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WITH SINGULAR DATALOREDANA CASO Received 22 December 2005; Revised 22 May 2006; Accepted 12 June 2006 We obtain some uniqueness results for the Dirichlet problem for second-order ellipti

WITH SINGULAR DATA

LOREDANA CASO

Received 22 December 2005; Revised 22 May 2006; Accepted 12 June 2006

We obtain some uniqueness results for the Dirichlet problem for second-order elliptic equations in an unbounded open setΩ without the cone property, and with data de-pending on appropriate weight functions The leading coefficients of the elliptic operator are VMO functions The hypotheses on the other coefficients involve the weight function Copyright © 2006 Loredana Caso This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

LetΩ be an open subset ofRn,n ≥3 Consider inΩ the uniformly elliptic differential operator with measurable coefficients

L = −

n



i, j =1

a i j ∂

2

∂x i ∂x j +

n



i =1

a i ∂

and the Dirichlet problem

Lu =0, u ∈ W2,p(Ω)∩ W o1,p(Ω), (D) withp ∈]1, +∞[

Suppose thatΩ verifies suitable regularity assumptions

Ifp ≥ n, a i j ∈ L ∞(Ω) (i, j=1, , n), and the coe fficients a i(i =1, , n), a satisfy

cer-tain local summability conditions (witha > 0), then it is possible to obtain a uniqueness

result for the problem (D) using a classical result of Alexandrov and Pucci (see [17] for the case of bounded open sets and [6, Section 1] for the unbounded case)

Ifp < n, some more assumptions on the a i j’s are necessary to get uniqueness results for the problem (D) IfΩ is bounded, problem (D) has been widely studied by several authors under various hypotheses on the leading coefficients In particular, if the coefficients ai j

Hindawi Publishing Corporation

Boundary Value Problems

Volume 2006, Article ID 98923, Pages 1 9

DOI 10.1155/BVP/2006/98923

belong to the spaceC o(Ω), then uniqueness results for problem (D) have been obtained (see [12–15]) On the other hand, when the coefficients ai j are required to be discon-tinuous, the classical result by Miranda [16] must be quoted, where the author assumed that thea i j’s belong toW1,n(Ω) (and consider the case p =2) More recently, a relevant contribution has been given in [11,22], where the coefficients ai j are supposed to be in the class VMO and p ∈]1,∞[; observe here that VMO contains both classesC o(Ω) and

W1,n(Ω) (see [10]) IfΩ is unbounded, uniqueness results for problem (D), under as-sumptions similar to those required in [16], have been for istance obtained in [4,18,19] withp =2 and in [5] withp ∈]1,∞[ Moreover, futher uniqueness results for (D), when thea i j’s are in VMO andp ∈]1,∞[, can be found in [6,9]

Suppose now thatΩ has singular boundary In [8], a problem of type (D) has been investigated, with (a i j)x k,a ianda singular near a nonempty subset S ρof∂ Ω, and p =2

In particular, the data are supposed to be depending on an appropriate weight functionρ

related to the distance function fromS ρ

The aim of this paper is to obtain uniqueness results for a Dirichlet problem of type (D) under hypotheses weaker than those of [8] on thea i j’s, and withp > 1 More precisely,

if there exist extensionsa o

i j of the coefficients a i j (i, j =1, , n) in VMO(Ωo)∩ L ∞(Ωo), whereΩois a regular open set containingΩ, and the functions ρa i(i =1, , n), ρ2a are

assumed to be bounded with ess infΩρ2a > 0, we can prove a uniqueness result for the

problem

Lu =0, u ∈ Wloc2,p

Ω\ S ρ

∩ W o1,locp

Ω\ S ρ

∩ L t p(Ω), (D1) whereL p t(Ω), t∈ R, is a weighted Sobolev space

Observe that ifS ρ = ∂Ω and Ω has the segment property, we are able to deduce from the above result that the problem

admits only the trivial solution

2 Notation and function spaces

Let G be any Lebesgue measurable subset of Rn and letΣ(G) be the collection of all

Lebesgue measurable subsets ofG If F ∈ Σ(G), denote by |F|the Lebesgue measure ofF

and byᏰ(F) the class of restrictions to F of functions ζ ∈ C o ∞(Rn) withF ∩suppζ ⊆ F.

