Solvability Properties of Linear Elliptic Boundary Value Problems with Non-smooth Data

Solvability Properties of Linear Elliptic Boundary Value Problems with Non-smooth Data

Lutz Recke

Fachbereich Mathematik der Humboldt-Universitat zu Berlin Unter den Linden 6, 10099 Berlin, FRG

In this paper linear elliptic boundary value problems of second order with non-smooth data (^L1-coecients, Lipschitz domain, mixed boundary conditions) are considered. It is shown that the weak solutions are Holder continuous and that they depend smoothly { in the sense of Holder spaces { on the coecients of the equation.

1 Introduction

In this paper we consider boundary value problems for linear elliptic equations of the type

N

X

ij=1

^;@j(^aij@iu+^bju) +^cj@ju+^du] = ^;XN

j=1

@

j f

j+^g in

N

X

ij=1

(^aij@iu+^bju)^j = ^XN

j=1 f

j

j on ;

u = 0 on ^@n;^:

9

>

>

>

>

>

>

=

>

>

>

>

>

>

(1.1)

In (1.1) is a bounded Lipschitz domain in^IRN, and ; is a subset of the boundary^@ of . By^@j we denote the partial derivative with respect to the^j-th component of the space variable (^x1:::xN)², and (^1:::N) :^@!IRN is the unit outward normal eld on^@. The coecients ^aij=^aji,^bj,^cj and^dare bounded measurable functions on , and it is supposed that there exists an^">0 such that

N

X

ij=1 a

ij(^x)^ij"XN

j=1

2

j for all (^1:::N)^2IRN and for almost all ^x2:

It is well-known that each weak solution^uof the boundary value problem (1.1) is Holder continuous up to the boundary if, for example,

f

j 2L

p() ^g2Lp2() with ^p>N (1.2)

and if ; satises some regularity assumption (see, e.g.,Gilbarg,Trudinger3] for the case of ; =,

Troianiello 13] for the case when ; is open and closed in^@ andStampacchia 12],Murthy,

Stampacchia7] for more general cases).

In the present paper we will prove that the weak solution ^uof (1.1) { if it is unique { depends smoothly in the sense of a Holder space ^C0() on the coecients ^aijbjcjd2L1() (if (1.2) is satised and if ; is regular in the sense of Denition 2.1 below). This result seems to be new (in case of ^{N >}2) even if ; =(pure Dirichlet boundary conditions) or if ; =^@ (pure natural boundary conditions). Moreover, this result is of some interest because it allows to apply theorems of the dierential calculus (Implicit Function Theorem, Sard-Smale Theorem, Liapunov-Schmidt Procedure in bifurcation problems) to quasilinear elliptic boundary value problems with non-smooth data (cf.

Recke 9] and 10]).

In the case of ^N = 2 the smooth (in the sense of ^C0()) dependence of the weak solution of (1.1) on the coecients follows from the work of Groger 4]. Moreover, in the case of ^N = 2 this result holds true for boundary value problems for linear elliptic systems as well, whereas in case of

N >2 there are examples of linear elliptic systems with bounded measurable coecients which have unbounded weak solutions (cf., e.g., Giaquinta1]).

In this work, we will show that the weak solutions of mixed boundary value problems for "weakly coupled" linear elliptic systems depend smoothly (in the sense of^C0()) on the ^L1-coecients of the equations (if the right-hand sides of the equations are of the type (1.2) and if ; is regular in the sense of Denition 2.1 below).

We do not consider the solution regularity for mixed boundary value problems for linear ellip- tic equations with smooth coecients, see Pryde 8], Simanca 11] and Liebermann 6] for that question.

The present paper is organized as follows.

In the remaining part of Section 1 we introduce some notation and some results about Campanato spaces.

In Section 2 we prove a regularity result for weak solutions of (1.1) in the case of ^bj = ^cj = 0,

d= 1.

In Section 3 we introduce two scales of Banach spaces^U=^Wo12(;) and^V(^N;2^<<N) such that there are continuous embeddings ^{U ,! Wo12}(;)^\C0() (with = ^;N+22 ) and

V

,!W

;12

o (;) and such that the operator associated with the boundary value problem (1.1) (with^bj=^cj= 0 and^d= 1) is an isomorphism from^Uonto^V(ifis suciently close to^N;2). ^U is the space of all elements^uof the Sobolev space^Wo12(;) such that^rubelongs to the Campanato space ^L2(^IRN), and ^V is the image of^U with respect to the duality map of the Hilbert space

W 12

o (;).

In Section 4 we consider the case of arbitrary coecients^bj,^cjand^d, and we show that the operator associated with (1.1) is a Fredholm operator (index zero) from^Uinto^Vand that it depends smoothly (in the sense of the operator norm in^L(^UV)) on the coecients^aij,^bj,^cj and^d(ifis suciently close to ^N;2).

Finally, in Section 5 we show that our results about the boundary value problems for linear elliptic equations of type (1.1) hold for "weakly coupled" linear elliptic systems as well.

Let us introduce some notation.

In this paper ^N 2 is a natural number. For subsets ^G of^IRN we denote by^G, ^@G and ^G the interior, the boundary, and the closure of^G, respectively.

A bijective map from one subset of ^IRN onto another is called a Lipschitz transformation if and ^;1 are Lipschitzian.

For^2IRN we writefor their Euclidean scalar product, and^jj:=^p is the Euclidean norm of .

By^SIN we denote the space of all real symmetric^{N N}-matrices. If ^A = (^aij) ^2SIN and = (^1:::N)^2IRN, then we write^A2IRN for the vector with components ^PN

j=1 a

ij

j (ⁱ= 1^:::n), i.e. for the application of^A on, and

jAj:= sup^fjAj:^{2IRN jj} 1^g is the Euclidean operator norm of^A.

Let be a bounded open subset of^IRN.

We write ^L1(),^L1(^IRN) and^L1(^SIN) for the spaces of bounded measurable maps from into^IR,^IRN, and^SIN, respectively. The norms of these spaces are denoted by ^kk1, for example

kAk

1:= inf^fr>0 :^jA(^x)^{j r} for almost all ^x2g for ^A2L1(^SIN)^:

Analogously, for 1 ^p<1we write ^kkp for the norms in^Lp() and ^Lp(^IRN). The gradient of ^u2Lp() and the divergence of^{f 2Lp}(^IRN) (derivatives in the sense of distributions) will be denoted by ^ruand div^f, respectively, and^W1p() is the usual Sobolev space with the norm

kuk

1p:= (^kukpp+^krukpp)^p1:

Finally, for 0 ^<< 1 we denote by ^C0() the space of all functions from into ^IRthat are Holder continuous with exponent . The norm of^u2C0() is

sup^fju(^x)^j:^x2g+ sup^nju(^x)^;u(^y)^j

jx;y j

:^{xy2 x6}=^yo:

Now, let us recall some notation and facts about Campanato spaces (cf. Kufner, John, Fucik

5],Troianiello13],Giaquinta 2]).

For 1 ^p<1and 0 ^N+^pwe denote by^Lp() the space of all^u2Lp() such that ^u]^p:=sup^{nr; Z}

(xr )

ju(^y)^;uxrjpd^y:^x2r>0^{o1p <1:} (1.3) In (1.3) we have used the notations

(^xr) := ^fy2 :^{jx;y j<r g}

u

xr := 1

j(^xr)^j

Z

(xr )

u(^y)d^y

where ^j(^xr)^j is the ^N-dimensional Lebesgue measure of (^xr). The space ^Lp() is a Banach space with the norm

juj

p:= (^kukpp+ ^u]^pp)^p1: (1.4)

Analogously,^Lp(^IRN) is the space of all^{f 2Lp}(^IRN) with components from^Lp(), and the norm in^Lp(^IRN) is dened similarly to (1.3), (1.4) and denoted by ^jjp, too. Finally, for the sake of simplicity, we will use the notations

L

2() := ^L2()

juj

2 := ^kuk2 for ^u2L2()

for ^<0

and, if necessary, we indicate the dependence of the norms on the domain by an additional index.

Theorem 1.1

(i) Let ^{p q} and ^{N;q N;p} . Then ^Lq() is continuously embedded into

L p().

(ii) Letbe a Lipschitz transformation fromonto ~. Then, for all ^N, there exists a^c>0 such that it holds for all ^u2L2(~) that^u2L2()and ^{kuk2 ckuk2~}.

Theorem 1.2

Lethave a Lipschitz boundary. Then the following is true:

(i) If ^{N <<N}+ 2, then ^L2() is isomorphic to ^C0() with = ^;N2 .

(ii) For all ^<N there exists a ^c>0such that for all ^u2L2() and ^v2L1() the product

uv belongs to ^L2() and ^{juv j2 cjuj2kv k1}.

(iii) For all ^<N there exists a ^c>0such that for all^u2W12() with^ru2L2(^IRN)it holds that ^u2L2+2() and ^{juj2+2 c}(^kuk2+^jruj2).

2 Regularity of the Solutions

In this section we consider mixed boundary value problems of the type

;div^Aru+^u = ^;div^f+^g in

Aru = ^f on ;

u = 0 on ^@n;^:

9

=

(2.1)

In (2.1) ^IRN is a bounded Lipschitz domain, ; is a part of^@,:^@!IRN is the unit outward normal eld on ^@,^f : ^!IRN and^g: ^!IRare given right-hand sides, and ^A2L1(^SIN) is a given matrix valued function which is positive-denite almost everywhere.

In order to prove regularity properties for the weak solutions^uof (2.1), one has to impose appro- priate conditions on ;. Following the work ofGroger4] we will formulate such conditions in terms of the set

G:= ;^: (2.2)

Denition 2.1

A bounded subset ^Gof ^IRN will be called regular if, for every ^{x2 @G}, there exist subsets ^U and ~^U of ^IRN and a Lipschitz transformation from ^U onto ~^U such that ^U is an open neighbourhood of ^xin^IRN and(^G\U) is one of the following sets:

E

1 := ^fx2IRN :^jxj<1 ^{xN <}0^g

E

2 := ^fx2IRN :^jxj<1 ^xN 0^g: (2.3)

Remark 2.2

The Denition 2.1 does not change if one supplies, e.g., the sets

E

3 := ^fx2E2:^{xN <}0 or ^x1>0^g

E

4 := ^fx2E2:^x1>0^g

to the list (2.3), because there exist Lipschitz transformations from ^E2 onto ^E3 and from ^E2 onto

E

4, respectively. Thus, the regularity of ^G means, roughly speaking, that ^G is bounded, ^@G is a Lipschitzian hypersurface in^IRN, and ; :=^GnG (cf. (2.2)) and ^{@G n}; = ^GnGare separated by a Lipschitzian hypersurface of^@G.

Denition 2.3

(i) Let^GIRN be regular. For 1 ^p<1we denote by^Wo1p(^G) the closure in

W

1p(^G) of the set of the restrictions on^G of all smooth functions^u:^{IRN !IR}with compact support supp^uand such that supp^u\( ^GnG) =. The space ^Wo12(^G) is equipped with the norm^kk12 of

W 12(^G).

(ii) Let ^IRN and ;^@ be such that ; is regular. A function^u2Wo12(;) is called a weak solution of the boundary value problem (2.1) if

Z

(^Arurv+^uv)d^x=^Z

(^{f rv}+^{g v})d^x for all ^{v 2Wo12}(^G)^:

(iii) For^GIRN and^">0 we denote by^A"(^G) the set of all^A2L1(^GSIN) such that 1

"

jj 2

>A(^x)^>"jj2 for all ^{2IRN nf}0^g and for almost all ^x2G:

(iv) A regular set ^GIRN is called admissible if, for each^">0, there exists a^">N;2 such that for all ^",^A2A"(^G),^{f 2L2}(^GIRN),^g2L2;2(^G) and^u2Wo12(^G) with

Z

(^Arurv+^uv)d^x=^Z

(^frv+^{g v})d^x for all ^v2Wo12(^G) (2.4) it holds that^ru2L2(^GIRN) and

jruj

2

c(^jfj2+^{jg j2;2}+^kuk12) (2.5)

where the constant ^cin (2.5) does not depend on^A,^f, ^gand^u, but only on^"and.

Remark 2.4

Let^GIRN be regular.

The Lax-Milgram Lemma yields that for all ^{A 2 A"}(^G), ^{f 2 L2}(^GIRN) and ^{g 2 L2}(^G) there exists exactly one weak solution ^u2Wo12(^G) of the boundary value problem (2.1) with :=^G and

; :=^GnG, and the linear map

(^fg)^2L2(^GIRN)^L2(^G)^7!u2Wo12(^G) is continuous.

Hence, ^G is admissible i for each ^{" >} 0 there exists a ^{" > N ;}2 such that for all ^",

A2A

"(^G),^{f 2L2}(^GIRN) and^g2L2;2(^G) the gradient^ruof the weak solution^uof (2.1) (with :=^G and ; := ^GnG) belongs to ^L2(^GIRN) and that the map

(^fg)^2L2(^GIRN)^L2;2(^G)^7!(^uru)^2Wo12(^G)^L2(^GIRN) is continuous.

Therefore, Theorem 1.2(i) and (iii) imply the following: If ^Gis admissible, then, for each ^{" >}0, there exists a ^{" > N ;}2 such that for all ² (^{N ;}2^"], ^{A 2 A"}(^G), ^{f 2 L2}(^GIRN) and

g 2 L

2;2(^G) the weak solution ^u of (2.1) (with :=^G and ; := ^GnG) belongs to ^C0( ^G) with

=^;N+22 , and the map

(^fg)^2L2(^GIRN)^L2;2(^G)^7!u2C0( ^G) is continuous.

In the remaining part of this section we will prove three lemmas which will lead to the

Theorem 2.5

Each regular set ^GIRN is admissible.

A crucial point for the proof of Theorem 2.5 is the following regularity result for weak solutions of the boundary value problem (2.1) (Troianiello13] Theorem 2.19).

Theorem 2.6

If ^IRN is a bounded domain with ^C1-boundary and if ;^@ is open and closed in ^@, then ; is admissible.

Remark 2.7

(i) In fact, we will use the assertion of Theorem 2.6 in the special case of =^fx2

IR

N :^jxj<1^gand ; =, only.

(ii) It is obvious that ; is regular if ^@ is ^C1-smooth and if ; is open and closed in ^@. Hence, in fact Theorem 2.6 asserts that (2.5) follows from (2.4) if^@ is ^C1-smooth and if ; is open and closed in^@. That is exactly the formulation of Theorem 2.19 ofTroianiello13].

Let us begin the sequence of the three lemmas with

Lemma 2.8

^E1 and ^E2 (cf. (2.3)) are admissible.

Proof.

Let^Eo:=^fx2IRN :^jxj<1^g. For^k= 12 and^u2L2(^Ek) we dene^Sku2L2(^Eo) by (^Sku)(^x) :=

u(^x) for ^x2Ek

(^;1)^ku(^x0;xN) for ^x= (^x0xN)^2EonEk:

Thus, ^S1uand^S2uare the extensions of^uto^Eo"by antireection" and "by reection", respectively.

It is well-known that^u2Wo12(^Ek) i^Sku2Wo12(^Eo), and in this case

kS

k uk

12Eo=^p2^kuk

12E

k :

Moreover, for 0^<<N we have^u2L2(^Ek) i^Sku2L2(^Eo), and, in this case,

jS

k uj

2E

o

p2^juj

2E

k

p2^jSkuj2Eo (cf. Troianiello13] p. 31 and p. 36).

Now, we extend vector valued maps ^f = (^f1:::fN)^{2 L2}(^EkIRN) to ^Skf = (^{f1(k ):::fN(k )}) ²

L

2(^EoIRN) and matrix valued maps^A= (^aij)^2A"(^Ek) to ^SkA= (^{a(k )ij} )^2A"(^Eo) by

f (k )

j (^x0xN) := (^;1)^kfj(^x0;xN) for ^j<N

f (k )

N (^x0xN) := (^;1)^{k +1fN}(^x0;xN)

)

a (k )

ij (^x0xN) := (^;1)^{k +1aij}(^x0;xN) for ^ij<N or ⁱ=^j =^N

a (k )

ij (^x0xN) := (^;1)^kaij(^x0;xN) otherwise

)

for (^x0xN)^2EonEk. Then we get

S

k(^Af) = (^SkA)(^Skf)

S

k(^ru) = ^r(^Sku)

for all^A2A"(^Ek),^ff~^2L2(^EkIRN) and^u2W12(^Ek).

Finally, for^k= 12 and^v2Wo12(^Eo) we dene^{Tk 2Wo12}(^Ek) by

(^Tkv)(^x0xN) := 12^v(^x0xN) + (^;1)^kv(^x0;xN)] for (^x0xN)^2Ek:

T

1

vand^T2vare the restrictions to^Ek of the antisymmetric and the symmetric part of^v, respectively, and we have

Z

Eo

^Skfrv+ (^Skg)^v]d^x= 2^Z

Ek

^{f r}(^Tkv) +^{g Tkv}]d^x for all^{f 2L2}(^EkIRN),^g2L2(^Ek) and^v2Wo12(^Eo).

Now, take^A2A"(^Ek),^{f 2L2}(^EkIRN),^g2L2;2(^Ek) and^u2Wo12(^Ek) such that

Z

Ek

(^Arurv+^uv)d^x=^Z

Ek

(^frv+^{g v})d^x for all ^v2Wo12(^Ek)^: Then, for all ^w2Wo12(^Eo) it follows

Z

Eo

(^SkA)^r(^Sku)^rw+ (^Sku)^w]d^x= 2^Z

Ek

^Arur(^Tkw) +^uTkw]d^x=

= 2^Z

Ek

^fr(^Tkw) +^{g Tkw}]d^x=^Z

Eo

^Skfrw+ (^Skg)^w]d^x:

However ^Eo is admissible (Theorem 2.6). Therefore, there exists^">N;2 such that for all ^"

we have^r(^Sku) =^Sk(^ru)^2L2(^EoIRN) (and, hence,^ru2L2(^EkIRN)) and

jruj

2E

k

jS

k ruj

2E

o

c(^jSkfj2Eo+^{jSkg j2;2Eo}+^kSkuk12Eo)

p2^c(^jfj

2E

k

+^{jg j}

2;2E

k

+^kuk

12E

k

)

where the constant ^{c >} 0 does not depend on ^A, ^f, ^g and ^u, but only on^" and . Hence, ^Ek is admissible.

The next lemma is

Lemma 2.9

Let^GIRN be admissible and be a Lipschitz transformation from^G onto ^H. Then

H is admissible.

Proof.

Obviously,^His regular.

Let us denote by ⁰(^x) the derivative of in ^x (⁰(^x) exists for almost all ^{x 2 G}, and ⁰ is a bounded measurable map from ^Ginto the space of the real ^{N N}-matrices). For the inverse and the inverse transpose of⁰(^x) we write⁰(^x)^;1and⁰(^x)^;, respectively. If^L+and^L; are Lipschitz constants of and^;1, respectively, then we have

j

0(^x)^{;1jL;1+ jj} for all ^2IRN

L N

;

jdet⁰(^x)^{j LN+}

9

=

for almost all ^x2G: (2.6)