a , b)) t h e r e e x i s t s a unique solution u(t, x) EWI(f) of p r o b l e m (4.22)-(4.24). Ok Proof. We choose a s y s t e m of s m o o t h functions v~(x), v2(x) . . . . which is c o m p l e t e in Wp(a, b). We s e e k an a p p r o x i m a t e solution Un(t , x) in the f o r m Un(t, x) = f(t, x) + D2rzn(t , x), w h e r e zn(t , x) = clv~(x) + . . . + CnVn(X). The unknown functions c v - cu(t) a r e d e t e r m i n e d f r o m the s y s t e m of nonlinear o r d i n a r y differential equations

Ou, v~ ) + ( - - 1 ) k ( A ( t , x , Dtun),D~v~ ) = ( h, v~ ) , (--1)~ ( -bT-' c~(0)=0,

v = 1. . . . . n.

(4.25) (4.26)

The solvability of s y s t e m (4.25), (4.26) follows f r o m L e m m a 1.1 of the p r e s e n t c h a p t e r , and t h e r e is the a priori estimate

[l Dtu,,] p-~, I D%~I] ~
(4.27)

519

o r the equivalent e s t i m a t e

(4.28)

I I D*z,l p-i, 1D%l ] < K, w h e r e K > 0 depends on f(t, x) and h(t, x).

To deduce e s t i m a t e s (4.27), (4.28) we multiply the relations (4.25) by e v (t), sum o v e r v f r o m i to n, and i n t e g r a t e o v e r t f r o m 0 to T. We then obtain

,~,pOu,, - "~ L--~' z"] -t'(-1)k[A (t' x,

Dltt~), D~z,]=lh,

z,]

o r , what is the s a m e , ( - - 1 ) ' [OS L0t L--D 2 , OZ, 0~' z~l+(--1)k[A(t' X, Dttt~), Dttt,,--Dk.fl=[h, z,]. F u r t h e r , integrating in the second t e r m o v e r x r t i m e s and taking account of condition II, we find that

[r). o,. D%] + ao[[ D%?-', I D%II ~<[4 + (-- 1).+' ~0y, z.] + (-- t)'[A if, x, D%), Dk/I+ Sf.

(4.29)

Since h + ( - 1 ) r + l o f / a t E Lp,(0, T; W ~ ( a , b)), by the Schwarz inequality

I[h+(- 1)r+l 0/

::]l-.
w h e r e e > 0 is an a r b i t r a r y constant mad C(e) > O. F u r t h e r , since I A(t, x , ~)1 -< K(i ~ Ip-i + 1), it follows that

I [A (t, x, Dtttn), Dkil I< s( [ t DZa.l"-' + 1, i D,$ tI < K IID%llgT/.ll Dkf II0,p+ K [i, I O~f I] ~<9 IID'u,ll0",p+ c (0 I10~fllg, p + K [1, I D~$ I], w h e r e again e > 0 is a r b i t r a r y . Finally, we note that

D r -~-, elzn u"r z~j"1= T 1 <' . D'zn, D'z~ ) tffir > O. Considering all these inequalities and choosing a > 0 sufficiently small, f r o m inequality (4.29) we deduce the d e s i r e d e s t i m a t e s (4.27) and (4.28). Ok It follows f r o m these e s t i m a t e s that t h e r e exists a function z(t, x) ~ Lp(0, T; Wp(a, b)) and a subsequence 0 Zs(t, x) such that zs(t, x) --" z(t, x) weakly in Lp(0, T; w k ( a , b)). This implies that the sequence us(t, x) = f(t, x) + 0 D2rzs(t, x) c o n v e r g e s to the function u(t, x) = fit, x) + D2rz(t, x) weakly in Lp(0, T; W/(a, b)). This function will be the d e s i r e d solution. T o , r o v e this, as in the p r o o f of T h e o r e m 4.1, it is n e c e s s a r y to establish the following: 1) a u / ~ t ~ Lp,(0, T; Wpl'(a, b)); 2) u(0, x) = go(x); 3) u(t, x) satisfies the equation (4.22), i.e., for any funeN.

tion v(t, x) ~ Lp(0, T; ~V~(a, b)) the following equality holds: (-- 1)r [ ~-~, v] q- (-- 1)* [A (t, x, OttO, D k v I = [n, v]. The proof of all these a s s e r t i o n s differs in no e s s e n t i a l way f r o m the proof of analogous a s s e r t i o n s in P a r e 1 ~ mad is t h e r e f o r e not presented. We only indicate one fact that simplifies the argument. This is that in the proof of the equality Dka(t, x) - DkA(t, x, D/u) it is possible to choose as functions v(t, x) functions of the f o r m v(t, x) = f(t, x) + D2rw(t, x), w h e r e w(t, x) = al(t)vl(x) + . . . + aN(t)VN(X) (N -> 1 is any number). The existence of a solution has thus been established. Uniqueness. We shall show that the solution found of p r o b l e m (4.22)-(4.24) is unique. Indeed, suppose t h e r e a r e two solutions ul(t , x) and u2(t , x) in the space Wi(f). This means that for any 0

function v(t, x) E Lp(0, T; Wpk(a, b)) the following equalities hold: (-- 1)' [ ~ / , ~] + (-- l) ~ [A (t, x, Dtttl), Dkv]

I4,

-- 1~, [ Ou, v ] + ( - - 1)k [A (t, x, Dttto), D*v]=[h, v]. "~ L at ' 520

F r o m this we have (--1) r [ ~ (a,--~2), v ] + ( - - 1 ) ~ t A r x, Dta~) - - A (t, x, D%), D~v]=0

(4.30)

0 for any function v(t, x) ~ Lp(0, T; w kl y( a , b)).

0 Since ul(t, x) and u2(t, x) belong to Wl(f), t h e r e exist functions zl(t , x) e Lp(0, T; w k ( a , b)) and z2(t , x) 6 0 Lp(0, T; w k ( a , b)) such that ul(t, x) = fit, x) + D2rzl(t, x), u2(t, x) = fit, x) + D2rz2(t, x). In (4.30) we set v - z 1 - z2; then D2rv = u 1 - u2, and we obtain (__ 1)r [D2, ~/0 (zl--z2), zl--ze] + (-- 1)~ [A (t, x, Dial) ~ A (t, x, Dial), D~ ( u l - ~)]. = 0. This, as in P a r t l ~ implies that ul(t, x) - u2(t, x), which we were r e q u i r e d to prove.

It has thus been established that the o p e r a t o r ~ (u) has a unique i n v e r s e o p e r a t o r ~-~ (h). The t h e o r e m has been completely proved. Regarding the continuity of the o p e r a t o r 9~-~(h) we have the following result. THEOREM 4.4. If the condition (-- 1)~Re [A (t, x, Dttt)--A (t, x, D~v), D z ( a - - v ) ] > ao [I Dt ( u _ v ) [p-i, [D t (tt--v)j] is satisfied, where a 0 > O, u(t, x) e Lp(O, T; W/(a, b)), v(t, x) E Lp(O, T; W/(a, b)), then the i n v e r s e o p e r a t o r 9l-1 (h} is continuous. In other w o r d s , the mapping

~, : W, (f) ~ Lp, (0, T; W-~3 (a, b)) is a h o m e o m o r p h i s m . T h e o r e m 4.4 is proved in exactly the s a m e way as T h e o r e m 4.2. R e m a r k 4.1. We c o n s i d e r the l i n e a r equation

(-- 1)~-~i-t + D 2 ~ u = h ( t , x).

(4.31)

Since D2mu can be r e p r e s e n t e d in the f o r m D2mu - Dk(D2m-ku), where k (0 -< k -< 2m) is a r b i t r a r y , for Eq. (4.31) it is possible to a s s i g n any boundary conditions of the type (4.3) when k -< m by r e q u i r i n g additionMly Ok that the solution have the f o r m u = f + D2rz, where z E Lp(0, T; Wp(a, b)). If we denote by D-lu a p r i m i t i v e of the function u(t, x), then in the case k -> m the l a t t e r r e q u i r e m e n t can be t r e a t e d as assigning to the boundary conditions

u]v = f o (~) . . . . . Dt-lulr = f t-1 (7) k - m additional conditions of the type

D -2(k-m) ( f - - u ) l r = O . . . . . D-(k+~+l) ( / - - u) [ r = 0 . 5 ~ A R e m a r k on Multidimensional P r o b l e m s . It is obvious that the r e s u l t s obtained in P a r t s 2 ~ and 3 ~ can be c a r r i e d over to nonlinear parabolic equations with a multidimensional spatial part if the symbol D is f o r mally r e p l a c e d by an elliptic o p e r a t o r A with cokernel zero. We c o n s i d e r , e.g., in the cylinder Q = [0, T] • G, G c Rn with l a t e r a l s u r f a c e S the following problem: ~ ( t t ) - - ( - - 1)~ 0~ + [ hmu[p-2Am~=h(t' x), u(O,x)=O,

D~

[~ol~
(4.32) (4.33)

THEOREM 4.4. The o p e r a t o r ~(u) maps the space Wp, p~'2m(Q) [with conditions (4.33)] onto the space LpT(Q) homeomorphically. The proof of this t h e o r e m is c a r r i e d out in exactly the same way as in the one-dimensional case, onlythe a p p r o x i m a t e solutions uk(t, x) a r e found f r o m the r e l a t i o n k

Amu (t, x) =

(t, x ) 521

u n d e r the conditions DWUkIS = 0, I ~I -< m -

I.

We r e m a r k in c o n c l u s i o n that analogous r e s u l t s hold for nondivergenee equations on the plane of g e n e r a l f o r m (cf. Dubinskii [20]). 5.

Nonlinear

Semigroups.

Komura's

Theory

In this s e c t i o n we study s e m i g r o u p s of nonlinear o p e r a t o r s Tt(u) in a H i l b e r t s p a c e H and t h e i r r e l a t i o n to nonlinear monotone o p e r a t o r s . In the c a s e of l i n e a r s e m i g r o u p s this r e l a t i o n is given by the well-known t h e o r e m of H i l l e - Y o s i d e - Phillips (el., e.g., Yoside [30]). H e r e an analogous nonlinear t h e o r y is c o n s t r u c t e d . H e r e the d e c i s i v e f e a t u r e in the c o n s t r u c t i o n of a c o m p l e t e theory is c o n s i d e r a t i o n of multivalued monotone o p e r a t o r s , which was f i r s t u n d e r t a k e n in the w o r k of K o m u r a [81]. 1~ Definitions and F o r m u l a t i o n of the P r o b l e m . L e t H be a H i l b e r t s p a c e , and l e t Tt(u) (t -> 0) be a f a m i l y of nonlinear o p e r a t o r s in H with c o m m o n d o m a i n D(Tt). Definition 5.1. The f a m i l y of nonlinear o p e r a t o r s Tt(u) f o r m s a s e m i g r o u p if the following conditions a r e satisfied: 1) Tt+ s (u) = Tt(Ts(u)), w h e r e u E D(Tt); t and s a r e a r b i t r a r y nonnegative n u m b e r s ; 2)

limTt(u)=Ts(U)

for any e l e m e n t u E D(Tt).

t--*8

Definition 5.2. The s e m i g r o u p Tt(u) is called c o n t r a c t i v e if for any e l e m e n t s u E D(Tt), v ~ D(T t) and any t_>0

!t T, (u)-- re (~o)II~< IIu-v llDefinition 5.3. O p e r a t o r A(u), defined by the f o r m u l a A (g)~ lim Th(u)--u h /~-*0

(5.1)

is called the operator generating the semigroup Tt(u). Domain D(A) consists of all those elements u E H for which the limit (5.1) exists. We shall study two questions: 1. Given a nonlinear operator A(u), what properties must it possess in order to generate a contractive semigroup Tt(u)? 2. Given a contractive semigroup Tt(u), what is its generator like? In the linear case the Hille-Yosida-Phillips theorem gives the answer to these questions: An operator A generates a contractive semigroup if and only if: a) The operator A is dissipative, i.e., for any u ED(A) l~e
Re<0. In other words, we call the operator A(u) dissipative if the operator -A is monotone. 2~ Construction of a Nonlinear Semigroup_. There exists the following theorem on the construction of a nonlinear semigroup on the basis of a given operator A. THEOREM 5.1. Suppose A(u) is a dissipative (in general, nonlinear) operator with R(I - A) = H. Then for any element u0 E D(A) there exists a unique solution of the problem

flu(t) ~--ACaCt)), t > 0 , dt u(o)=u0.

522

(5.2)

(5.3)

By a s o l u t i o n of p r o b l e m (5.2), (5.3) we m e a n a function u(t) s a t i s f y i n g Eq. (5.2) and p o s s e s s i n g the following p r o p e r t i e s : 1) u(t) is s t r o n g l y continuous in H with r e s p e c t to t; 2) u(t) ~D(A) f o r a n y t - > 0; 3) u(t) is w e a k l y d i f f e r e n t i a b l e in H; 4) A(u(t)) is w e a k l y continuous in H and ]1A(u(t))II is n o n i n c r e a s i n g . the c o n d i t i o n (5.3).

The f u n c t i o n u(t) m o r e o v e r s a t i s f i e s

R e m a r k 5.1. It is not h a r d to s e e that in the H i l b e r t s p a c e H any w e a k l y continuous function v(t) with noni n c r e a s i n g n o r m II v(t)tl is s t r o n g l y c o n t i n u o u s f r o m the right. This and (5.2) i m p l y that the function u(t) is s t r o n g l y d i f f e r e n t i a b l e f r o m the r i g h t f o r all t -> 0, and

D+u (t) = A (u (t)) for all t -> 0 (D + is the r i g h t d e r i v a t i v e ) . M o r e o v e r , the c o n d i t i o n IIA(u(t))II -< tt A(u 0) II i m p l i e s that the function u(t) s a t i s f i e s a L i p s c h i t z condition. Therefore,

du (t) dt=

A (a(t))

a l m o s t e v e r y w h e r e in the s t r o n g s e n s e . R e m a r k 5.2. We s e t Tt(u 0) --- u(t), w h e r e u(t) is the s o l u t i o n of p r o b l e m (5.2), (5.3). By the a s s e r t i o n of T h e o r e m 5.1 the f a m i l y of t r a n s l a t i o n o p e r a t o r s Tt(u 0) f o r m s a n o n l i n e a r s e m i g r o u p (which is c o n t r a c t i v e , as will be s e e n below), and lira Th (ao)--uo h~o h = A (u0). The p r o o f of T h e o r e m 5.1 will be c a r r i e d out in s e v e r a l s t e p s . Step 1 (study of Rh, c o n s t r u c t i o n of a p p r o x i m a t e solutions). We denote by R)~ the o p e r a t o r Rh = ff - ~ ) - 1 . We have the following r e s u l t . L E M M A 5.1. The o p e r a t o r R?~ is defined f o r any ~ > 0, s a t i s f i e s a L i p s c h i t z condition, and IIRhllLi p _< 1. P r o o f . It is obvious that (I - hA) = ?~(~I - A), w h e r e ~ = h -1. It t h e r e f o r e s u f f i c e s Lo p r o v e the i n v e r t i b i l i t y on the e n t i r e s p a c e H of the o p e r a t o r (~I - A) f o r ~ > 0. We note f i r s t of all that f o r ~ = 1 the o p e r a t o r I - A is i n v e r t i b l e . I n d e e d , s i n c e the o p e r a t o r A is d i s s i p a t i v e Re ( ( ! - - A ) ( u ) - - ( I - - A ) ( v ) , u - - v > - ~ l [ u - - v I [ 2 - - R e

( A ( u ) - - A (v), u - - v } >[[ u--vii2.