Moreover, forp ∈[1, +∞], letLlocp (F) be the class of functions g such that ζg ∈ L p(F) for

allζ ∈ Ᏸ(F).

LetΩ be an open subset ofRn We put

whereB(x, r) is the open ball of radius r centered at x.

Denote byᏭ(Ω) the class of all measurable functions ρ : Ω → R+such that

γ −1ρ(y) ≤ ρ(x) ≤ γρ(y) ∀y ∈Ω, ∀x ∈Ωy, ρ(y)

whereγ ∈ R+is independent ofx and y For ρ ∈Ꮽ(Ω), we put

S ρ =z ∈ ∂Ω : limx → z ρ(x) =0

It is known that

ρ ∈ L ∞loc(Ω), ρ −1∈ L ∞loc

Ω\ S ρ

and, ifS ρ ,

ρ(x) ≤dist

x, S ρ

(see [7,20])

Ifr ∈ N, 1≤ p ≤+∞,s ∈ R, andρ ∈ Ꮽ(Ω), we consider the space W r,p

s (Ω) of distri-butionsu on Ω such that ρ s+ | α |− r ∂ α u ∈ L p(Ω) for|α| ≤ r, equipped with the norm

u W r,p

s (Ω)= 

| α |≤ r

ρ s+ | α |− r ∂ α u

Moreover, we denote byW o r,p s (Ω) the closure of C∞

o(Ω) in Wr,p

s (Ω) and put W0,p

s (Ω)=

L s p(Ω) A detailed account of properties of the above-defined function spaces can be found in [21]

IfΩ has the property

whereA is a positive constant independent of x and r, it is possible to consider the space

BMO(Ω,t) (t∈ R+) of functionsg ∈ L1

loc(Ω) such that

[g]BMO(Ω,t)= sup

x ∈Ω

r ∈]0,t]

−

Ω(x,r)



g −

−

Ω(x,r) g

d y < +∞, (2.8)

where− Ω(x,r) gd y =(1/| Ω(x,r) |) Ω(x,r) g d y We will say that g ∈VMO(Ω) if g∈BMO(Ω)=

BMO(Ω,tA), where

t A =sup

t ∈R+

⎛

⎜

x ∈Ω

r ∈]0,t]

r n

Ω(x,r)  ≤ A1

⎞

⎟

and [g]BMO(Ω,t) →0 fort →0+

3 Some density results

Letρ ∈ Ꮽ(Ω) We consider the following conditions on ρ.

(i1) There exists an open subsetΩoofRnwith the segment property such that

(i2)H =infΩρ − n(x)| Ω(x,ρ(x)) | ∈ R+

Remark 3.1 If condition (i2) holds, then it is possible to find a function σ ∈Ꮽ(Ω)∩

C ∞(Ω)∩ C0,1(Ω) which is equivalent to ρ and such that

∂ α σ(x)  ≤ c α σ1−| α |(x) ∀x ∈Ω,∀α ∈ N n

wherec αis independent ofx (see [20])

Fixr ∈ Nandp ∈[1, +∞[ We denote byW o r,p(Ω\ S ρ) the space of distributionsu on

Ω such that

Lemma 3.2 Assume that condition (i1) holds ThenᏰ(Ω\ S ρ ) is dense in W o r,p(Ω\ S ρ ) Proof Fix u ∈ W o r,p(Ω\ S ρ) and denote byu o the zero extension ofu toΩo It is easy

to prove that u o belongs toW r,p(Ωo) It follows from (i1) that there exists a sequence

{u k } k ∈N ⊂Ᏸ(Ωo) such that

u k −→ u o inW r,p

Ωo



(3.4) (see [1, Theorem 3.18])

Letψ ∈Ᏸ(Ω\ S ρ) such thatψ =1 on suppu Observe that {ψu k } k ∈N ⊂Ᏸ(Ω\ S ρ) and

ψu k − u

W r,p(Ω)≤ψ

u k − u o

W r,p(Ωo)≤ c1 u k − u o

W r,p(Ωo), (3.5) wherec1depends onn, ψ Thus the statement is a consequence of (3.4) 

Lemma 3.3 Assume that conditions (i1) and (i2) hold ThenᏰ(Ω\ S ρ ) is dense in W s r,p(Ω) Proof It follows from (i1), (i2), and [20, Theorem 4.1] that there exists a sequence{δ k } k ∈N

⊂Ᏸ(Ω\ S ρ) such that

lim

k →+∞ ∂ α

1− δ k(x)

=0 ∀x ∈Ω,∀α ∈ N n

sup

k ∈N

∂ α δ k(x)  ≤ c α ρ −| α |(x) ∀x ∈Ω,∀α ∈ N n

wherec αis independent ofx.

Fixu ∈ W s r,p(Ω) Observe that condition (3.7) implies thatδ k u ∈ W s r,p(Ω) for all k∈

N Moreover, by (3.6) we have that

On the other hand, using (o 2.4), it is easy to show that δ k u ∈ W r,p(Ω), and so δk u ∈

W r,p(Ω\ S ρ) For eachk ∈ N,Lemma 3.2yields that there exists a sequence{u k

h } h ∈N ⊂

Ᏸ(Ω\ S ρ) such that

Moreover, letψ k ∈ C ∞ o(Rn) such thatψ k =1 on supp(δ k u) Thus by (2.4), we have

ψ k u k

h − δ k u

W s r,p(Ω)≤ c1 u k

h − δ k u

wherec1∈ R+depends onρ, r, s, k It follows from (3.9) that there existsh k ∈ Nsuch that

ψ k u k

h k − δ k u

W s r,p(Ω)≤1

Ifϕ k = ψ k u k

h k,k ∈ N, we obtain from (3.8) and (3.11) that

Ifr ∈ N, 1≤ p < +∞, we will denote byW o r,ploc(Ω\ S ρ) the set of distributionsu onΩ such thatζu ∈ W o r,p(Ω) for any ζ∈Ᏸ(Ω\ S ρ)

Lemma 3.4 Assume that conditions (i1) and (i2) hold Then

o

W r,ploc

Ω\ S ρ

∩ W s r,p(Ω)= W o r,p s (Ω). (3.13)

Proof It is clearly enough to show that

o

W r,ploc

Ω\ S ρ

∩ W s r,p(Ω)⊆ W o r,p s (Ω) (3.14) Letu ∈ W o r,ploc(Ω\ S ρ)∩ W s r,p(Ω) and consider a sequence{δ k } k ∈N ⊂Ᏸ(Ω\ S ρ) satis-fying (3.6) and (3.7) Since eachδ k u belongs to W o r,p(Ω), for any k ∈ N, there exists a sequence{u k

h } h ∈N ⊂ C o ∞(Ω) such that

u k

Letψ k ∈ C ∞ o(Rn) such thatψ k =1 on supp(δ k u) Since ψ k u k h ∈ C ∞ o(Ω), the same argument used inLemma 3.3allows to deduce from (3.15) that for everyk ∈ N, there existsh k ∈ N

such that

ψ k u k

h k − δ k u

We putϕ k = ψ k u k h kfor eachk Therefore it follows from (3.16) that

ϕ k − u

W s r,p( Ω)≤1k+δ k u − u

As the sequence{δ k } k ∈Nsatisfies (3.8), (3.17) yields that the sequence{ϕ k } k ∈Nconverges

4 Main results

LetΩ be an open subset ofRn,n ≥3, with the segment property Fixρ ∈Ꮽ(Ω)∩ L ∞(Ω) and consider the following condition onΩ

(h1) There exists an open subsetΩoofRnwith the uniformC1,1-regularity property, such that

Remark 4.1 If condition (h1) holds andρ ∈Ꮽ(Ω)∩ L ∞(Ω), then Ω satisfies (i2) (see [20])

Letp ∈]1, +∞[, and letL be the differential operator in Ω defined by

L = −

n



i, j =1

a i j ∂2

∂x i ∂x j +

n



i =1

a i ∂

Consider the following conditions on the coefficients of L:

(h2) there exist extensionsa o i jofa i jtoΩosuch that

a o i j = a o ji ∈ L ∞

Ωo



∩VMO

Ωo



, i, j =1, , n,

∃ν ∈ R+:

n



i, j =1

a o i j ξ i ξ j ≥ ν|ξ |2 a.e inΩo,∀ξ ∈ R n, (4.3)

(h3)

a i ∈ L ∞1(Ω), i =1, , n, a ∈ L ∞2(Ω),

a o =ess inf Ω



σ2(x)a(x)

whereσ is the function defined inRemark 3.1

Moreover, we suppose that the following hypothesis onρ holds:

(h4)

lim

k →+∞



sup

Ω\Ωk



σ(x)

x+σ(x)

σ(x)

xx



where

Ωk =



x ∈ Ω : σ(x) >1

k



In the proof of our main theorem, we need the following uniqueness result.

Lemma 4.2 Assume that conditions (h1)–(h4) hold and also that p > n/2 Then the problem

Lu =0, u ∈ Wloc2,p(Ω), lim

x → x o



σ s u

(x) =0, ∀x o ∈ ∂Ω, lim

| x |→+∞



σ s u

(x) =0, if Ω is unbounded,

(4.7)

admits only the zero solution.

Proof The statement can be proved as [2, Corollary 5.4] In fact, the proof of that result also works if the conditionS ρ = ∂Ω is replaced by the assumption (h1) 

Theorem 4.3 Suppose that conditions (h1)–(h4) are satisfied Then for any t ∈ R, the prob-lem

u ∈ Wloc2,p

Ω\ S ρ

∩ W o1,locp

Ω\ S ρ

admits only the zero solution.

Proof Let u be a solution of the problem (4.8) It follows from [3, Theorem 5.2] that

u ∈ W t+22,p(Ω) Moreover, u belongs to W1,p

t+1(Ω), and henceLemma 3.4 yields thatu ∈

W t+22,p(Ω)∩ W o1,t+1 p(Ω) UsingRemark 3.1, it is easy to prove that

Putv = σ t+2 u and denote by v othe zero extension ofv toΩo Then

v o ∈ W2,p

Ωo



∩ W o 1,p

Ωo



(4.10)

byLemma 3.3 Suppose first thatp > n/2 By the Sobolev embedding theorem, v obelongs

toC0(Ωo)∩ W o 1,p(Ωo), and hencev o | ∂Ωo =0 On the other hand,v o ∈ W2,p(Ωo), so that another application of the Sobolev embedding theorem gives that lim| x |→+∞ v o(x) =0 Thus by (h1), we have that

lim

| x |→+∞



σ t+2 u

(x) =0, 

σ t+2 u

In this case the statement follows now fromLemma 4.2

Assume now thatp ∈]1,n/2] Then by the Sobolev embedding theorem, we have that

v o ∈ L q(Ωo), where 1/q ≥1/ p −2/n It follows from [3, Theorem 5.2] thatv o ∈ W22,q(Ωo), and hencev o belongs toW2,q(Ωo) by (2.4) Ifq > n/2, the previous case can be used to

complete the proof If finallyq ≤ n/2, an iterated application of [3, Theorem 5.2] yields thatv o ∈ W2,q 

(Ωo) withq  > n/2 Thus the first case applies again to complete the proof.



As an application ofTheorem 4.3, we consider the case S ρ = ∂Ω (examples of such situation can for instance be found in [20]) The condition (h ) is obviously satisfied by

eachΩo ⊃ Ω with the uniform C1,1-regularity property; in this case, condition (h2) means that the coefficients ai jadmit extensions outsideΩ in the class L ∞(Ωo)∩VMO(Ωo)

Corollary 4.4 Assume that (h2), (h3), (h4) hold and that S ρ = ∂ Ω Then the problem

admits only the zero solution.

Proof The statement follows fromTheorem 4.3observing that, in this case,u belongs to

o

References

[1] R A Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2] L Caso, Bounds for elliptic operators in weighted spaces, Journal of Inequalities and Applications

2006 (2006), Article ID 76215, 14 pages.

[3] , Regularity results for singular elliptic problems, Journal of Function Spaces and

Applica-tions 4 (2006), Article ID J171, 17 pages.

[4] L Caso, P Cavaliere, and M Transirico, Existence results for elliptic equations, Journal of

Mathe-matical Analysis and Applications 274 (2002), no 2, 554–563.

[5] , Solvability of the Dirichlet problem in W2,p for elliptic equations with discontinuous

coef-ficients in unbounded domains, