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Solvability of general elliptic problems in Holder

spaces

V. Volpert

, A. Volpert

Laboratoire de Mathematiques Appliquees, UMR 5585 CNRS,

Universite Lyon 1 69622 Villeurbanne, Prance

Department of Mathematics, Technion, 32000 Haifa, Israel

Email : volpert@numerix.univ-lyonl.fr

Abstract

The paper is devoted to general elliptic operators in Holder spaces in bounded

or unbounded domains. We discuss the Fredholm property of linear operators and

properness of nonlinear operators. We construct a topological degree for Fredholm

and proper operators of index 0.

1 Normal solvability of linear problems

Normal solvability of linear elliptic problems in bounded domains is well studied. In the

case of unbounded domains it is studied for some classes of operators in [11], [21] for

the whole R", in [8], [16] for conical and in [12], [19] for cylindrical at infinity domains

(see also [9]). In this section we consider general elliptic problems in arbitrary domains

with some natural regularity conditions. We generalize the notion of limiting problems

introduced already in the previous works, and formulate conditions of normal solvability

through solvability of the limiting problems.

1.1 Operators and spaces

Let j3 = (/?!, ...,/3„) be a multi-index, p

be nonnegative integers, \/3\ = (3

+ ... + /?„,

= D

...Dn ', Di = d/dxi. We consider the following operators:

Atu = J2 Y,

ffc(*)£"V (i

l,...,p), leilcR",

*=1 \P\<P

BiU

= Z) Z)

)

D/}uk

(i = l,...,r), xedSl.

*=l l/5|<7i*

Here /3

, -y

are given numbers. As in

[1],

we suppose that there exist integers si,...,s

ti,...,t

u\,...,

such that

Pij < Si + tj, i,j = l,...,p; %j <

+ tj, i =

l,...,r,j

= l,...,p, s

< 0.

291

292

We suppose that the problem is elliptic in the sense of [4], [1], i.e., the system is ellip-

tic and supplementary and complementing (Shapiro-Lopatinskii) conditions are satisfied.

Moreover we assume that the system is uniformly elliptic.

We consider the space E

—

i+tl+Q!

(fi) x ... x

t+t

(Q)

of vector valued functions

u(x) =

(MI(S),

...,Up(x)), Uj e C'

(f2), j = 1,

...,p,

where / and a are given numbers,

I > max(0, a

), 0 < a < 1. The space

,+t

+a,

(Q)

is a Holder space of functions bounded

together with their derivatives up to the order I + tj, and the latter satisfy the Holder

condition uniformly in x.

The domain

is supposed to be of class

l+x+a

where A = max(—s;, — a

tj),

afj £ C'-

Si+a

{Q,), bfj £ C

ai+a

{dQ,). We assume that

> 1.

Operator A acts from E

into C'~

Si+a

(fi). Denote L

= (A

...,A

Then L\

acts from E

into E

C'-

sl+a

(Q.)

x ... x C'-

(fl). Operator B

{

acts from E

into

ai+a

(dQ,). Denote L

= (B

...,B

Then L

acts from E

into E

l+a

{dn)

... x C'-

{dn). Let E = E

x E

. Then L = (L

) acts from E

into E.

1.2 Limiting domEiins

In this section we define limiting domains for unbounded domains in R

, show their

existence and study some of their properties. We consider an unbounded domain Q C K",

which satisfies the following condition:

Condition 1.1 For each

6 dQ, there exists a neighbourhood

U{XQ)

such that

1. U{XQ)

contains a sphere with the radius S and the center

XQ,

where 6 is independent

of x

;

There exists a homeomorphism ip(x; x

) of the neighbourhood

U(XQ)

on the unit

sphere B = {y : \y\ < 1} in R

such that the images of fl f] U(x

) and dQ n U(x

)

coincide with B

= {y : y

> 0, \y\ < 1} and B

= {y : y

= 0, \y\ < 1}

respectively;

The function ip(x; x

) and its inverse belong to the Holder space

k+OL

with k = l + X.

Their || • ||

fc+Q

-norms are bounded uniformly in x

To define convergence of domains we use the following Hausdorff metric space. Let

A and B denote two nonempty closed sets in R". Denote q{A, B) = svp

a€A

p(a, B),

<;(B,

A) = sup,,

p(b, A), where p(a, B) denotes the distance from a point a to a set B,

and let

g{A,B) = max(<;(A,B),<;{B,A)). (1)

We denote H a metric space of bounded closed nonempty sets in R" with the distance

given by (1). We say that a sequence of domains Q

converges to a domain Q in H

ioc

if -

g(Cl

n B

, Q n B

) -» 0, m -> oo

for any R > 0 and B

= {x : \x\ < R}. Here bar denotes the closure of domains.

293

Definition 1.2 Let

C W

be an unbounded domain, x

6 H, \x

\ —* oo as m —>

oo;

x(z) ^e a characteristic function of fi, and f2

fee a shifted domain defined by the

characteristic function X

(

) = x{

m)- We say that Q* is a limiting domain of the

domain fi if O

—>

f2„ in E(

as m

—»

oo.

Theorem 1.3 If a domain Q satisfies Condition 1.1, then there exists a function f(x)

defined in W such that:

f{x)&

k+a

(W)

;

f(x)>Oiffxen

;

|V/(a;)| >l forxedQ ;

4- min(d(x), 1) <

|/(x)|,

where d(x) is the distance from x to d£l.

Theorem 1.4 Let fi be an unbounded domain satisfying Condition 1.1, x

e fi, \x

—>

oo,

and f(x) be the function constructed in Theorem 1.3. Then there exists a subsequence

and a function f

{x) such that

{x)

= f(x + x

.) —> /»(a) in C

), and the

domain fl* = {x : f,(x) > 0} satisfies Condition 1.1, or fi, = R

. Moreover Q

. —> fi„

in E

loc

, where Q.

= {x :

(x)

> 0}.

1.3 Limiting problems

In the previous section we have introduced limiting domains. Here we will define the

corresponding limiting problems. Let Q be a domain satisfying Condition 1.1 and x(^)

be its characteristic function. Consider a sequence x

€ Q, \x

—•

oo and the shifted

domains Q

defined by the shifted characteristic functions Xm{

)

m)- We

suppose that the sequence of domains Q

converge in S;

to some limiting domain f2».

Definition 1.5 Let u

(x) 6

(Q,

),m

1,2,...

. We say that u

(x) converges to a

limiting function u„(x) G

(Q.

)

(Q.m

—>

fi,) if there exists an extension v

(x) €

(E") of u

(x),m =

1,2,...

and an extension v,(x) g C

) of u,(x) such that v

—>

v,inCtM

Definition 1.6 Let u

(x) e C

(dQ

), m =

1,2,...

. We say that u

(x) converges to

a limiting function u,(x) £

(dQ,

)

(dQ

dQ.*)

if there exists an extension

{x) e

(M.

)

of u

(x),m =

1,2,...

and an extension v,(x) e C

(E") of u„(x) such

thatv

inCi

Remark 1.7 In these definitions u

(x) does not depend on the choice of the extensions

(x) and v

(x).

Indeed,

in Definition 1.5 for any point x G f2» there exists a sequence

€ fi

suh that x

—>

x. Therefore

u,(x) = v,(x) = lim v

) = lim u

Similarly it can be

checked

for Definition 1.6.

294

Theorem 1.8 Let u

e C

k+a

), ||u

j:+o < M, where the constant M is indepen-

dent of m, 0 < k < I + A. Then there exists a function u, £ C

k+a

(Q,*), ||w»||

fc+

< M

and a subsequence u

such that u

—>

u* in

(fl

—»

fi,).

Let u

€ C

k+a

(dQ.

), ||u

k+a < M. Then there exist a function u, G C

k+a

(dO,

\\u

< M and a subsequence u

such that u

—»

{dQ,

—>

9fi*).

We recall that the operator L consists of a pair of operators, L = (L

), where the

operator Lj acts inside the domain and L

is a boundary operator. So we can represent

the boundary problem as

Liu = /i, L

u = f

, (2)

where u e

(Q),

h e -Ei(ft), f

e £

(#n), £ = £i x E

. The coefficients a

(a;) of

the operator Li are defined in Q and the coefficients bij(x) of L

in 9fi. The shifted

coefficients a^ix + x

) and bij(x + x

) satisfy conditions of Theorem 1.3. Therefore we

can define the limiting problem

= f, L

u = g,

where u 6

f € Ei(£l

), g € E

(d£l

), L\ and L

are operators with limiting

coefficients a^(x) € C'-

5i+Q

(0»), b^{x) € C-

ai+a

{d9.

). The operator L = {L

) :

)

—*

E(Q,

) will be called limiting operator.

We note that for a given problem (2) there can exist a set of limiting problems corre-

sponding to different sequences x

and to different converging subsequences of coefficients

of the operators.

1.4 Normal solvability

We introduce limiting domains and limiting operators defined above.

Condition 1.9 For any limiting domain £2, and any limiting operator L the problem

Lu = 0, u G E

(£l

) has only zero solution.

Theorem 1.10 Let Condition 1.9 be satisfied. Then the operator L is normally solvable

and its kernel is finite dimensional.

We suppose now that a little more smoothness of operators and domains is required:

C'-

Si+6

(ty,

i+s

(dQ)

and the domain Q is of class

l+X+S

(3)

with a < 6 < 1.

Theorem 1.11 Let the operator L be normally solvable and its kernel be finite dimen-

sional. Then Condition 1.9 is satisfied.

295

1.5 Dual spaces. Invertibility of limiting operators

We consider now the space E = E(Q) defined above and the space E°, which consists of

functions u e E converging to 0 at infinity in the norm E, i.e., ||w||s(nn{|x|>Ar}) ~* 0 as

—•

oo. We say that u

—»

in 25i

(Q) if this convergence holds in fi n {\x\ < N} for

any N.

Lemma 1.12 Let

be a functional in the dual space

)*,

u 6 E and u

E°, u

e E°,

n|U < 1; and u

—*uin Ei

. Then there exists a limit

<j>=

lim <p{u

(4)

n—*oo

It does not depend on the sequence u

. Ifu

—>

0 in Ei

, then

<p{u

)

—•

We can extend now the functional

<fi

to the space E(Q.). For any u e E(Q) we put

4>(u)

if u e E°(Q) and

4>(u)

= lim

KX)

(/>(u

), where u

G E°(Q.) is an arbitrary

sequence converging to u in E\

. This is a linear bounded functional on

E(fl).

Denote all such functionals E. It is a linear subspace in E*. Suppose that E ^ E*.

We take a functional

£ E*, which does not belong to E. Let ip

be a restriction of

E°. Then V'o S

)*.

As above we can define the functional ip

€ (-E)*. By assumption

i/i

T^ 4>

Denote ip = ip

—

xji

Then

V-

= 0, Vu 6 £°. (5)

Thus we have the following theorem.

Theorem 1.13 The dual space E* is a direct sum of the extension E of (E

)* on E and

of the subspace E consisting of all functionals satisfying (5).

Lemma 1.14 Suppose that the operator L :

EQ —»

E is normally solvable with a finite

dimensional kernel, and the problem Lu = /, / € E is solvable if and only ifipi(f)

—

i = 1,..., N, where

ipi

are linearly independent functionals in E*. Then

ipi

€ E.

These results allow us to prove invertibility of limiting operators (cf. Theorem 1.4).

Theorem 1.15 If the operator L is

Fredholm,

then any its limiting operator is invertible.

Corollary 1.16 // an operator L coincides with its limiting operator, and it is Fredholm,

then it is invertible.

The last result shows in particular that the spectrum of operators with constant,

periodic or quasi-periodic coefficients in unbounded cylinders does not contain eigenvalues

and consists only of points of the essential spectrum.

296

2 Properness and topological degree

It is known that topological degree can be constructed for Fredholm and proper operators

with index 0. The construction is based essentially on two properties: density of regular

values in the image of the operator [14], [17] and invariance of the orientation in homotopy

classes of invertible operators. Different approaches to define the orientation determine

some difference in constructions and in conditions on operators and spaces. The degree is

constructed for abstract operators in Banach spaces (see [2], [5], [6], [7], [10] and references

therein), and applied for elliptic operators of the second order in the one-dimensional

spatial case [3], [18], for cylindrical domains [19], in R" [15].

In this work we construct the degree for general elliptic problems. For this we use

conditions of normal solvability obtained for general linear elliptic problems in unbounded

domains (Section 1); we introduce weighted spaces, without which elliptic operators are

not generically proper; the orientation is adapted to elliptic problems. This allows us to

remove the limitations in application of abstract degree constructions.

2.1 Operators and spaces

We consider general nonlinear elliptic operators

(x,D

«u

,...,D

ISi

) =0, i = l,...,p, refi (6)

with nonlinear boundary operators

Gj(x,D^u

...,D

) = 0, j = 1, ...,r, x e dQ (7)

in a domain fl C K", which can be bounded or unbounded. Here D^^Uk is a vector with

the components

—

/dx"

...dx°

where the multi-index a = (ai,...,a„) takes

all values such that 0 < \a\ — a

+ ... + a

are given integers. The vectors

are defined similarly. The regularity of the functions Fi, Gj, u = (ui,

...,u

and

of the domain Q is determined by

, i,k = 1,..., p, j = 1,..., r (see below).

In what follows we will use also the notations

Fi{x,Vi) = Fi{x,D^u

...,D^u

Gjix^) = Gj(x,D^

...,D^u

The corresponding linear operators are

; =

E E

<^)D

l,...p, xeQ (8)

*=i M<ftk

E E

%(x)D

, j =

l,...r,

xedQ, (9)

*=1 M<7jk

where

**w

drj?k

. W -

dC%

297

and (^ are the vector with the components nf

and C°j., respectively, ordered in the

same way as the derivatives in (6), (7).

The system (6), (7) is called elliptic if the corresponding system (8), (9) is elliptic in the

sense of [1] for all values of parameters r^, £.. When we mention the Shapiro-Lopatinskii

condition for operators (6), (7) we mean the corresponding condition for operators (8),

(9) for any ^ e R

and

€ K

We introduce Banach spaces where operators F

and G, act. Denote by E

0ili

a space

of vector valued functions u(x) = (ui(x), ...,u

(x)), Uj e C

+t;

'(fl), j = 1,

...,p,

where I

and a are given numbers, I > max(0,<Tj), 0 < a < 1. The space C

' "(fi) is a weighted

Holder space with a weight function

fi(x).

By definition u €

{Q)

if \m e C

fc+C

"(fi).

0(I

= C^+'

1+Q

(f2) x ... x C^+^n). (10)

We suppose that the weight function

is a positive infinitely differentiable function

defined for all x £ K

, /j(a;)

—>

oo as |x|

—>

oo, tef! and

|il»^|-+0 as \x\-> oo (xSfi,

|/?|

>0) (11)

for. any multi-index (3.

We suppose that Ft (G;) satisfies the following conditions: for any positive number M

and for all multi-indices

and 7: |/3 + 7| < I - s

+ 2 (\0 +

< I - a

+ 2),

\/3\

< I - s;

(\/3\

< I-in) the derivatives D^D^F

{

(x,ri) (DfDJ'G^x.C)) as functions of x e Sl.tjel"',

\r}\

< M {x G dU, (

, ICI < M) satisfy Holder condition in x uniformly in

(Q and

Lipschitz condition in

(£) uniformly in a; (with constants depending possibly on M).

The domain

is supposed to be of class

l+x+a

where

= max(—s

—<Ti,tj),

and to

satisfy Condition 1.1.

Operator F

acts from E

Qtll

into C

Si+a

(Q.). Denote F =

(i^,...,

). Then F acts

from E

0tji

into £i,

= C^

si+a

(n) x ... x G|r

Sp+a

(n). Operator Gi acts from E

0i)l

into

_<Ti+a

(9n). Denote G = {G

..., G

). Then G acts from E

0:ll

into £

,^ =

1+a

{dn)

... x C

{dtt). Let £„ = Si,

x E

2lli

. Then (F, G) acts from E

0ill

into E

2.2 Properness

We mean properness of the operator A

—>

E in the sense that the restriction of A on

any closed bounded subset of E

is proper. Properness of elliptic operators in bounded

domains follows from the Schauder estimate (see [22]). In unbounded domains they are

not generically proper. The simplest example is provided by the problem u + F(u) = 0,

u(±oo) = 0, which can have a bounded for all x € R solution. Since the solution is

invariant with respect to translation in space, the inverse image of 0 is not compact in

the Holder space

2+a

(R)

or in Sobolev spaces.

To get properness of elliptic operators in unbounded domains we have introduced

weighted spaces.

298

Theorem 2.1 Suppose that the system of operators (6) is uniformly elliptic and for the

system of operators (6), (7) Shapiro - Lopatinskii conditions are satisfied. Assume further

that all limiting

operators

for the operator (8), (9) satisfy Condition NS. Then the operator

(F, G) : E

0ill

—•

Ep is proper.

To prove the homotopy invariance of the topological degree we will consider the op-

erators depending on parameters. Similar to the result on properness in Theorem 2.1 we

obtain properness of the operators with respect to two variables.

2.3 Orientation

Let E

, Ei and E

be Banach spaces. We suppose that E

C E\. Denote E = E

x E

We consider linear operators A

: E

—>

: E

—*

, A = (Ai,A

) : E

—*

E, and

the following class of operators.

Class O is a class of bounded operators A

—>

E satisfying conditions:

(i) Operator (A

+ \I, A

) : E

—>

E is Fredholm with index 0 for all

> 0,

(ii) Equation A\U = 0, A

u = 0(«£ E

) has only zero solution,

(iii) There exists A

= Ao(^4) such that the equation

+ \I)u = 0, A

u = 0 (ti€ E

) (12)

has only zero solution for all

> A

. Here I is the identity operator in E

Definition 2.2 The number

o(A) = (-1)",

where v is the sum of multiplicities of all positive eigenvalues A of the problem (12),

is called orientation of the operator A. Operators A belonging to Class 0 are called

orientable.

Definition 2.3 Operators A

e O and A

£ 0 are said to be homotopic if there exists an

operator A(T) : E

x [0, l]->£ such that A(r) e O for all r €

[0,1],

A(T) is continuous

in the operator norm with respect to r,

AO(J4(T))

is bounded, and A(0) = A

, A(l) = A

Theorem 2.4 If A

and A

are homotopic, then o(A°) =

o(A

2.4 Topological degree

Let E

, Ei, E

and E = E\ x E

be Banach spaces as in Section 2.3, G C

be an open

bounded set. We consider the following classes of linear ($) and nonlinear (F) operators.

Class $ is a class of bounded linear opeators A = (A\,A

) : E

—>

E satisfying condi-

tions:

299

(i) Operator (Ai + IX, A

) : E

-> E is Predholm for all

> 0,

(ii) There exists A

= X

(A) such that operators (A\ + IX, A?) : E

—>

E have inverse

which are uniformly bounded for all

> Ao-

Class F is a class of proper operators / € C

(G, E) such that for any x e G the Frechet

derivative f'(x) belongs to $.

Class H is a class of proper operators f(x, t) € C

(G x

[0,1],

E), which belong to class

F for any t E

[0,1].

Two operators /o(x) : G

—>

25 and f\{x) : G

—>

E are said to be

homotopic if there exists f(x,i) £ H such that : /o(a;) = f{x,0), /i(z) = /(s, 1).

In what follows D will denote an open set which belongs to G with its closure D. Let

a 6 E, f <E C

(G,E), f(x) j= a (x e 3D), where <9D is the boundary of D. Suppose

that the equation f(x) = a (x € D) has finite number of solutions Xi, ...,x

and

f'{x

)

(fc = 1, ...,m) are invertible operators belonging to the class

4>.

Then the orientation o of

these operators is defined. Denote

(f,D;a) = Y,o(f'(x

)). (13)

fc=i

If f(x) / a, x € D, then it is supposed that -y(f, D; a) = 0.

Lemma 2.5 Let f(x,t) e H, a£ E be a regular value o//(.,0) and

f(.,l).

Suppose that

f{x,t)^a{xedD, te[0,l]). rten

(/(.,0),D;a)=7(/(.,l),D;o).

From this lemma we easily obtain the following theorem.

Theorem 2.6 £ei / 6 F and B be a ball \\a\\ < r,a e E such that f(x) ^ a (x e dD)

for all a 6 B. Then for all regular values a

B, -y{f, D; a) does not depend on a.

Using this theorem we can give the following definition of topological degree j(f, D).

Definition 2.7 Let f e F and f(x)

=fi

0 (x e dD). Let B be a ball \\a\\ < r in E such

that f(x) 7^ a (x G dD) for all a £ B. Then j(f, D) = 7(/, D; a) for any regular value

aeB.

Existence of regular values a e B of / follows from Sard-Smale's theorem [17], [14].

Theorem 2.8 (Homotopy invariance). Let f(x,t) e H. Suppose that for an open set

D,DCG, f(x,t) ± 0 (x e 3D,t € [0,1]). Then

(f(;0),D)=

(f(-,l),D).

We note finally that f{x,t) is a Fredholm operator of index 1. To conclude that the

set of regular points for it is dense, we need to assume that f(x,t) e C

(G x

[0,1],

[14],

[17]. This regularity condition is weakened due to [13] where it is shown how to

approximate a C

-mapping by another one for which a is a regular value.

The topological degree constructed here is applicable for elliptic operators satisfying

the conditions imposed above. In particular it is used to prove existence of travelling wave

solutions of reaction-diffusion systems (see [18], [20]).

The proofs of results presented above are given in the complete version of the paper

which will be published elsewhere.