This i m p l i e s that the o p e r a t o r R i - (I - A) -1 e x i s t s and l] R~ IILip _< 1. We now have

(~I - - A)(a) ~ v, v E H ,

(5.4)

o r , w h a t is the s a m e , (I -- A)(a)

~- (r - -

1) u = v ,

u = R , [v -- 0 --1) .]. We c o n s i d e r the o p e r a t o r

T (,)= R1 [v--(~-- 1) u]. O b v i o u s l y , T(u) is a c o n t r a c t i o n o p e r a t o r if 1~ - 11 < 1, i.e., if I~ - II < 1 it has a fixed point. This i m p l i e s the s o l v a b i l i t y of Eq. (5.4) in the i n t e r v a l I$ - If < 1. R e p l a c i n g now R 1 by R~ f o r I~ - lU < 1, w e obtain the a s s e r t i o n of the l e m m a f o r ]~ - 11 < 2, etc. The e s t i m a t e ft R)~IILip < 1 is obvious. The p r o o f of the l e m m a is complete. We c o n s i d e r the o p e r a t o r Ah -= A(I - XA) -1 - A R x , which p l a y s an e s s e n t i a l r o l e below. L E M M A 5.2. The o p e r a t o r AX s a t i s f i e s a L i p s c h i t z condition and is d i s s i p a t i v e .

523

Proof. We obviously have Ax = -- ),-' ( I - - Rx), whence the Lipschitz condition follows immediately. Further,

Re
au~ (t) at = Ax (ux (t)), ux (0)= ricED (A).

(5.5)

(5.6)

Since the operator A~ satisfies a Lipschitz condition, problem (5.5), (5.6) has a unique solution. For what followswe observe that if u/,(t) and uA(t) are two solutions of Eq. (5.5), then J]u~(t) -u~(t)I[ is nonincreasing, i.e.,

Ilu~ (t) -~x (t)l[ -.< i!u~ (s)- }7~ (s)l[

(,)

for t_> s. The inequality (,) follows i m m e d i a t e l y f r o m the condition that the o p e r a t o r A~(u) is dissipative (see also the p r o o f below of the uniqueness of the solution in P a r t 3~ Step 2 (estimates, compactness). F o r solutions u},(t) of p r o b l e m (5.5), (5.6) we have the e s t i m a t e s IIAx(u~ (t))[] < ]IA (u~ (t))iI < iIA (u0)ll,

(5.7)

Proof. We o b s e r v e f i r s t of all that for any u E D(A) t h e r e exist the inequality [lAx(u)[]4 IIA (u)]!.

(5.9)

Indeed, ]lAx (u)fl= X-1 Ilu" R~ (u)tl- x-, ItR~ (u q, ZAx (u)) -- Rz (u)]l< IIA~(u)ii. We now make use of inequality (*), setting tx~(t) = uh(t + s); we obtain 1

q- [lu~(t q-s) --ux (011~< ~ llux (s)--ux (0)lI. Using (5.9), we find f r o m this as s - - 0 that

t clux (t)

au~ (0)

--77-- ~<[ ~

I = IIA~(ux (0))11-.
Inequality (5.8) and hence also inequality (5.7) have been proved. The p r o o f of the l e m m a is complete. The e s t i m a t e s obtained a r e , however, inadequate for passing to the limit as X -- 0, since AT, is a nonlinear o p e r a t o r . The possibility of passing to the limit is contained in the next lemma. LEMMA 5.4. F o r any t -> 0 the sequence ux(t) converges strongly in H to some function u(t). The conv e r g e n c e u},(t) --u(t) is h e r e b y u n i f o r m in t on any compact set K c RI+. P r o o f . We have d

-dllux(t)--u~(t)[I 2=2Re<

du x (t)

at

du~ (t)

at

,ux(t)--u~(t) ) = 2 R e < Ax(ux(t))--A~(u~. (t)), u~(t)--u~(t) >

=---2 Re + R~.t, where

524

(5.10)

F u r t h e r , it is not h a r d to s e e that II Rx,~. [[ ~<4 tl A (u0) II (ll Rx (uz) - - a~l[ + i! R~ (ui) - u~ II) --< S (~ + l) II A (u0) 1!2, s i n c e by L e m m a 5.4 II Rz (u~) - u~ I[= ), ]] A~ (uz)!I ~< ~ I[ A (u0)I[ [ s i m i l a r l y f o r R~ (u~)]. Since the o p e r a t o r A is d i s s i p a t i v e , f r o m inequality (5.10) we find that d d--t-l[u~ (t)--u~ (t)[I2 ~< s, w h e r e e > 0 is a r b i t r a r y ,

and X > 0 and ~ > 0 a r e s u f f i c i e n t l y s m a l l .

Hence

Hux( t ) - - u i (t)t[ 2 < ~t, f r o m w h i c h the a s s e r t i o n of the l e m m a follows. The p r o o f of the l e m m a is c o m p l e t e . Step 3 ( p a s s a g e to the limit). H e r e we let h ~ + 0 and e s t a b l i s h that the l i m i t f u n c t i o n is a s o l u t i o n of the o r i g i n a l p r o b l e m (5.2), (5.3). We f i r s t i n t r o d u c e Definition 5.5. The o p e r a t o r A(u) is c a l l e d m a x i m a l if when the ineqaality

Re < A ( v ) - - w , v - - u > ~<0 is valid f o r e v e r y e l e m e n t v E D(A), then it follows that uED(A) and w =A(u). L E M M A 5.5. O p e r a t o r A(u) is m a x i m a l . P r o o f . Suppose that f o r e v e r y v E D(A)

Re ..<0. Subtracting the e x p r e s s i o n IIu - v II2 f r o m this inequality, we obtain

Re< A ( v ) - - v - - w + u , v--a> ~<0. I n this inequality w e s e t (A - I ) v - w

(5.11)

+ u = tG w h e r e t > 0 and ~ E H is an a r b i t r a r y e l e m e n t , i.e., we s e t

v = Rt (tr.+ w--u). F r o m (5.11) we then o b t a i n the inequality Re < t~,

RI(t~,H-w--u)--u

-...<0

o r , w h a t is the s a m e , Re < 5, Rt ( t S + c v - - u ) - - u

..<0.

L e t t i n g t - - + 0 , w e obtain Re < ;, R, (w - - u) -- u > -.
Since ~ E H is a r b i t r a r y , this m e a n s t h a t Rl(w - u) = u. The l a s t r e l a t i o n shows that u E D(A) and w = Ar p r o o f of the l e m m a is c o m p l e t e .

The

L E M M A 5.6. If ux(t) - - u ( t ) s t r o n g l y in H and A(uk(t)) ~ w w e a k l y in H, then u(t) E D(A) and w(t) = A(u(t)) (here t -> 0 is fixed and X ~ + 0). P r o o f . Since A(u) is d i s s i p a t i v e , we have Re < A (v) - - A (u~), v - - a~ > ~<0 f o r any h > 0. F r o m this w e obtain as ~ - - 0 the inequality Re< A ( v ) - - w , v - - u > ...<0. Since the o p e r a t o r A is m a x i m a l , w e find that u E D(A) a n d w =A(u). T h i s is w h a t w e w e r e r e q u i r e d to p r o v e . L E M M A 5.7. F o r any t -> 0 u(t) ~ D(A), and the function A(u(t)) is w e a k l y continuous in t.

525

Proof. Indeed, s i n c e IIR~(u h) - u)~tt -< Xll A(u 0) II, it follows that R},(u)~(t)) - - u x ( t ) s t r o n g l y in H locally u n i f o r m l y with r e s p e c t to t. F u r t h e r , the v a l u e s Ax(uh(t)) - A[RX(u)~(t))] a r e u n i f o r m l y bounded in n o r m for all X > 0 and t -> O. T h e r e f o r e , by L e m m a 5.6 for any fixed t > 0 and any s u b s e q u e n c e of ~ such that A)~(u)~(t)) c o n v e r g e s weakly to w(t) we find that u(t) E D(A) and w(t) = A(u(t)). In exactly the s a m e w a y , s i n c e u(t) is s t r o n g l y continuous and IfA(u(t))t] -< IIA(u 0) II, it follows that A(u(t)) is weakly continuous in t. The p r o o f of the 1 e m m a is complete. We shall now c o m p l e t e the p r o o f of the t h e o r e m . We m u s t e s t a b l i s h that the function u(t) is weakly diff e r e n t i a b l e f o r t > 0 and s a t i s f i e s the d i f f e r e n t i a l - o p e r a t o r equation

da(t)-~A(u(t)), dt

/>0.

(5.12)

We have f o r any X > 0 t 0

w h e n c e f o r X---0, using L e m m a s 5,4 and 5.6j we find that t

u(t)=Uo+IA

(u ('0) d':

0

[ p a s s a g e to the l i m i t u n d e r the i n t e g r a l sign is justified, s i n c e It Ah(uh(t)) II -- HA(u 0) tl]. T h e r e f o r e , u(t) is weakly d i f f e r e n t i a b l e and the r e l a t i o n (5.12) holds. T h e uniqueness of the solution found follows i m m e d i a t e l y f r o m the f a c t that the o p e r a t o r A(u) is d i s s i p a t i v e . M o r e o v e r , il A(u(t)) il -< IIA(u 0) II. In c o n c l u s i o n we o b s e r v e for t -> s the validity of the inequality Jl A(u(t))II -< ll A(u(s))II. Indeed, let Us(t) be the solution of Eq. (5.12) u n d e r the condition Us(S) = u(s). By the uniqueness of the solution we have us(t) - u(t) for t - s. Hence

II A (u (t))II = l[ A (us(t))!l--< li A (us(s))[i = I1A (u (s))If, which we w e r e r e q u i r e d to show. T h e t h e o r e m is c o m p l e t e l y proved. 3 ~ C o n s t r u c t i o n of a Nonlinear S e m i g r o u p in the C a s e of a Multivalued O p e r a t o r . In c o n t r a s t to the linear" e a s e , the t h e o r e m pr0ved in 2 5 does not a d m i t a d i r e c t c o n v e r s e . A t h e o r e m extending the previous r e s u l t s to the c a s e of multivaiued o p e r a t o r s A(u) does a d m i t a c o n v e r s e . Definition 5.6. A multivalued mapping A(u) (defined in a H i l b e r t s p a c e H) is called d i s s i p a t i v e if for any u ED(A) and v ~ D ( A ) and any gl EA(u) and g2 ~A(v)

Re ( gl--g~, u - - v )

~0.

Definition 5.7. The o p e r a t o r A(u) is called m a x i m a l if Re ( g l - - w , v - - u > ~ 0 , w h e r e v E D(A), gl E A(v) a r e a r b i t r a r y i m p l i e s that u E D(A) and w E A(u). The folIowing r e s u l t is e s t a b l i s h e d as in the p r o o f of L e m m a 5.6. LEMMA 5.8. If uh --" u s t r o n g l y in H and gx - - w weakly in tI, w h e r e gx E A(ux) , then u E D(A) and w E A(u). Definition 5.8. A s i n g l e - v a l u e d o p e r a t o r A 0 : D(A) - - H is called the canonical r e s t r i c t i o n of the o p e r a t o r A if A 0 ( ~ ) ~ "/~Jt

w h e r e w E A(u) and is such that H~[I--=- in[ IIg, ilR e m a r k 5.1. It can be shown that for a m a x i m a l d i s s i p a t i v e o p e r a t o r A the s e t A(u) is convex and closed. This i m p l i e s the e x i s t e n c e and univalence of the o p e r a t o r A 0. We shall now e s t a b l i s h the m a i n t h e o r e m of this section.

526

THEOREM 5.2. L e t A(u) be a d i s s i p a t i v e (in g e n e r a l , multivalued) o p e r a t o r with R(I - A) = H. Then for any u 0 E D(A) t h e r e e x i s t s a unique solution of the p r o b l e m

dd~) ~A(u(t)),

t >0,

(5.13)

u (0) = u0.

(5.14)

M o r e o v e r , for any t -> 0 the function u(t) is r i g h t - d i f f e r e n t i a b l e , and

D+u ( t) = Ao(u ( t)).

(5.15)

Proof. The f i r s t p a r t of the a s s e r t i o n of the t h e o r e m is e s t a b l i s h e d along the s a m e lines as in the c a s e of a s i n g l e - v a l u e d o p e r a t o r A(u). We shall note only d i f f e r e n c e s . Namely, we define the o p e r a t o r A?~(u) by the following formula: def

Ax(u) = A (I -- }~A)-'(a)~- - - ),-' [ ( I , XA) - - I] (I - - XA)-*(u) ----- - ).-~(u-- R~.(u)). We have thus c h o s e n a b r a n c h defined by the condition

(I - - XA) ([ --).A)-'(u) ~ u,

uEH.

The o p e r a t o r Ah(u) is defined on all of H, s a t i s f i e s a Lipschitz condition, and is d i s s i p a t i v e . The a p p r o x i m a t e solutions u~,(t) of p r o b l e m (5.13), (5.14) a r e defined as solutions of the p r o b l e m

~T . = ~ ( u ~ ( t ) ) ,

t>0,

(5.16)

u~ (0) = v.o-- kAo (u0),

(5.17)

w h e r e A 0 is the canonical r e s t r i c t i o n of the o p e r a t o r A. We a r g u e f u r t h e r as in the p r o o f of T h e o r e m 5.1, using L e m m a 5.8 in place of L e m m a 5.6. We shall now show that the solution of p r o b l e m (5.13), (5.14) is unique. Indeed, let u 1 and u 2 be two s o l u tions of the original p r o b l e m . We have

du,(O =gtEA(u~(t)), dt

ui(0)=uo,

du2 (t) at =g~EA (~ (t)), us (0) =uo. F r o m this we obtain

ddt t t u ~ _ u , - ll2__2Re ( d~, dt

au, u~--~)=--2Re(gt--g,,at--u2) dt '

~<0

by the fact that the o p e r a t o r A(u) is d i s s i p a t i v e . Hence

It ul ( t ) - u2 (t) tl ~< tl ul ( o ) - ~ (0)It = o, i.e., ul(t) - u2(t). Uniqueness has been d e m o n s t r a t e d . It r e m a i n s to e s t a b l i s h Eq. (5.15), f o r w h i c h it obviously suffices to p r o v e that

D*u (0) = Ao (u~).

(5. is)

To this end we c o n s i d e r the sequence ux(t) of solutions of p r o b l e m (5.16), (5.17). We have

lI A~ (u~ (t))ii ~< It Aa (u~ (0)) [l==II Ao (u0)it, since A~ (u~ (0)) • - - ~-' [u~ (0)-- R~ (u~ (0))] ------ - k-1 [u0-- ~A0 (u0)-- (I -- ~A)-1 ( I - - kAo) (u0)] ~--A0 (u~). This i m p l i e s that

h-, li u~.(t+h)--u~(t)lt==yt

A~(u~.(s))ds --
T h e r e f o r e , as h - - 0 we obtain for a l m o s t all t -> 0 the inequality

527

and hence a l s o the inequality t du (0 []~< 11A0 (u0)[[-

(5.19)

F u r t h e r , since u(t) E D(A), t h e r e e x i s t s a s u b s e q u e n c e t n - - 0 such that

lz (tn)fiD (A),

du (tn)

a----?---~" w weakly in

H.

Since [du(t)/dr] e A(u(t)) and the o p e r a t o r A(u) is m a x i m a l , it follows that w E A(u 0) and by (5.19) that l t w , -- itAo(u0)lt. On the o t h e r hand, by definition of the canonical o p e r a t o r A 0 we have the inequality []w II --I[ A0(u 0) I[. Hence [Iw II = I]Ao(uo)]1. T h e r e f o r e , w = Ao(u 0) and d u ( t n ) / d t - w strongly in H as t n ---0. By the uniqueness of the l i m i t we obtain d~ (t) dt ~"Ao (uo) as t ~ 0 along a s e t of full m e a s u r e . F r o m this we find t h a t h

h-,

(h)-

(0)l = h-1

et-

A0

0

This m e a n s that D+u(0) = A0(u0). The p r o o f of the t h e o r e m is complete. 4 ~ C o n v e r s e of the Main T h e o r e m . T h e o r e m 5.2 c a n be f o r m u l a t e d as follows: F o r any m a x i m a l , d i s s i pative o p e r a t o r A t h e r e e x i s t s a r e s t r i c t i o n A 0 (single-valued) g e n e r a t i n g a c o n t r a c t i v e s e m i g r o u p (we point out that the o p e r a t o r A 0 is d i s s i p a t i v e but not n e c e s s a r i l y m a x i m a l ) , H e r e we shall p r o v e the c o n v e r s e . THEOREM 5.3. L e t Tt(u) be a c o n t r a c t i v e s e m i g r o u p , and let A(u) be its g e n e r a t o r : We a s s u m e that the d o m a i n D(A) is nonempty. Then the o p e r a t o r A is d i s s i p a t i v e and a d m i t s a m a x i m a l , d i s s i p a t i v e extension (in g e n e r a l , multivalued), i.e., a d i s s i p a t i v e extension for which the o p e r a t o r R1 = ~ - ~ ) - i is defined on all of H and is bounded. H e r e A = A 0 [for all u E D(A)], i.e., A is the canonical r e s t r i c t i o n A 0 of the o p e r a t o r A, and the s e m i g r o u p T t is an extension of the s e m i g r o u p T t. Proof. We f i r s t p r o v e that A is a d i s s i p a t i v e o p e r a t o r . We have Re ( A ( u ) - - A (v),u--v ) - - R e lim < ra(u)--u h--~0

h

rh(~--v, u - - v ) = R e l~m~- ( Ta (u)--Ta(v)--(u--v), u - - v ) ~<0,

s i n c e the s e m i g r o u p T t is c o n t r a c t i v e . It cannot, h o w e v e r , be a s s e r t e d that the o p e r a t o r A(u) is m a x i m a l , i.e., R(I - A) = H [ m a x i m a l i t y in the s e n s e of Definition 5.5 and R ~ - A) = H a r e equivalent]. C o n s t r u c t i o n of the o p e r a t o r .~ c o n s i s t s in "lifting" the mapping (I - A) -1 defined on D(A) to a mapping .~)-1 defined on all of H [in other w o r d s , .~ is defined f r o m the condition of solvability of the equation u A(u) = v f o r any v E H]. (I -

F o r this we c o n s i d e r the a u x i l i a r y o p e r a t o r A h = h ' l ( T h - I). Obviously, A h -~ A for all u E D(A). It is a l s o c l e a r that A h is d i s s i p a t i v e . F u r t h e r R(I - A h) = H. Indeed, we c o n s i d e r the equation

u-- Aa (u)=v or

u+~u--h-Th or

(5.20) Since the o p e r a t o r

u-~ V ~h ~ + V~I Th(u)

528

is o b v i o u s l y a c o n t r a c t i o n o p e r a t o r with c o n s t a n t l(+h) -1 < 1, Eq. (5.20) is uniquely solvable. T h u s , the r e s o l v e n t R h = (I - Ah )-1 e x i s t s , and by the fact that A h is d i s s i p a t i v e the o p e r a t o r R h s a t i s f i e s a L i p s c h i t z c o n d i t i o n w i t h HRhlILi p -< 1. Definition 5.7. F o r any u E H w e s e t .4 (u) = {u-- v : u = w-I im (I - - Ah)-'v}, h~0

w h e r e w - l i r a d e n o t e s all p o s s i b l e w e a k l i m i t s . This is the d e s i r e d (multivalued) o p e r a t o r A(u). We s h a l l v e r i f y that .~(u) is d i s s i p a t i v e . L e t u 1 E D(.~), u 2 E D(~.) and v~ E .~(ul) , v 2 E . ~ ( U 2 ) o By definition of the o p e r a t o r A(u) we have u 1 - v 1 = w~, w h e r e ul = w - l i r a (I--Jib)-1 (w~). In e x a c t l y the s a m e w a y , u 2 - v 2 = w2, ~w-lim ( I - - A h ) -1 (w2). F u r t h e r , h~o h~0

Re < vl--v~, ul--g~. ) ~ R e < u l - - w l - - ~ - w 2 ,

u i - - ~ . ) ~ll u l - u ~ H"~-Re < w i - - w 2 , u l - - u_) ) -----[Iu l - - ~ f]~

- - R e w-lira < w l - w2, ( I - - A h ) -i ( w , ) , ( I - Ah)-1 (w~)) .

(5.21)

h--*0

We s e t Ulh = (I - Ah)-l(wl), U2h = (I - Ah)-l(w2). F r o m (5.21) w e then obtain the inequality Re ( vl - - v2, ul - - ~2 ) - - l l u l - - u 2 [ p - R e w - l i m < ( l - - A~)(u~) h-+0

--

(I - - Aa)(~a), ul~-- u~ ) ----]Iu l - - ~2 II 2- Re w-lira (ll u l a - ~2a [[~ h-~0

- -

< Aa(ulh)--Aa(~a),

ula--~2h > ~
uih--~_a ) .<~0,

which we w e r e r e q u i r e d to p r o v e . We now show that A(u) is a n e x t e n s i o n of the o p e r a t o r A(u) in the s e n s e that if u E D ( A ) , then A(u) E ,~(u). F o r this it m u s t be s h o w n t h a t f o r any e l e m e n t u E D(A) w e have the equality u--

A (u) =

v,

w h e r e u = w - l i m ( I - - A h ) -1 (v). We have h-~O

[l u-- (I -- Ah)-' (v) l[--=-!I

u - (I - A/z) -1 ([

-

-

A)(u) l[ < II q -

A,,)-'

[(~"- Ah)(u) -- ([ -- A)(u)l [[ ~ N(A - - Aa)(a)[1 -+ O,

as r e q u i r e d . It r e m a i n s to e s t a b l i s h the m a x i m a l i t y of the o p e r a t o r .~(u), i.e., to e s t a b l i s h the s o l v a b i l i t y o f the e q u a tion

u-2(u)=v

for any v ~ H. Since operator Ah is dissipative, the operator (I - Ah)-i satisfies a Lipsehitz condition which implies the boundedness of the norms If(I- Ah)-i(v)I[, since by assumption D(A) is nonempty. This means that there exists an element u such that u = v~-Iim (I - - A~)-' (v). h~0

This is the d e s i r e d e l e m e n t , s i n c e by definition of the o p e r a t o r .~(u) we have (I

-

~)(u) = v.

The o p e r a t o r A(u) has thus b e e n c o n s t r u c t e d . It is now obvious ~hat the s e m i g r o u p ~?t(u) g e n e r a t e d by the o p e r a t o r A(u) (cL T h e o r e m 5.2) is an e x t e n s i o n of the o r i g i n a l s e m i g r o u p Tt(u) , s i n c e f o r any u ~ D(A) c D(A) by the u n i q u e n e s s t h e o r e m ~?t(u) = Tt(u). F u r t h e r , f o r u ~ D(A) we have, on the one hand, d r , (~) = A ( T ~ (u)), dt

and, on the o t h e r hand, dt

529

This i m p l i e s that ,Z,0(u) = A(u), i.e., A is the canonical r e s t r i c t i o n of the o p e r a t o r ,~. In conclusion, we p r e s e n t an e x a m p l e which i l l u s t r a t e s the preceding r e s u l t s . On the axis R 1 we c o n s i d e r the s e m i g r o u p

Tt (u)

(max (0, u - - t), u > 0 ,

it,

tt~
It is not h a r d to v e r i f y that Tt(u) is a c o n t r a c t i v e s e m i g r o u p . F u r t h e r , D(A) = R i, and the g e n e r a t o r is defined b y the f o r m u l a

The extension ,~(u) c o n s t r u c t e d in T h e o r e m 5.3 is

X(u)---/[- 1, 01, u--0,u>~ [

0,

u
Indeed, a d i r e c t c o m p u t a t i o n shows that

[ u, u~h, whence

u=(I--An)-t

(v)=

Vl~_~ v~
v>l+k,

i.e., (I -- ~)-' (v)----~iom(I -- An)-' (v)

9 =t

~max (v-- I, 0), v > 0,

v,

v<0.

F r o m this we obtain the f o r m u l a for ~,(u). 5 ~ Applications to Nonlinear P a r a b o l i c P r o b l e m s . On the b a s i s of T h e o r e m 5.1 we obtain in this s e c t i o n a t h e o r e m on the solvability of the f i r s t b o u n d a r y - v a l u e p r o b l e m for a q u a s i l i n e a r p a r a b o l i c equation of o r d e r 2m. L e t G c R n be a d o m a i n with boundary F, and let Q = [0, T] x G be a cylinder with l a t e r a l s u r f a c e S =

[0, T] x F. In the c y l i n d e r Q w e c o n s i d e r the following p r o b l e m :

_$7+ (u)=h(t, x), (t, x)EQ, O%ls=0,

I~l~<m'l,

u O, x)=Uo (X).

(5.22) (5.23) (5.24)

In Eq. (5.22) the o p e r a t o r A(u) is a q u a s i l i n e a r elliptic o p e r a t o r of o r d e r 2m: l~l~m

The functions Ac~ (x, }y) a r e continuous and s a t i s f y the inequality l~l~m

w h e r e K > 0, p - 2 a r e s o m e constants. We c o n s i d e r the o p e r a t o r A(u) as an o p e r a t o r in the s p a c e L2(G) with d o m a i n 0

D (A) = {u (x):u (x) EW~ (O), A (u) EL2 (O)}. Definition 5.8. A function u(t, x) is called a solution of p r o b l e m (5.22)-(5.24) if it s a t i s f i e s the following conditions:

530

1) The mapping u(t, x) :[0, T] ~ L2(G) is continuous, i.e., the function u(t, x) is continuous into L2(G) as a function of the v a r i a b l e t; 9 0 2) u(t, x) ~D(A) for any t -> 0, i . e . , for any fixed t E [0, T] we have the inclusions u(t, x) E W~n(G), A(u) E L2(G); 3) the mapping u(t, x) : [0, T] ~ L2(G) is weakly continuously differentiable with r e s p e c t to t; 4) the mapping A(u)(t, x) : [0, 2"] ~ L2(G) is weakly continuous, and the n o r m il A(u)]IL2(G ) is n o n i n c r e a s i n g in t. M o r e o v e r , the function u(t, x) s a t i s f i e s Eq. (5.22) and the initial condition (5.24) [the boundary conditions 0 (5.23) a r e included in the fact that u(t, x) ~ W~(G) for any t]. A c o r o l l a r y of T h e o r e m 5.1 is the following result. 0 THEOREM 5.4. We a s s u m e that the o p e r a t o r A(u) as an o p e r a t o r f r o m W~(G) to wp,m(G) is c o e r c i v e and monotone (for the definitions s e e Chap. II, Sec. 3). Then t h e r e e x i s t s a unique solution u(t, x) of p r o b l e m (5.22)- (5.24). The p r o o f c o n s i s t s in a d i r e c t appeal to T h e o r e m 5.1 if we s e t H = L2(G) and c o n s i d e r the function u(t, x) as a function of the v a r i a b l e t taking values in H. Only the condition R(I - A) = H r e q u i r e s verification. The p r o p e r t y follows i m m e d i a t e l y f r o m the basic t h e o r e m of the t h e o r y of monotone o p e r a t o r s and, in p a r t i c u l a r , f r o m T h e o r e m 3.1 of Chap. II, since for p -> 2 0 we have w~n(G) c L2(G) and hence L2(G) ~ wp,m(G). CHAPTER

SOBOLEV OF

SPACES

OF

NONLINEAR

INFINITE

IV

ORDER

PROBLEMS

AND

WITH

ORDER

OF

THE

BEHAVIOR

UNBOUNDED

THE

GROWTH

OF OF

SOLUTIONS THE

EQUATION

Introduction Sobolev s p a c e s of infinite o r d e r

0

I

ro~

}

a r i s e n a t u r a l l y in c o n s i d e r i n g nonlinear equations of infinite o r d e r ; h e r e a s -- 0, Poz -> 1, r e _ i a r e n u m e r i c a l s e q u e n c e s . T h e s e s p a c e s c o r r e s p o n d to the D i r i c h l e t p r o b l e m of infinite o r d e r ~(--

t)I~ID~A~ (x, D~a) = h (x),

D~,lr=0,

]~[=0,1,.o.

(0.1) (0.2}

(it is n a t u r a l to call this p r o b l e m a C a u c h y - D i r i c h l e t problem). Obviously, it is f i r s t of all n e c e s s a r y to c l a r i f y under what conditions on a s , p a , and rc~ the s p a c e s ~V~ao~, p~} a r e nontrivial, i.e., contain at l e a s t one function not identically z e r o . This question is c o m p l e t e l y solved in Sec. 1, and the a n s w e r turns out to be v e r y c l o s e l y r e l a t e d to the 0oo c l a s s i c a l t h e o r y of q u a s i a n a l y t i c c l a s s e s of H a d a m a r d . Namely, the s p a c e W {aoz , p a } is nontrivial if and only if s o m e sequence MN > 0 uniquely d e t e r m i n e d by the n u m b e r s a a and p~ with I ~ I = N g e n e r a t e s a oned i m e n s i o n a l nonquasianalytic c l a s s . F u r t h e r , in Sec. 2 the nonlinear C a u c h y - D i r i c h l e t p r o b l e m (0.1), (0.2) is studied under the n a t u r a l c o n d i t i o n s 0o o of c o e r c i v i t y and monotonicity. Solvability of this p r o b l e m in the s p a c e W. {ac~, p a } is established. The method of p r o o f c o n s i s t s in c o n s i d e r i n g the D i r i c b l e t p r o b l e m for a n equation of o r d e r 2m, which is a p a r t i a l s u m of Eq. (0.1), a n d p a s s i n g to the l i m i t as m ~ ~.

531

The s o l v a b i l i t y of p r o b l e m (0.1), (0.2) is a v e r y s p e c i a l c a s e in the study of the l i m i t b e h a v i o r of nonl i n e a r b o u n d a r y - v a l u e p r o b l e m s of finite o r d e r as the o r d e r of the equation tends to infinity. Namely, in Sec. 3 we study the l i m i t b e h a v i o r of a solution of the p r o b l e m trl

Z (-- 1)I=[D~'A~ (X, OVum)= ttm (x), laI~o D'~u~ lr---_O; l~ l~
(0.3) (0.4)

w h e r e in a p a r t i c u l a r s e n s e Aam(X, ~/) - * A a ( x , ~7) as m - - ~. H e r e a g e n e r a l t h e o r e m on the e x i s t e n c e of a l i m i t of p r o b l e m (0.3), (0.4) is proved. It is h e r e b y found that the e a s e s of a l i m i t equation of infinite o r finite o r d e r a r e e s s e n t i a l l y different. While in the e a s e of a l i m i t equation of infinite o r d e r the condition of c o e r c i v e n e s s alone is sufficient for the e x i s t e n c e of a l i m i t point of the f a m i l y of solutions of p r o b l e m (0.3), (0.4) and the investigation is r e l a t i v e l y s i m p l e , in the c a s e of a l i m i t equation of finite o r d e r the additional condition of monotonieity is n e c e s s a r y ( f o r m a l l y due to the method of proof), and in this c a s e the justification of p a s s i n g to the l i m i t as m ~ ~ is c o n s i d e r a b l y m o r e c o m p l i c a t e d . H e r e full use is m a d e of the techniques of nonlinear monotone o p e r a t o r s , o r , m o r e p r e c i s e l y , the m o d i f i c a t i o n of t h e s e techniques to the c a s e of s p a c e s depending a l s o on a p a r a m e t e r m. In this connection we focus attention on the s i m p l e , but useful l e m m a 3.2 (a v e r s i o n of the c l a s s i c a l Fatou t h e o r e m ) , which, in our opinion, is of independent i n t e r e s t . F u r t h e r , in Sec. 4 on the b a s i s of the r e s u l t s and ideas of See. 3 we obtain a v e r y g e n e r a l t h e o r e m on the s o l v a b i l i t y of p r o b l e m (0.1), (0.2) u n d e r the condition of e o e r e i v i t y alone. In this r e g a r d nonlinear elliptic equations of infinite o r d e r a r e found to be akin to s y s t e m s of nonlinear a l g e b r a i c equations for the s o l v a b i l i t y of which, a s is known, a c o e r c i v i t y condition suffices and no monotonetype conditions a r e r e q u i r e d . At the end of this s e c t i o n we e s t a b l i s h the solvability of a mixed p r o b l e m for the nonlinear hyperbolic equation of infinite o r d e r

tt't + ~ (-- t)I~I D~A~(t, x, DVtt)~ h (t, x). lc~l=o

(0.5)

It is useful to note that the mixed p r o b l e m for an equation of hyperbolic type (0.5) of finite o r d e r is, g e n e r a l l y speaking, not s o l v a b l e in Sobolev s p a c e s . Finally, in the concluding s e c . 5 p e r i o d i c p r o b l e m s a r e c o n s i d e r e d f r o m a s i m i l a r point of view. N a m e l y , we e s t a b l i s h a c r i t e r i o n for the nontriviality of the s p a c e s ~r a a ' Pa } on the t o r u s T n and p r o v e a n u m b e r of t h e o r e m s on the b e h a v i o r of solutions of p e r i o d i c p r o b l e m s of o r d e r 2 m a s m --'- ~ which a r e analogous to those obtained in Sees. 2-4. 0 1. A C r i t e r i o n for the Nontriviality of the Spaces W~{aa, p~} L e t G ~ R n be a d o m a i n of finite m e a s u r e with boundary which we denote by F; l e t Co(G) be the s p a c e of infinitely d i f f e r e n t i a b l e functions u(x) which on F s a t i s f y the conditions DWuIF = 0, I wl = 0, 1 . . . . . We shall a s s u m e that the functions u(x) E C~~ have b e e n continued by z e r o to the e x t e r i o r of the d o m a i n G and call t h e s e functions c o m p a c t l y supported. Suppose f u r t h e r that a a - 0, Pa --> 1, r a - 1 a r e a r b i t r a r y n u m e r i c a l s e q u e n c e s (we a l s o a d m i t the c a s e r ~ = + ~o). We c o n s i d e r the s p a c e

W ~ {a~, p=} = u (x) EC~ (G) : O(u)-~

as ItD~u ll~:< o~ ,

I~zl=o

w h e r e II- lira denotes the n o r m in the L e b e s g u e s p a c e L r a (G).* I f a m o n g the n u m b e r s a s infinitely m a n y a r e n o n z e r o then the q u e s t i o n a r i s e s of the nontriviality of the s p a c e ~V~{aot, Poz }. 0oo Definition 1.1. The s p a c e W {a a , P a } is called nontrivial if t h e r e e x i s t s a function u(x) ~ 0, u(x) E C~(G) s u c h that p(u) < ,o. 0 In o r d e r to f o r m u l a t e a condition for the nontriviality of the s p a c e s W*~ P a } , we introduce a s e quence of positive n u m b e r s M N. Namely, we define MN, N = 0, 1 , . . . , a s a solution of the equation *As witl be evident below, the values r a a r e of no consequence and a r e t h e r e f o r e omitted in the notation.

532

a~M~~ - 1,

Z

(1.1)

l~t=N

s e t t i n g M N = + ~o if all ac~ = 0 f o r [ a ] = N. Obviously, the r e l a t i o n (1.1) uniquely defines the n u m b e r s M N. 0

T H E O R E M 1.1. The s p a c e W ~ { a a , Pa } is n o n t r i v i a l if and only if the s e q u e n c e {M N} defines a nonq u a s i a n a l y t i c c l a s s of functions of a s i n g l e r e a l v a r i a b l e . 0

P r o o f . To p r o v e n e c e s s i t y it o b v i o u s l y s u f f i c e s to show t h a t if u(x) E W ~ 1 7 6 Pa } and the s e q u e n c e {MN} d e f i n e s a q u a s i a n a l y t i c c l a s s , t h e n u(x) -= 0. I n d e e d , s i n c e u(x) E C~(G), f o r any a and ~ = ( ~ . . . . ~n) w e have

w h e r e u(~) is the F o u r i e r t r a n s f o r m of the f u n c t i o n u(x) and K > 0 is a c o n s t a n t d e p e n d i n g only on the m e a s u r e of the d o m a i n G. We s e t 8 = ~?0, w h e r e ~ ~ R t, 0 = (0t . . . . .

On), t 0j I -> 1 is fixed. F r o m (1.2) w e then obtain the inequality

[u(~o)l~[(t[m u[]~=, ]~]-~N. Since N is a r b i t r a r y and u(x) is c o m p a c t l y s u p p o r t e d , this i m p l i e s that f o r all n a t u r a l n u m b e r s N >- 2

(1-K'q2)I~IA'-21u('qo)]4KI!D%II~=,

I~[=g,

w h e r e K > 0 is a c o n s t a n t . * T h e r e f o r e ,

Icq=N

I~zl~Ar

0

Since u(x) E W ~ { a a ,

pa } it then follows that FLI~I't

and h e n c e f r o m the definition of the n u m b e r s M N (1 ~-~u)[ ~ izr

I u-(~0) I < KMN, g = 2 ,

3......

T h e s e i n e q u a l i t i e s i m p l y that f o r the function v(y), y E R t, w h i c h is the i n v e r s e F o u r i e r t r a n s f o r m of the f u n c t i o n ~(~ 0) on the v a r i a b l e ~, t h e r e a r e the i n e q u a l i t i e s [ DN-2~ (g)] ~ [ ( M y ,

N=2, 31 . . . ,

i . e . , the f u n c t i o n v(y) belongs to the H a d a m a r d c l a s s C { M N+2}, which b y a s s u m p t i o n is q u a s i a n a l y t i c . By the P a l e y - W i e n e r t h e o r e m v(y) E C oeo ( R 1) and h e n c e v(y) --- 0. F r o m this we find that on any line ~ = T0, w h e r e 0 = (01 . . . . . On), I 0j I -> 1, the function u(~) = 0 ; h e n c e u(~) = 0 i n R n, i.e., u(x) =- 0 in the d o m a i n G. This is w h a t w e needed to show. Sufficiency. T h e p r o o f of s u f f i c i e n c y is b a s e d on a w e l l - k n o w n l e m m a r e g a r d i n g c o m p a c t l y s u p p o r t e d functions of a s i n g l e r e a l v a r i a b l e (el. M a n d e l b r o i j t [38, p. 105]). L E M M A 1.1. L e t #0 = 1, # N > 0 (N = 1, 2 , . . .

) be a s e q u e n c e of n u m b e r s s a t i s f y i n g the c o n d i t i o n

~ , + ~ 2 + 9 9 9 + ~ N - t - 9 9 9 < ~-, a > 0 . T h e n t h e r e e x i s t s a f u n c t i o n v(r) E C~~

(1.3)

a), r E R t, w h i c h s a t i s f i e s the following c o n d i t i o n s :

1) v(0) = 1, D N v ( - a ) = DNv(a) = 0, N = 0, 1 . . . .

;

2) f o r any r E ( - a , a)

I DWv (r)] ~ (~o~ . . . PN) -1, N = O, 1 . . . . .

(1.4)

P r o o f . We c h o o s e a continuous function u0(r) s a t i s f y i n g the conditions: 1. 0 ~ uo(r) -

1,

2. u0(r) = I f o r r E

(-a/3, a/3),

3. uo(r) -- 0 f o r r E (--a, ' 2 / 3 a ) , r E (2/3a, a). * H e r e and below c o n s t a n t s w h o s e values a r e of no i n t e r e s t a r e denoted by the single l e t t e r K. 533

S u p p o s e , m o r e o v e r , that u0(r) is e v e n with r e s p e c t to the point r = 0. F u r t h e r , f o r m = 1, 2 . . . . f u n e t i o n s . u m ( r ) r e c u r r e n t l y by the f o r m u l a

we define

r-Him

um(r)=~

I a~,(t)dt. r--it m

Since condition (1.3) is s a t i s f i e d , for any m the functions Um(r) a r e c o m p a c t l y s u p p o r t e d in the i n t e r v a l ( - a , a); m o r e o v e r , u(0) = 1. It is a l s o c l e a r that the functions um(r) a r e even and a r e d i f f e r e n t i a b l e at l e a s t to o r d e r m. We shall show that the s e q u e n c e urn(r) ( s t r i c t l y s p e a k i n g , a f t e r c h o i c e of a s u b s e q u e n c e ) c o n v e r g e s as m - - = to s o m e function u(x) E C~(--a, a). To this end we f i r s t e s t a b l i s h the e s t i m a t e max a) I Dmum (r)] ~< (t~olh... t~m)-1, m = O, 1, .

,E(-a.

"'"

(1.5)

"

Indeed, f o r m = 0 the inequality (1.5) is obvious. F u r t h e r , f o r any n _ m w e have

D"u,. (r) = ~

[D*-'u,._l (r + ~ ) - - D"-'u,._I (r

l~m)],

(1.6)

w h e n c e f o r n = m, in p a r t i c u l a r , we obtain max r~(--a,a)

I Dmum (r) I-.< ..1-- max I Dm-'um_, (r)I. t~m r ~ ( - a , a )

This inequality p r o v e s the v a l i d i t y of the e s t i m a t e (1.5) if it is t r u e f o r m - 1. The e s t i m a t e (1.5) is thus t r u e f o r all m. Suppose now that n _< m - 1 is a r b i t r a r y . points r ~ , . . . , r m _ n s u c h that

Using Eq. (1.6), we then find that f o r any point r t h e r e e x i s t

Dnttm(r) = Dnttn_l (r O . . . . .

Dntt~ (rm_n).

H e n c e , a c c o r d i n g to inequality (1.5)

]Dnttm (r) l ~< max I Dntt~ (r) j ~<(~o~l . . . ~,)-'.

max r~(--a,a)

(1.7)

r~(--a,a)

F r o m this e s t i m a t e , u s i n g A r z e l a ' s t h e o r e m and a d i a g o n a l i z a t i o n p r o c e s s , w e find that s o m e s u b s e q u e n c e [we a g a i n d e n o t e it by Um(r) ] c o n v e r g e s u n i f o r m l y with all d e r i v a t i v e s to s o m e function u(r) E C ~ ( - a , a). Obv i o u s l y , u(r) is the d e s i r e d function; we r e m a r k only that inequality (1.4) follows f r o m inequality (1.7) as m ~. The p r o o f of the l e m m a is c o m p l e t e . We now c o n t i n u e the p r o o f of the t h e o r e m ; n a m e l y , on the b a s i s of the l e m m a we shall find a function u(x) E r~/~~ Per}. A c c o r d i n g to the c r i t e r i o n of n o n q u a s i a n a l y t i c i t y of M a n d e l b r o i j t - B a n g , {MN} s a t i s f i e s the c o n d i t i o n s 1

oo

ltmM~

the s e q u e n c e

Mr

~ 1 ~ .,V--1 < c~,

oo,

g--~oo

w h e r e M~g is a convex r e g u l a r i z a t i o n of the s e q u e n c e MN by m e a n s of l o g a r i t h m s .

L e t N o be s u c h that

,•e+

M~v_l a M~ < Y "

N--

1

It is obvious that t h e r e e x i s t p o s i t i v e n u m b e r s #1, ~2, 9 9 9 # N and a p o s i t i v e n u m b e r q < 1 s u c h that t~

I

~+"

+~.+

~

M~-I a M 7 <-X"

N=No-I-1

We set ~N

534

~ .~, qI Ml~lc

,

N=No+I

....

(1.8)

By c o n d i t i o n (1.8) and the l e m m a t h e r e e x i s t s a f u n c t i o n of a s i n g l e r e a l v a r i a b l e v(r) E C ~ ( - a , f i e s the e s t i m a t e (1.4). R e c a l l i n g the d e f i n i t i o n of the n u m b e r s # j , w e o b t a i n

a) w h i c h s a t i s -

l D% (r) l

(1.9)

w h e r e K > 0 i s a c o n s t a n t not d e p e n d i n g on N. Now i t is not h a r d to c o m p l e t e t h e p r o o f of the t h e o r e m . N a m e l y , l e t x ~ = (x ~. . . . . x~ be a n i n t e r i o r p o i n t of G, and l e t a > 0 b e s u c h t h a t t h e c u b e U(x~ a) = {x : Ix - x~ < a} l i e s e n t i r e l y i n the d o m a i n G. W e s e t 0

u(x) = v(x 1 - x ~ ) . . . V ( X n - X ~ [a[= Nwehave

w e s h a l l s h o w t h a t u(x) E W ~ { a a ,

pa}. I n d e e d , f r o m (1.9) f o r ce = ( a l . . . . .

an),

]D~tt (x) [~ O. S i n c e the s e q u e n c e M~q i s l o g a r i t h m i c a l l y c o n v e x , M e 1 . l o s s of g e n e r a l i t y t h a t M e = 1). Therefore,

.

.

M e n < M c( 2 1 + " - -

. .+Or n

:

M~T (we m a y a s s u m e w i t h no

t h e f u n c t i o n u(x) s a t i s f i e s the i n e q u a l i t i e s

]D=tt(x) !~
[:~!-- N,

w h e r e K > 0 is a c o n s t a n t , and h e n c e a l s o the i n e q u a l i t i e s

s i n c e M~q -< M N. R e c a l l i n g the d e f i n i t i o n of the n u m b e r s MN, f r o m t h i s w e find t h a t

N=0

[
N~0

[~zl=N

N=0

]ccl=N

N--0

w h e r e q < q0 < 1 and K1 > 0 i s a c o n s t a n t . 0

T h u s u(x) ~ W ~ { a ~ , 2.

Cauchy-Dirichlet

of

Infinite

Order

p~}, and the t h e o r e m is c o m p l e t e l y p r o v e d . Problem

(the

Case

for of

Nonlinear

a Monotone

Elliptic

Equations

Equation)

L e t G c R n be a b o u n d e d d o m a i n w i t h b o u n d a r y F. I n t h e d o m a i n G w e c o n s i d e r t h e C a u c h y - D i r i c h l e t p r o b l e m f o r t h e n o n l i n e a r e l l i p t i c e q u a t i o n of i n f i n i t e o r d e r :

L(tt)-- ~ (--1)I~ID~A~(x, DVtt)=h(x),

I-/]~<[~l,

D~ulr=O, ] ~ 1 = 0 , 1. . . . .

(2.1) (2.2)

H e r e Am (x, }y) a r e c o n t i n u o u s f u n c t i o n s o f t h e v a r i a b l e s x E G and a l l p o s s i b l e }~,, I y I - I a I. W e a s s u m e t h a t the f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d : a) F o r a n y x ~ G , ~y, and ~?a, w h e r e I~$ = m , [ y l -< m , m = 0, 1 . . . . .

[~[=tn

we have the inequalities

lal=m

w h e r e K > 0, a a > 0, Pa > 1, b m - 0 a r e c o n s t a n t s , and the s e q u e n c e o f P a is b o u n d e d , ~vl'dle b 1 + b 2 + . . . < ~; b) f o r a n y x E G, ~y, and C a , l a l = m , w e h a v e the c o e r c i v e n e s s i n e q u a l i t i e s

[~[=ra

w h e r e 5 > 0, 6 m > 0 a r e c o n s t a n t s w i t h 61 + 5 2 + . . .

leq=ra

< ~;

c) the c o n d i t i o n of s t r i c t m o n o t o n i c i t y i s s a t i s f i e d , i . e . , Re ~ ,

[A~(x, ~v)--A~(x, ~ v ) ] ( ~ = - - ~ ) > 0 ,

[~l=m

and is e q u a l to z e r o i f and only i f ~ ~ = ~ ,

I c~ I = m.

535

o

We m o r e o v e r a s s u m e that the n u m b e r s a a -> O, Pa > 1 a r e such that the s p a c e W ~ { a a ,

The space of right sides W-~~ 0

W*~

Pa} is nontrivial.

Pa}, P'a = Pa/(Pa - 1) is defined as the f o r m a l dual space to the space

Pa}. Namely, co

w h e r e ha (x) E Lp~(G) and

0

Pa,1 and W~176

[Hae duality of the spaces W-~176

pa} is by definition given by the relation

which is well defined, as is easily verified. Two elements hl(x) EW-~176 '

p~} and h2(x) EW-~176

p~} are

0

considered equal if for every function v(x) ~W~~ from condition a)

L

p~) the values (hl, -vY and (112,v) are equal. Obviously, O

uz

p3}-

0 Definition 2.1. A funetion u(x) E W { a a , Pa} is called a solution of the C a u c h y - D i r i c h l e t problem (2.1), 0Qo

(2.2) if for e v e r y function v(x) E W { aa, Pa}

(L(u),v)=(h,v). THEOREM 2.1. L e t conditions a)-c) be satisfied. Then for any right side h(x) E W - ~ 1 7 6 0oo e x i s t s a unique solution u(x) E W {a a , pa} of p r o b l e m (2.1), (2.2).

p~} t h e r e

Proof. We o o n s i d e r the D i r i c h l e t p r o b l e m f o r an equation of o r d e r 2 m which is a p a r t i a l s u m of the series" (2.1):

L2m(ttm)~ Z (-- 1~'I~'"D~A~ (x, Drum)=hm (x),

(2.4}

1~1=o D~U~]r~---O, ttol~O, 1..... m - - l ,

(2.5)

w h e r e hm (x) = ~ ( - - 1)1~ta~D~h~(x). In view of conditions a)-c) t h e r e exists a unique solution of this p r o b l e m t~i=0 0m --~ Um(X) E W~.:(G), p = { P a , {a{ _< m} (of. Chap. II). H e r e for the solution Um(X) we have the e s t i m a t e

I~{=o

2

{~i--o

'

w h e r e K > 0 is a constant. Using the S o b o l e v - K o n d r a s h e v imbedding t h e o r e m s and a diagonalization p r o c e s s , we find f r o m this that the sequence urn(x) (a p r i o r i , a subsequence) c o n v e r g e s uniformly in the domain G with all d e r i v a t i v e s to some 0 function* u(x)EC~(G). By (2.6} u(x) E W ~ { a a , Poz}, and 0(u) -~ KO'(h). We will show that u(x) is the d e s i r e d 0

solution; to this end we pass to the l i m i t as m ~ ~ in p r o b l e m (2.4), (2.5), taking v(x) E W~'{aa, p j .

Obviously,

lira (L(ttm),v) = l i m ( h re,v> = ( h , v ) *It is obvious that the a p p r o x i m a t e solutions urn(x) have continuously increasing, finite smoothness; t h e r e f o r e , in speaking about the c o n v e r g e n c e Um(X) --u{x) in C~~ we imply that Daum(X) - - D a u ( x ) uniformly in G beginning with some sufficiently l a r g e m depending on ~. This is intended below. 536

0

for any function v e W ~ 1 7 6

pm}. F u r t h e r , let r% be a fixed n u m b e r , and let m > m 0. We have m0

( L (u), v ) - - ( L2m(u~), v ) ~ __a~ ( A~ (x, lai~0

DVu)

--Ac,(x, DVum), D~v) + ~ (Ac~(x,D~'u),DC*v) - ~ ( A=(x,D~'um), Dr162 ]ai=mo+l

la[=rno+I

It is c l e a r that the middle t e r m is a r b i t r a r i l y s m a l l for sufficiently l a r g e m 0. F u r t h e r , using Young's inequality and the e s t i m a t e (2.6), we find (here it is e s s e n t i a l that Pa is a bounded sequence) that f o r any e > 0

12 (A~(x, DVttm),D''v ) ~<:2 ~[(zl~m0+l

~

a~!lD'~u,~[i~:+ K ( ~ )

Ir

~<,Kp'(h)+K(r

2 a~.!lD~vii~+K 2 ~,~

Iq.i= m,~+ 1

m=mo+l

~ a.[ID'~v '~:+K ~ a,~, K (~-)>0. [rz{~mo+l

m=mo+l

T h e r e f o r e , this e x p r e s s i o n is a l s o a r b i t r a r i l y s m a l l for sufficiently l a r g e m 0. Finally, for any fixed m 0 ra o

lira ~

(

As(x , D~u)--A~(x, D~'am),D~v )

-----0,

since urn(x) ~ u(x) u n i f o r m l y in G together with all d e r i v a t i v e s . 0

In s u m m a r y , we find that for any function v(x) EW~176 (L(u), v)

=lira

Pa}

(L2m(u~), v )

=

( h, v ) ,

rrt,-+oo

i.e., u(x) is a solution of p r o b l e m (2.1), (2.2). The uniqueness of the solution found follows i m m e d i a t e l y f r o m the monotonicity condition c). The p r o o f of the t h e o r e m is complete. E x a m p l e . We c o n s i d e r the p r o b l e m ~ , ] ( - - 1)I~-ID~(a~ [ D~'uf~-2D~u)= D~u]r-0,

h (x), xEG,

tool=0, 1. . . . .

0

I f the coefficients a r e such that the s p a c e W ~ { a a , p~} is nontrivial (for e x a m p l e , a a = [ ( 2 a ) ! ] - P , p > 1 in the c a s e d i m G = 1), then all the conditions of T h e o r e m 2.1 a r e satisfied. C o u n t e r e x a m p l e . We c o n s i d e r the p r o b l e m exp (--

~) u (x) = h (x), xfiO, D~ I~[=0,1,. 0

I t is not difficult to v e r i f y that the c o r r e s p o n d i n g s p a c e W~1769 , 2} is empty, and hence this p r o b l e m t r a n s f o r m s into the t r i v i a l i t y 0 --- 0. 3.

Behavior

of Order

2m

of Solutions as

of Nonlinear

Elliptic

Equations

m-

In a d o m a i n G c R n with boundary F we c o n s i d e r a family of D i r i c h i e t p r o b l e m s for the nonlinear equations m

L2,,,(urn)= ~ ( - - 1)'~ID~A~m(x, Drum)= lal=O D~ ]r = 0 , [told<m--I,

hm(x),

(3.1) (3.2)

w h e r e m = 1, 2, . . . . 537

We a r e i n t e r e s t e d in the behavior of the solution of p r o b l e m (3.1), (3.2) as m - - ~o. Two c a s e s a r e possible: the. ease of a limit equation of infinite o r d e r and the ease of a limit equation of finite order. As will be s e e n below, they a r e e s s e n t i a l l y distinct. 1~ Case of a L i m i t Equation of Infinite Order. We a s s u m e that the following conditions a r e satisfied: a) The functions Aam(X , ~Tm) a r e continuous i n x and ~Tm and for any values of the v a r i a b l e s x E G, ~Tm, ~ a m they s a t i s f y the inequalities

It~l=O

1o:1=o

w h e r e K > 0 is a constant, a a m -> 0 is s o m e n u m e r i c a l sequence, and P a m -> 1 is s o m e bounded n u m e r i c a l s equ enc e. b) F o r any ~Tm and x ~ G we have the inequality //~

?/Z

l~I=O

Is I=0

w h e r e 6 > 0 and K > 0 a r e constants. c) If m - - 4 , then a a m - - a a , P a m - - P a , and a a > 0 for an infinite set of values of a. M o r e o v e r , if in this p r o c e s s ~Tm-'* ~T' then

A~m (x, ~vm)-+ As (x, ~v) u n i f o r m l y with r e s p e c t to x E G, w h e r e A a (X, ~T) a r e continuous functions of their a r g u m e n t s . 0 d) We denote b~=supa~m, q~=supp~m and a s s u m e that the space W ~ 1 7 6 qa} is nontrivial (cf. Sec. 1) m m 0 000 (it is not hard to see that in this case the space W ~ 1 7 6 Pa} is also nontrivial, and ~V~176 qa} c W { a a , Pa}). e) Let the right sides hm(X) have the f o r m ?/z

h m (x) ----Z asn~D~h~ (x), i~l-0

w h e r e ham(X) E Lp~m(G), and for any m and values of P a m > 1

w h e r e K > 0 is a constant [in the case P a m = 1 the inequality v r a i m a x I ham(X) I < 5 must be satisfied, where 6 > 0 is the constant of condition b]. M o r e o v e r , suppose that for m - - oo the sequence hm(x) c o n v e r g e s to h(x) E W-~176

I

0 r

Pa} i n t h e s e n s e that ( h m, v) ~ (h, v) for any function v(x) E W { b a , qa}.

THEOREM 3.1. Let conditions a)-e) be satisfied. Then the sequence urn(x) of solutions of p r o b l e m (3.1), 0o0 (3.2) has a limit point u(x) in the s e n s e of the p r e v i o u s l y noted c o n v e r g e n c e in Co(G), and u(x) E W { a a , Pa} and is a solution of the C a u c h y - D i r i e h l e t p r o b l e m co

L (u) -----~ ( - - 1)t=ID~A=(x, Dvu) = h (x),

(3.3)

lal=0

Do,u l r = 0 ,

1o,1=0, 1. . . . .

(3.4)

The p r o o f of the t h e o r e m e s s e n t i a l l y parallels the p r o o f of T h e o r e m 2.1. Namely, we o b s e r v e f i r s t of all the solutions urn(x) by conditions b) and e) satisfy the e s t i m a t e P~n, 1~1=o w h e r e K > 0 is a constant. T h e r e f o r e , s o m e subsequence of Um(X) (we r e t a i n the s a m e notation) converges in C~~ to some c o m p a c t l y supported function u(x). F r o m inequality (3.5) and condition c) it follows that

538

(3.5)

0

u(x) 6W~176 W~~

Pa}. We shall show that u(x) is a solution of p r o b l e m (3.3), (3.4). Suppose H r s t of atl that v(x)

qc~}" We have ( L~,. (uA, v ) -

( ~m, V ) ~ ( ~, V )

by virtue of condition e). On the other hand, if m 0 is a fixed number and m > m 0, then

( L (tt), v ) -- ( Lo.m(t~m), v ) ~ ~.a

< A~ (x, Dvu), D'~'v )

t~

-- Z


l~l=m,+l

+ ~

(A~(x'D~u)--A~'~(x'Dvam)'D~v>

~l'-kt'2+I~O

t~l,Kmo

as m--- ~o Indeed, by condition a)

~a~mo+l

l~l>mo+I

is a r b i t r a r i l y small for sufficiently l a r g e n~, since u{x) 6W F u r t h e r , by condition a) and the estimate (3.5) [or any e > 0

{ace, Pa}, v(x) E~V~{bce, qc~} C W ~ { a ~ ,

Pc~}.

t~ I'zI=rno+l

I~i~,no+I

t~i=m.+ t

lai~m.+t 0

iS also a r b i t r a r i l y small for large m0, since v(x) ~ W~176a , qa}. Finally, for any fixed m 0 by oor~titfon c) 13 ~ O, since urn(x) ~ u(x) uniformly with all derivatives. 0

In s u m m a r y , we have f o u n d t h a t for any function v(x) ~ W~{ ba, qa} ( L(u),v > = ( h,~>.

It r e m a i n s to pass to the c l o s u r e on v(x) to an a r b i t r a r y function v(x) ~ ~V~ { a a , pa}, F o r this we r e q u i r e the following lemma. 0

0~

LEMMA 3.1. Let v(x) ~W~{ac~, pa}. Then t h e r e exists a sequence of functions v~(x) ~W that as e -- 0 ve(x) --" v(x} in C~(G), and p(v e) __
{b~, qa} such

co

Iczl~O

F o r the proof of the l e m m a it suffices to take ve{x) =-- we{x),v(x), w h e r e we(x) is a 6-type sequence in the space 0 W~~ h a , qa} (the l a t t e r obviously exists by L e m m a 1.1 and subsequent constructions). It is now not hard to see that if re(x) is a sequence chosen according to the l e m m a , then as e if-- 0 the equation < L(u), v e ) =
I~xl--o

D~tt, It-----0, ]~]~<m--1. We a s s u m e that the sequences ac~m >- 0 and P a r e -> 1 a r e nondecreasing in m, and acem --* ave, P~m ~ P~. 0 Suppose that these limits a r e such that the space W ~ { a ~ , Pc~} is nontrivial. With the comlitions of T h e o r e m 3 . 1 539

in mind, we note that in our e x a m p l e

b~.=supa~m=a~, q~=sup pr /75

0

, and hence {V~~ ~, q~} : W{ ac~, Pc~}.

//$

Thus, by T h e o r e m 3.1 the solutions Um(X) c o n v e r g e in C~(G) to a solution of the C a u c h y - D i r i c h i e t p r o b l e m Z ( - - 1)iulD~ (a~ I D~u

I,a-2Dau) = h (x),

icr

D~

=0,

I ~ 1 = o , 1. . . .

2 ~ C a s e of a L i m i t Equation of Finite Order. We a g a i n c o n s i d e r p r o b l e m (3.1), (3.2). We a s s u m e as b e f o r e that conditions a)-e) a r e s a t i s f i e d , towhieh, h o w e v e r , we now m a k e the following addition. Namely, we a s s u m e that the n u m b e r s ac~ = O for t a I - r + 1, w h e r e r -> 0 is s o m e natural n u m b e r . We f u r t h e r a s s u m e that the functions hm(x) have the f o r m r

I~1=0 w h e r e h~m(X)E LsJ~(G) and s ~ = s u p p ' m ,

p'~,n=p~,nl(P*,r,,--1). H e r e as m - - ~ o , Ilhc~m(X) - h~ (x) lIs~ - - O, w h e r e

h~ (x) E Ls~(G) a r e c e r t a i n functions. Finally, we a s s u m e that for any m the o p e r a t o r s L2m(U) a r e monotone, i . e . , f o r any functions u(x)E ira

0 m

W - ~ (@),

--*

v (x)~W~,,n (G), pm= {p~m, Ir ]~<m} we have the inequality < L ~ (u)-- L:~ (~), u - - v ) > 0.

THEOREM 3.2. L e t the conditions e n u m e r a t e d above be satisfied. T~en the family of solutions Um(X) of problem'~ ~ . ~ h a s a w e a k l i m i t point u(x) in the m e t r i c of the s p a c e w r ( G ) , w h e r e s = {sa=inf p~,n, [~ [..< S m Or -.. r} . H e r e u(x) 6 W ~ ( G ) , w h e r e p = {pC~, I~[ -< r } , is a solution of the D i r i c h l e t p r o b l e m L (u) ~ X ( - - 1)I~ID~A~(x,, D~u)=h(x),

D~

Iol~r -1,

(3.6) (3.7)

w h e r e h (x) = ~ D~h: (x). I~l<:r

P r o o L The following l e m m a , which is a v e r s i o n of the w e l l - k n o w n F a t o u t h e o r e m , will be u s e d in the proof. LEMMA 3.2. Suppose that the sequence of functions Um(X) E Lqm(G) , q m -> 1 and as m ~ oo Um(X) ~ u ( x ) w e a k l y in LI(G), i.e., for any bounded m e a s u r a b l e function v(x)


IIu Uq~
[1r

(3.8)

Proof. L e t v(x) be any bounded, m e a s u r a b l e function. Then by hypothesis
(urn, v )

and hence

] ( u, v )I~ N; u(x), if lu(x)l < N}; N is a natural number. We then a r r i v e at the inequality

IIu ~' U~~ lj._~_~ I)umII~"Uu '~ IIF', i.e.,

II uNU~--
540

whence, letting N ~ + ~, we find that u(x) E Lq(G) and inequality (3.8) holds. The p r o o f of the l e m m a is c o m plete. We now t u r n to the p r o o f of the t h e o r e m . F r o m condition b) and the r e s t r i c t i o n s on the function hm(x) we find that the solutions urn(x) s a t i s f y the e s t i m a t e a=m IID=u,~ll~::
(3.9)

tc~t=0

w h e r e K > 0 does not depend on m = 1, 2 . . . . .

Since a a m ~ a a > 0 for l a l -< r , f r o m this we obtain r

lcq--0

UD ~rn ll~rn < :(

and hence it m a y be a s s u m e d that for all l a l -< r , D a u m ~ D a u weakly in L s a (G), w h e r e s ~ = i n f P ~ m ) 1 is s o m e number. By L e m m a 3.2 (and this is essential) u{x) ~ X~

w h e r e p ---- { p ~ : p ~ : lim p ~ , [ ~ ] ~ r } . m-~oa

'We shall show

that u(x) is a solution of p r o b l e m (3.6), (3.7). To this end we note f i r s t of all that by the e s t i m a t e (3.9) and c o n dition a) the functions Aam(X , DYum) E Lp~m(G), and the n o r m s of t h e s e functions in L P ~ m ( G ) a r e u n i f o r m l y bounded. M o r e o v e r , f o r ] a l -< r it m a y be a s s u m e d (by again choosing a s u b s e q u e n c e if n e c e s s a r y ) that as m~o

A ~ (x, t ) ~ ) - ~ g~ (x) weakly in the s p a c e Ls~(G), w h e r e s"~= inf p ~ . g a (x) E Lp~' (G). "~

It is i m p o r t a n t to note, however, that in view of L e m m a 3.2 0

C o n s i d e r i n g what has been said, we find that for any function v(x) E W ~ { b a , qa} lira ( L~rn(Urn), 'v >= lira m-~oo

~_j

( A~.~(x,

DVtt~), D% > + Z

[al~.r

rn-+c~ l~:l= r + I

( g~ (x), D% >.

(3.10)

The r e s t of the p r o o f c o n s i s t s e s s e n t i a l l y in v e r i f y i n g the following two a s s e r t i o n s . 0 A s s e r t i o n 1. F o r any function v(x) E W ~ 1 7 6 qa} lira

~

< A~rn(x,

Drum), D% > : 0 .

m ~ o o ickier + 1

A s s e r t i o n 2. F o r any function v(x) E ~Vp(G) t h e r e is the identity

( g~(x), D% ) ~ ~ Ivq
( A~(x, DYu), D% ) .

1cr

P r o o f of A s s e r t i o n 1. F o r any fixed r 0 -> r + 1 and m > r 0 we have by condition a)

I~I-r-I'l

I~l--r+l

+K

~

( a~,n(lD'~urnlP~rn-'+l), ID~] ) ~I~+12.

ltzl--ro+l

Obviously, for any e > 0 ro

ro

g

rn Pff.rtl 9 l~I--r+l

Icq~r+l

whence by virtue of the e s t i m a t e (3.9) we s e e that I s is a r b i t r a r i l y s m a l l , since a a m ~ 0 for all I c~l -> r + 1. Analogously,

a~rn(llDurnll.=rn-t-1)-t-K(e)~ I{zI--ro+I

I~[~ro+l

a~rnHD Vllp~rn <

(~) Itzl--r,+l

541

T h e r e f o r e , 12 is a r b i t r a r i l y s m a l l if r 0 is sufficiently large. In s u m m a r y , 11 + 12 ~ 0 as m ~ 0% as we w e r e r e q u i r e d to show. Before p r o c e e d i n g to the p r o o f of the second a s s e r t i o n ~ve note that we now find f r o m (3.10) that for any 0 function v(x) E W~~ q~} t h e r e is the identity

( h, v ) =~.a (g,~(x), D~v ).

(3.11)

l(xl~r

0, o

It is not h a r d to s e e , h o w e v e r , that the s p a c e W { b a , qoL} is d e n s e i n ~ v r ( G ) (of. L e m m a 3.1). Thus, the e q u a l i t y (3.11) is t r u e f o r any function v(x) E

(G) (tNs is r e q u i r e d below).

The p r o o f of A s s e r t i o n 2 m a k e s u s e of the monotonieity p r o p e r t y of the o p e r a t o r s L2m(Um). Namely, for

0

any function v(x) E W e { b e ,

q,~} we have the inequality Re ~ (A~m(x, I~1-0

Dvum)--A~m(x, Dvv), D~ (u,, - - v) )

>0.

Since ( I ~ m ( U m ) , u m - v ) = (h m, u m - v ) , f r o m this we find that

Re{ ( h~' u~--v ) -- ~ ( A~'~(x'D~'v)'D'~(am--v) ) ] Further,

(3.12)

m

Z ( a~m (x, Dvv), D ~ (ttm - v) ) I~l-O

~ ( A~,n(x, Dvv), D~ (am--v) ) + ~

( .4~m(x,Dye,), D~ (u,~--~,) > -- ~, + :2.

Ir

I~.l-o It is c l e a r that as m ~

L.

I1~ ,~ ( AcL(X,Dvv), D~(tt--v) ). l~l-O We shall show that 12 ~ 0 as m ~ ,o. Indeed, if r 0 >- r + 1 is a fixed n u m b e r , then for m > r 0 '"

I~- ~

( A~m(x, Dr,o), D~(um--~) ) +

[~]--r+X

( A~(x, Ov~), D~(um--~) ) --t2x+:22. ]~]~ra+l

As b e f o r e , I2~ ~ 0 as m---~o. Finally, applying Young~s inequality and the e s t i m a t e (3.9), we find that

Iu2~< ~

( a~m(lD%l~247

)

lal--ro+l -

o , o ~

[color,+1

lal=ra+! 0

w h e r e s > 0 is a r b i t r a r y and K(e) > 0 i s a constant depending on s is a r b i t r a r i l y s m a l l for l a r g e r 0. This m e a n s that 12 - - 0 .

Since v(x) E W ~ { b a , q~}, it follows that I22

F u r t h e r , by the conditions on the b e h a v i o r of the functions hm(x) ( h~, u . , - - v

) -~ ( h, u - - v

) .

In s u m m a r y , f r o m inequality (3.12) we find that Re{ ( h, , - - v ) - - s

542

(A~(x'D'~v)'D~"--v))}>O"

(3.13)

0 00

The last inequality was obtained for v(x) E W { b a , qa}, but, as already mentioned, the imbedding ~V~{bce, 0

q~} c W~~

p~} is dense, and t h e r e f o r e (3.13) is valid for any function v E W ~ { a ~ , Pc~}.

Comparing now the inequality obtained with the inequality (3.11), we find that for any function v(x) E

Re X@~ 0 . Or

Standard arguments complete the proof. Namely, setting v(x) = u(x) - (w(x), w(x) E W ~ ( G ) , we find that as ~ - - + 0 Re

~

( g~ (x)-- A s (x, D~u), D~w ) > O.

Sinee the function w(x) E W ~ (G) is a r b i t r a r y , it follows that

"~_~ ( g~ (x) -- A s (x, D vu), O ~ ) = O. The proof of A s s e r t i o n 2 is complete. In view of the r e l a t i o n (3.11) the t h e o r e m has also been proved. Example. In a domain G c R n we consider the problem ,g,,,i

ai=r+l

w h e r e h(x) E W

i~i<:r

(G), m -> 0. We a s s u m e that the coefficients a a m -- + 0 a r e such that the space W { b~, 2 },

b~----sup%,~ is nontrivial. All the conditions of T h e o r e m 3.2 a r e then satisfied, and hence the solutions urn(x) c o n v e r g e weakly in

(G) to a solution of the limit p r o b l e m

~

(_l)I~IO~(lD~ttlP~-~O~u)=h(x),

1~1~o

D~lr=O, 4.

Nonlinear

Hyperbolic

Equations Type

(the

of Infinite General

Order

l ~1-.< r - - t . of Elliptic

and

Case)

The techniques used in Sec. 3 enable us to establish the solvability of the C a u c h y - D ! r i c h l e t p r o b l e m for nonlinear elliptic equations of infinite o r d e r which satisfy only a c o e r c i v e n e s s condition Ca monotonlcity condition is not required). M o r e o v e r , f r o m the s a m e point of view it is possible to prove a t h e o r e m on the soivability of a mixed problem for e s s e n t i a l l y nonlinear hyperbolic equations of infinite o r d e r . 1 ~ C a u c h y - D i r i c h l e t P r o b l e m for Nonlinear E l l i p t i c E q u a t i o n s of Infinite Order (the General Case)_. In a domain G c R n with boundary F we consider the p r o b l e m

L (u) -~ ~ (-- 1)I[~ID~A~(x, DVu)= h (x),

(4.1)

l~i=O

D~'ttlr~-O, l,~I=0, 1. . . . . .

(4.2)

We a s s u m e that the functions A~(x, 87) satisfy conditions a) and b) of See. 2, i.e., the conditions of nonlinear power growth and the c o e r c i v e n e s s condition. THEOREM 4.1. Under the conditions indicated for any right side h(x) EW-~~ has at l e a s t one solution u(x) E~V~176

p~} p r o b l e m (4.1), (4.2)

Pa}.

Proof. The idea of the proof consists in the following. F i r s t a "truncated" equation of o r d e r 2m - a partial s u m of the s e r i e s (4.1) - is p e r t u r b e d by a sufficiently " s m a l l " linear equation of o r d e r 2m + 2. The

54:~

c o r r e s p o n d i n g b o u n d a r y - v a l u e p r o b l e m is always solvable. We then p a s s to the l i m i t s as m --- + oo. T h u s , we c o n s i d e r the D i r i c h l e t p r o b l e m for the following equation of o r d e r 2 m + 2:

Z

cc*D2~'~m+t7-~ ~

[~i=m+l

(--l)i~'lDC~A~(x, OVum+l)=hm(x)'

(4.3)

[r=0, Io[
(4.4)

Ice)- 0

D~

w h e r e c a > 0 a r e c o n s t a n t s and h m (x) = ~ ao:D~h~(x) 9 This p r o b l e m is a weakly nonlinear p r o b l e m (the right i~l-0 side of the equation is linear); c o e r e i v i t y t h e r e f o r e suffices for its solvability. Thus, a solution Um+l(x) of p r o b l e m (4.3), (4.4) e x i s t s and a d m i t s the e s t i m a t e

:Z co Ho%+,

+

lal--m+l

tl

li;:-< K.

I~1--0

w h e r e K = K0a) > 0 is a constant. It is t h e r e f o r e p o s s i b l e to a s s u m e that Um+l(x) c o n v e r g e s in C~~

0oo

to s o m e function u(x) E W { a a , Pa}.

0o ~

0

0

We c h o o s e the coefficients c a > 0 such that the s p a c e W { c a , 2} and theintersectionW~176 2} ~q W~{ao~, Pex} a r e nontrivial [this is e a s i l y achieved, for e x a m p l e , by d i r e c t c o n s t r u c t i o n of a function u(x) belonging to the s p a c e s indicated; in p a r t i c u l a r , if Pa >- 2 then it is possible to s e t c a - a s ] . Then for any function v(x) E 000

000

W '{a a, p a } n w { c a , 2 } w e h a v e ~.~ c~ ( D~u,,+,, D~v ) + ~.~
Ic~l.<:m

Repeating the a r g u m e n t s c a r r i e d out in Secs. 2 and 3, we a r r i v e at the identity

< L(u),v>

=

(h,v>,

000 w h e r e v(x) is an a r b i t r a r y function in the i n t e r s e c t i o n W {a a , Pa} N ~r Ca, 2 }. 000 000 By L e m m a 3.1 W { a a , Pa} ~ W {c a , 2} is d e n s e l y imbedded in the s e n s e of c o n v e r g e n c e in C~~ in 0,0 000 the s p a c e W {a a , Pa}. T h e r e f o r e , the l a s t identity is valid for any function v(x) EW { a a , Pa}- This m e a n s that u(x) is a solution of p r o b l e m (4.1), (4.2). The p r o o f of the t h e o r e m is complete. R e m a r k 4.1. In the c a s e Pa - 2 the p r o o f of T h e o r e m 4.1 can be c a r r i e d out by d i r e c t appeal to T h e o r e m 3.1.

R e m a r k 4.2. P a r a b o l i c equations with m - - oo c a n be studied in a s i m i l a r way. 2 ~ A Mixed P r o b l e m for Nonlinear H y p e r b o l i c Equations of infinite O r d e r . In the cylinder Q = [0, T] • G with l a t e r a l s u r f a c e S = [ 0 , T] x r we study the mixed p r o b l e m

u"+L(u)=h(t, x), D%[s=O, loJ[=O, 1. . . . . u(0, x ) = 0 ,

u'(0, x ) = 0 ,

where co

L(u)--- X(--l)l~lD~A~(t,x, DVu), I'~1~
is a nonlinear d i f f e r e n t i a l o p e r a t o r of infinite o r d e r . We a s s u m e that the following conditions a r e satisfied: a) The functions A(x(t, x, ~y) a r e continuous in all a r g u m e n t s and for any m s a t i s f y the inequality

w h e r e K > 0, a a -> 0, Pa >- 1, 6 m -> 0 a r e constants. H e r e Pa -< P < 00, 51 + 52 + 9 9 9 < oo.

544

(4.5)

(4.6) (4.7)

b) F o r any sufficiently s m o o t h function u(t, x) (u(0, x) = 0, W(0, x) = 0) t h e r e is the inequality t

Re I -~ ( As (t, x, DVu), D~u" ) d~ > ~ ~ 0 I~zI=tn

p~ (t)-- b~, a~ [1D ~~ I1,~

Iczi=m

w h e r e t E [0, T] is a r b i t r a r y ; 6 > 0, b m-> 0 a r e c o n s t a n t s , and b 1 + b 2 + . o . < ~ . c) The function h(t, x) has the f o r m

h (t, x) ~- ~ a~D~h~(t, x), w h e r e h a (t, x) E Lp~ (Q), h~ (t, x) ~ Lp~ (Q), and

z_~a~sup[l/z~J)(t,x)ll I~i=O

<~,

j = o , 1.

t

THEOREM 4.2. Let conditions a)-c) be satisfied. Then t h e r e exists at l e a s t one solution of p r o b l e m (4.5)(4.7). M o r e o v e r , t h e r e exists a function u(t, x) which satisfies the following conditions: 1) u(t, x) has a finite " e n e r g y integral" D~

p~

;

2) for any function which is compactly supported in x having a finite energy integral I with t - - T the fol-

lowing identity holds:

--f ( u ' , v ' ) d t + 0

~'~

(A=(t,x, DVtt),O~v)dt -- ( h , v ) d t ;

Icq=0 o

(*)

0

3) conditions (4.6) and (4.7) a r e satisfied, and the condition u'(O, x) = 0 is satisfied in the s e n s e of the 0

t h e o r e m of distributions on W~{ ac~, Pc~}. Proof. We c o n s i d e r the auxiliary p r o b l e m

~,~+i +

~

( - 1)~+~ c~n~m+~ + &m(~+~)--h~+~ (t, X),

(4.S)

l~l=m+I

D~

Is=O, t |

ttm+l(O, x)=O,

tt;~+1(O,x ) = O .

(4.9) (4.10)

where

L2m(Um+t)~ ~=~ (--I)I~ID~A~(t, x, DVtt~+l), Icq~m

h~ (t, x)= ~, ~o~h~(t, x). Ic~l~
The coefficients c a > 0 a r e chosen as in the elliptic case. As a weakly nonlinear hyperbolic equation, p r o b l e m (4.8)-(4.10) is solvable: The solutionum+t(t , x) can be obtained, for example, by the G a l e r k i n - F a e d o - H o p f method (cf. Lions [1]). Here f r o m conditions b) and c) by e s t i m a t e s standard in the theory of hyperbolic p r o b lems we have the inequality 2

sup ]um+l 2(t)+sup ~] as IID ~u,,+llt~ P~(t) ~ 4, t

t

icq<;m

w h e r e K = K(h) > 0 is a constant not depending on m. T h e r e f o r e , we see that there exists a function u(t, x) having a finite integral I to which the sequence Um+t(t, x) (in g e n e r a l , a subsequence) c o n v e r g e s uniformly with all d e r i v a t i v e s with r e s p e c t to the variables x. It is c l e a r that uT(0, x) = 0, D W u / s = 0, I wl -< m. M o r e o v e r , repeating the a r g u m e n t s g i v e n i n P a r t l ~ we see that the function u(t, x) satisfies the identity (,). Since no r e s t r i c t i o n s as t ~ 0 a r e imposed on the function v(t, x) in this identity, it follows also that ul(0, x) = 0. Thus, u(t, x) is a g e n e r a l i z e d solution of the initial-value problem. The p r o o f of the t h e o r e m is complete.

545

Example. The conditions of T h e o r e m 4.2 a r e satisfied for the p r o b l e m

u"~- ~

(--1)t~176176176

t, x),

1~[~o

tt (0, x) = tt" (0, x) = O, D~

= O,

I o) I = O, 1 . . . . .

T h e r e f o r e , this p r o b l e m has at l e a s t one g e n e r a l i z e d solution. 5.

Periodic

Order

and

Solutions Related

of Nonlinear

Equations

of Infinite

Topics

In this s e c t i o n periodic p r o b l e m s a r e considered. We f i r s t c o n s i d e r the question of nontriviality of Sobolev s p a c e s of infinite o r d e r on the torus T n. 1~ Nontriviality of the Spaces W*~ Ps} on the T o r u s . Suppose as p r e v i o u s l y that a s > 0, Ps -> 1, rot - 1 a r e a r b i t r a r y sequences. We c o n s i d e r the s p a c e as IIO~u II~ < ~

IV ~' {a~, pc,} = u (x)~C ~ ( r , ) : p (u)-~ ~

'

laid0

which is a s p a c e of infinitely differentiable, periodic functions u(x)0 x = (x 1, . . . . Xn); to be specific, we shall a s s u m e these functions to be 27r-periodic functions. As p r e v i o u s l y , the question a r i s e s as to whether the space W*~ Ps} is nontrivial, i.e., w h e t h e r t h e r e exists at l e a s t one function u(x) ~ const having a finite "integral" p(u). We shall be i n t e r e s t e d only in those s p a c e s W*~ act, Ps} which a r e infinite-dimensional, i.e., which contain an infinite set of l i n e a r l y independent functions u(x) E W~{ a s , pot}. THEOREM 5.1. The space W~176 Ps} is nonempty and is infinite-dimensional if and only if t h e r e exists a sequence of distinct normegative multiindices qu = (qw . . . . , qnu), u = 0, 1, . . . such that co

nPo:

~ a~q~ a= (2r.)~ < ~ .

(,)

17.1=o

Proof. The sufficiency of condition (,) is obvious. Indeed, in this case exp (iq~x) EW*~ p (exp (iq, (xl) ~

pot}, since

a~q; p~ (2~) ~ < ~ . i~k=O

(ql . . . . . W~{a~, 1, . . . .

To prove n e e e s s i t y we a s s u m e the c o n t r a r y : The s e r i e s (*) c o n v e r g e s for a finite s e t of values q = qn), viz., for lqj I < Nj, j = 1 . . . . . n, w h e r e Nj > 0 a r e some numbers. We shall show that then pot} c L(exp (iqx)), w h e r e L(exp (iqx)) is the linear span of the functions exp (iqx) for I q j l - < Nj, j = n. The l a t t e r c o n t r a d i c t s the infinite dimensionality of the space W ~ { a s , Ps}.

Indeed, if u(x)EW"~

Ps}, then ~ ( x ) = 'z....l ~ cqexp(iqx), Iql~O

w h e r e Cq = (21r)-n(u, exp (-iqx)). Here for any s oo

D ~u (x)-~ ~

Cq (iq) ~ exp (iqx).

Iq[~O

We have

n

I cq (iq)~l= (2~)-~ [ (D~u, exp (--iqx)) [ ~ (2~) r= liD ~allr~, whence nPcr

P~> I c~q~ Ip42~) r~ IID~u IIr~

(5.1)

Our a s s u m p t i o n m e a n s , in p a r t i c u l a r , that the s e r i e s (*) diverges also at points of the f o r m (0, . . . , Nj, . . . , 0}. It is c l e a r that any multiindex c o m p l e m e n t a r y to the multiindices q with I qj I - Nj m a j o r i z e s an index of the type noted, i.e., 546

([q~ ]. . . . . ] qj[ . . . . . I q~ l) > (0 . . . . . N] + 1. . . . . 0), w h e r e 1 --- j --- n is s o m e number. Hence, for any fixed index q = (ql, 9 9 9 qn) c o m p l e m e n t a r y to the indices q with I qj i --< Nj t h e r e exists the inequality

~176

[al=0

w h e r e the indices ~ r u n through the values (0 . . . . .

~j . . . . . npc~

C~] ~0

0). F r o m this, by inequality (5.1) we find that ~

npGr

c,.j ~ 0

provided that Cq ~ 0. Thus, if u(x) ~ W ~ { a ~ , pa} all the F o u r i e r coefficients Cq = 0 if at l e a s t for one j we have j I qil > Ni. In s u m m a r y , we find that with c o n v e r g e n c e of the s e r i e s (,) at only a finite number of points the space W { a s , p~ } is finite-dimensional, which c o n t r a d i c t s the hypothesis of the theorem. The p r o o f of the t h e o r e m is c o m plete. R e m a r k 5.1. Condition (*) does not exclude the case in which the space W ~ { a ~ , PaJ c o n s i s t s of functions depending only on a fewer number of a r g u m e n t s . To exclude this degenerate c a s e below we r e q u i r e that, for example, the space W ~ { a ~ , Pa} be dense in L2(Tn). It is easy to see that for this it is n e c e s s a r y and sufficient that condition (,) be satisfied for s o m e sequence of qv such that man (q~v. . . . . qnv) ~ ~ as , ~ ~. R e m a r k 4.2. In condition (.) the factor (2~r)n p a / r a can be dropped. Indeed, in the p r o o f of n e c e s s i t y this is obvious. In the p r o o f of sufficiency it is n e c e s s a r y to c o n s i d e r the functions exp (iqx), w h e r e q = (O, . . . . qj . . . . . 0), qj ~ • In conclusion, we c o n s i d e r an important special c a s e , namely, the c a s e of a bounded sequence p~. COROLLARY 5.1. If the sequence p~ -> 1 is bounded, then the space W'~ if the function

Pc~} is nontrivial if and only

I~1=0

is an entire function of a complex variable. 2 ~ P e r i o d i c Solutions of Nonlinear Elliptic Equations of Infinite Order. On the torus T n we consider the equation oo

(5.2)

L (u) = Z (-- 1)I~ID~A~(x, D~u) = h (x), ] 7 1 ~ [ 9 ], lat----O

w h e r e the functions A a ( x , iv) satisfy conditions a) and b) of Sec. 2 of the p r e s e n t chapter, i.e., the o p e r a t o r L(u) maps the s p a c e W~r a ~ : pa}, a s s u m e d to be nontrivial, into the space W r162a a , p~} (the notation m obvious), and this mapping is coercive. We point out that monotonlcity of the o p e r a t o r L ( u ) i s , in general, not assumed. THEOREM 5.2. If the above conditions a r e satisfied, then for any right side h(x) E W-~{ a~, p~} there exists at least one solution u(x) E W~{ao~, p~} of Eq. (5.2). The p r o o f of this t h e o r e m can be c a r r i e d out in exactly the s a m e way as the p r o o f of T h e o r e m 4.2 by p e r t u r b a t i o n of the equation - a partial s u m of the s e r i e s (5.2) -

L2~(um)=--f~ (-- 1)I.~ID~A~(x, Dram)=h~(x)

(5.3)

lal=O

to a weakly nonlinear c o e r c i v e equation of o r d e r 2m + 2. In the p r e s e n t c a s e , however, it is s i m p l e r to solve Eq. (5.3) approximately by the Galerkin method using the fact that the s y s t e m exp (iqx), Iq I = 0, 1. . . . forms a basis in W ~ { a a , Pa} in a p a r t i c u l a r sense. H e r e , to be t r u e , it is n e c e s s a r y to make the minor additional a s s u m p t i o n that Pa -> P > 1. We set tt~ (x) = ~

cq~ exp (iqx),

[ql= 0

547

w h e r e the Cqm a r e d e t e r m i n e d f r o m the s y s t e m of nonlinear a l g e b r a i c equations

~

( A~ (x, D~u,,,), (iq)~exp (iqx) ) = ( hm, exp (iqx) ) .

(5.4)

[al=O

By the c o e r c i v e n e s s of the o p e r a t o r L2m(U m) the s y s t e m (5.4) is s o l v a b l e , and for the solutions urn(x) t h e r e is the e s t i m a t e m

I~1=o

w h e r e K = K(h) > 0 does not depend on m. It is t h e r e f o r e p o s s i b l e to a s s u m e without changing notation that as m - * ~ the sequence Um(X) c o n v e r g e s in C ~ ( T n) to s o m e function u(x) ~ W ~ { a a , p~}. This function is the s o l u tion of Eq. (5.3). Indeed, letting m --- oo and r e p e a t i n g a r g u m e n t s a l r e a d y c a r r i e d out, we s e e that the function u(x) s a t i s f i e s the s y s t e m of equations

( A~(x, DVu),(iq)C~exp(iqx) )

( k, exp(iqx) )

(5.5)

lal=O

for any i n t e g e r s q = (ql . . . . .

qn). It r e m a i n s to c l o s e Eqs. (5.5) to the identity

~

( A= (x, OVu), D~v ) = ( h, v ),

(5.6)

[~l~O

w h e r e v(x) t? W~{ a a , Pa} is an a r b i t r a r y function. Indeed, let v(x) be s u c h a function. Then

v (x) = ~ Cqexp (iqX), Iql=O

and the s e r i e s c o n v e r g e s u n i f o r m l y with all d e r i v a t i v e s . It is known (see, e.g., B a r i [5, p. 594]) that the p a r tial s u m s of the F o u r i e r s e r i e s s a t i s f y N

S~v (X) ~ ~ Cqexp (iqx) tq[-0

with the bounds

[IS~-(x) llr~Kll'V(x)[lr,

r>l,

w h e r e K = K(r) is a constant depending only on r. Since by hypothesis the sequence of exponents s a t i s f i e s pa -> p > 1 and is bounded, we find t h a t for any function v(x) ~ W*~ a s , p a } the sequence SN(X) - - v(x) in C*~ and for any N = 0, 1 . . . . . p we have (SN) -< Kp(v), w h e r e K > 0 is a constant. This m e a n s that in closing the r e l a tions (5.5) along the b a s i s it is again p o s s i b l e to use our s t a n d a r d a r g u m e n t s . Thus, u(x) s a t i s f i e s the identity (5.6) and is thus a solution of Eq. (5.2). The p r o o f of the t h e o r e m is complete. E x a m p l e s . 1. We c o n s i d e r the equation

exp(--A)u(x)=h(x),

xET",

(5.7)

o r the equation

u(x)=h(x), xesl.

(5.8)

All the h y p o t h e s e s of the p r e c e d i n g t h e o r e m a r e c l e a r l y s a t i s f i e d , and t h e r e f o r e Eqs. (5.7), (5.8) a r e solvable. M o r e o v e r , the o p e r a t o r s exp (--A), Cos d / d x r e a l i z e h o m e o m o r p h i s m s of the c o r r e s p o n d i n g s p a c e s . 2. Let ~(z)~- z.~ a~z~ be an e n t i r e function, and let p~ -> 1 be a bounded sequence. Then the following lal=0

o p e r a t o r s a t i s f i e s the h y p o t h e s e s of T h e o r e m 5.2: L (u) ~ ~ ( - - 1)I~'ID= (as [ D~u {P=-2D~u). 3 ~ B e h a v i o r of P e r i o d i c Solutions of Nonlinear Elliptic P r o b l e m s of O r d e r 2 m as m ~ ~o. On the nd i m e n s i o n a l t o r u s T n we c o n s i d e r the nonlinear equation

L2m(um)~ ~X~( -1)I~ID~A~m(x' DVum)=hm(X)' lal=o

548

17]~[~['

(5.9)

w h e r e the functions Aam(X, }Tin) satisfy the s a m e conditions a)-c) as in Sec. 3 of the p r e s e n t chapter. M o r e o v e r , we r e c a l l again condition d) of Sec. 3. Namely, suppose that b,~=supa~m, q~-~supp~,n, w h e r e a ~ m >- 0, m

?Tt

P a m -> 1 a r e defined in conditions a) and b). Condition d) r e a d s : T h e s p a c e W ~ { b a , qa} is nontrivial. Finally, we a s s u m e that the right sides hm(x) have the f o r m rn

hrn (x)------~ (-- l)l~lD~h~m(x),

where hmn(X) E Lp~m(Tn) and converge as m see condition e) of Sea. 3].

=o to a function h(x) e W-~{ac~, p~} [for the precise formulation

THEOREM 5.3. If the conditions indicated are satisfied, then the sequence of solutions urn(x) has in the sense of convergence in C~176n) a limit point u(x) EW~~ Pc~}which is a solution of the limit equation

( - - 1)I~'D~A=(x, DVu)--/~ (x). luL=0

The p r o o f of this t h e o r e m can be c a r r i e d out by repeating the proof of T h e o r e m 3.1. In the exaetly the s a m e way a t h e o r e m analogous to T h e o r e m 3.2 can be formulated and proved r e g a r d i n g the behavior of the solutionS Um(X) in the e a s e w h e r e the limit equation is a nonlinear equation of finite o r d e r 2r. Namely, suppose that the functions hm(x) have the f o r m r

I~l=0

where

h~,~(x)CLs,(Tn), s'~ =

sup P'~m'P'~m= p~rn/(p~m--l), Here as m -* ~o, IIh~ m - ha lls~ -~ 0, where h~ (x) ~ Ls~(T D)

are some functions. THEOREM 5.4. Suppose that conditions a)-d) of See. 3 are satisfied and moreover that the operators L2m(Um) are monotone. Suppose further that hm(x) -* h(x) in the sense just indicated. Then the family of solutions urn(x) of Eq. (5.9) has a weak limit point u(x) in the space W r ( T n) which is a solution of the equation P

@ (-- I)I~ID~A (x, D~u)----h(x). lui=O

Example. Let a m -> 0 with limv am = 0. We consider the family of problems

(-- l)'~amAmurn--AUm-l-U,,==hm(X),

xeT',

w h e r e hm(x) - - h ( x ) in W ~ ( T n ) . By T h e o r e m 5.3 the solutions urn(x) converge wealdy in W21(Tn) to a solution u(x) of the limit p r o b l e m --au§

x E T ~.

R e m a r k [regarding condition d)]. D i r e c t calculation of the solutions of the equation d~m

d2

( -- 1)"arn-d--2r~ Urn(x) - - d ~ 2 Um (x) -t- um (x) = tt (x),

xfiT',

w h e r e h(x) E L 2 (TJ), shows that if urn(x) c o n v e r g e s to a solution of the limit equation d~

e ~ u (x) + u (x) = h (x),

then lira ~/am = 0, i.e., the function ~ (z) = __~ amzm is an e n t i r e function. This example shows thaL condition d) is essential, m=0 In conclusion we r e m a r k that as in See. 4 we have solvability- of the Cauchy p r o b l e m in a class of functions periodic in x for nonlinear hyperbolic equations of infinite o r d e r . COMMENTS CHAPTER I The r e s u l t s of Sec. 1 a r e among those which a r e "weil knowm" We n e v e r t h e l e s s begin the exposition with these simple r e s u l t s , since they i l l u s t r a t e the s c h e m e of the investigation which is r e p e a t e d l y encountered: 549

1) finding a p p r o x i m a t e solutions; 2) e s t i m a t e s and c o m p a c t n e s s ; 3) p a s s a g e to the limit. As a l r e a d y noted, the " a c u t e - a n g l e " l e m m a is exceptionally important. This l e m m a c a n be found in v a r i o u s v e r s i o n s in the w o r k of K r a s n o s e l ' s k i i [32, p. 314], Sobolevskii [54], Vishik [12], B r o w d e r [6], and o t h e r s . H e r e we have p r e s e n t e d a v e r s i o n of the l e m m a t a k e n f r o m the book of Lions [36, p. 66]. Sections 2 and 3 contain r e s u l t s of the author [23~ 24] on n o n c o e r e i v e p r o b l e m s . Nonlinear equations which a r e not c o e r c i v e w e r e f i r s t c o n s i d e r e d by P o k h o z h a e v [44-46] under the hypothesis of c l o s e d n e s s or w e a k c l o s e d n e s s of the range. V e r i f i c a t i o n of the l a t t e r p r o p e r t y c o n s i s t s in p r a c t i c e in obtaining an a p r i o r i e s t i m a t e . It is e a s y to s e e that the p r o b l e m s studied in See. 2 have n e i t h e r the c o e r c i v e n e s s p r o p e r t y nor do they a d m i t an a p r i o r i e s t i m a t e . Investigations of n o n c o e r c i v e p r o b l e m s can be found in the w o r k of B r o w d e r [70, 71], Ne~as [94], Skrypnik [51, 52], Babin [3, 4], P o r t n o v [42, 43], and o t h e r s . CHAPTER

II

Monotone o p e r a t o r s a r i s e in v a r i o u s b r a n c h e s of a n a l y s i s and have been used ( p r i m a r i l y in connection with nonlinear i n t e g r a l equations) by Vainberg [7, 8], Vainberg and K a c h u r o v s k i i [10], L a v r e n t ' e v [34], Minty [90, 911, and Z a r a n t o n e l l o [98]. In the t h e o r y of monotone o p e r a t o r equations p r e s e n t e d h e r e the b a s i c r o l e was played by the w o r k of Minty indicated and the w o r k of B r o w d e r [6, 65, 69], and o t h e r s , while the l a t t e r in t u r n continue the w o r k of Vishik [11, 13] on the s o l v a b i l i t y of nonlinear p r o b l e m s for p a r t i a l d i f f e r e n t i a l equations of elliptic and p a r a b o l i c type. A s i m p l e p r o o f of the r e s u l t s of M. I. Vishik was found in 1965 by L e r a y and Lions [83] and the author [19]; t h e s e a r g u m e n t s w e r e p r e s e n t e d h e r e in the p r o o f of the b a s i c t h e o r e m 1.1. After 1965 t h e r e followed a l a r g e n u m b e r Of investigations in the t h e o r y of monotone o p e r a t o r s (see, in p a r t i c u l a r , the w o r k of B r o w d e r [66] and K a e h u r o v s k i i [31]; d i s c u s s i o n of this w o r k goes beyond the s c o p e of the p r e s e n t r e m a r k s ) . We r e m a r k only that the d e v e l o p m e n t of the theory of the solvability of o p e r a t o r equations b a s e d on the method of monotonieity is p r i m a r i l y connected with an a x i o m a t i c extension of the c l a s s of monotone o p e r a t o r s (see the w o r k cited of L e r a y and Lions and the w o r k of B r e z i s [63],where the concept of pseudomonotone o p e r a t o r s , etc. a r e used). In p a r t i c u l a r , in applications the c l a s s of o p e r a t o r s having semibounded v a r i a t i o n has been found v e r y useful; this c l a s s of o p e r a t o r s is studied in P a r t 2 ~ of Sec. 1. The s t r o n g c o n v e r g e n c e of the G a l e r k i n a p p r o x i m a t i o n under conditions of d e f i n i t e n e s s of the v a r i a t i o n was e s t a b l i s h e d in the w o r k of the author [21]. The r e s u l t s of See. 2 p e r t a i n i n g to s t r o n g l y elliptic equations a r e due to Vishik [11-13]; the elliptic equations with semibounded v a r i a t i o n d i s c u s s e d in Sec. 3 a r e c o n s i d e r e d in the w o r k of the author [19-21]. The c o u n t e r e x a m p l e p r e s e n t e d in R e m a r k 3.2 w a s c o m m u n i c a t e d to the author by Yu. S. Kolesov (verbal communication). In connection with this r e m a r k we point out that in c o n t r a s t to elliptic equations of second o r d e r (see, e.g., the book of L a d y z h e n s k a y a and U r a l ' t s e v a [35]) the c l a s s i c a l s m o o t h n e s s p r o b l e m has a n e g a tive solution; the f i r s t c o u n t e r e x a m p l e s in this d i r e c t i o n w e r e c o n s t r u c t e d by De Giorgi [75], Giusti and Miranda [77], M a z ' y a [37], and o t h e r s . An exception is the c a s e of q u a s i l i n e a r equations in the plane, w h e r e the s m o o t h n e s s t h e o r e m s a r e valid (see Ne~as [94] and Skrypnik [50]). In the c a s e of a r b i t r a r y d i m e n s i o n of the d o m a i n M o r r e y [93] e s t a b l i s h e d that a g e n e r a l i z e d solution is s m o o t h for c o r r e s p o n d i n g s m o o t h n e s s of the data of the p r o b l e m e v e r y w h e r e in G with the exception of a closed s e t of m e a s u r e z e r o . The s m o o t h n e s s p r o b l e m has been thoroughly investigated in the r e c e n t book of Skrypnik

[3]. Q u a s i l i n e a r equations of second o r d e r and, in p a r t i c u l a r , the question of the s m o o t h n e s s of e a c h nonl i n e a r t e r m of an equation of second o r d e r w e r e studied by methods of monotone o p e r a t o r s in the w o r k of D z h a b r a i l o v [15-17] and Yakovlev [57, 58]. We note the w o r k of Yakovlev [59], inwhich he studies the Neumann p r o b l e m for a q u a s i l i n e a r equation of second o r d e r and obtains p r e c i s e e s t i m a t e s of the Neumann data in c l a s s e s of s u m m a b l e functions. The r e s u l t s of P a r t 4 ~ w e r e obtained jointly by the author and Pokhozhaev in [28], inwhich the r e l a t i o n of q u a s i l i n e a r d i f f e r e n t i a l o p e r a t o r s to o p e r a t o r s s a t i s f y i n g a duality p r o p e r t y is studied. Dual o p e r a t o r s w e r e studied e a r l i e r in the w o r k of Vainberg [9], Beurling and Livingston [62], B r o w d e r [68], and others. CHAPTER

III

The d e v e l o p m e n t of p a r a b o l i c evolution equations with a monotone nonlinear o p e r a t o r in many r e s p e c t s p r o c e e d e d in p a r a l l e l to the d e v e l o p m e n t of s t a t i o n a r y monotone equations. The f i r s t w o r k in this d i r e c t i o n was 550

the work of Vishik [13], inwhich, however, monotonicity was not explicitly used. In the evolution case the p r o per ty of monotonicity was introduced by Browder [67]. It was then repeatedly used to study various nonlinear problems by Lions [85, 87], the author [19, 20], -and others. Parabolic nonlinear problems with a decomposable nonlinear operator w e r e considered by Agaev [2] (see also the elliptic case [1]). The results of Sec. 1 are a "compiling" of the results of the work cited of Browder, Lions, and the author. We r e m a r k here that the proof of Lemma 1.1., which inthe evolution case replaces the "acute-angle" lemma, can be c a r r i e d out by a classical means: the C a u c h y - P e a n o theorem plus an a priori estimate. We present a topological proof with the intent of emphasizing the parallelism of the methods of monotonicity in the stationary (elliptic) and evolution (parabolic) cases. The method presented of solving nonlinear evolution equations (we catl it the method of G a l e r k i n - F a e d o Hopf) was applied in the linear case by Faedo [76] and for nonlinear equations (for the N a v i e r - S t o k e s equations) by Hopf [78]. The strong convergence of the G a l e r k i n - F a e d o - H o p f approximations (Part 3~ in the method of monotonicity is apparently presented here for the f i r s t time. Evolution equations with stationary part having semibounded variation (see the work of the author f19, 20], and others) a r e considered in Sec. 2 by the method of monotonicity. Boundary-value problems for parabolic partial differential equations (See. 3, P a r t 1~ a r e studied in the work cited of Vishik, Browder, Lions, and the author, and others; for equations with semibounded variation see the work of the author [20-22]. The results of Sec. 4 on the formulation and solvability of boundary-value problems for equations of parabolic type which a r e not of divergence form belong to the author [20]. We note that quasilinear equations of elliptic type having nondivergence form were considered by Simonov [49] and Krein and Simonov [33]. General homogeneous boundary-value problems for quasilinear systems in the sense of D o u g l i s - N i r e n berg-Solonnikov a r e considered in the work of Pokhozhaev [47]. A theorem is proved on the solvability of the problems posed with an a priori condition for the existence of an estimate of possible solutions. In the last s e c t i o n results of Komura [81, 82] are presented on nonlinear semigroups; the proof of Theor e m 5.1 was borrowed from Browder [72]. The deep results of Komura (in particular, the introduction of multivalued monotone mappings) were fundamental for a large cycle of subsequent papers. At the present time there is a large journal l i t erat ure devoted to various questions of nonlinear se m igroups and their applications (cf. Kato [79, 80], Browder [72], Brezis, Pazy [64], Crandall, Pazy [74], Barbu [60, 61], Khazan [55, 56], Webb [97], and others). CHAPTER

IV

In this chapter results a r e presented of the author [26, 27] on the solvability of nonlinear equations o f infinite o r d e r and the more general questions of the behavior of solutions of nonlinear boundary-value problems with unbounded growth of the o r d e r of the equation. The Sobolev spaces of infinite order which a rise here a re of independent interest. LITERATURE 1. 2. 3. 4. 5. 6. 7.

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9

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V

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i.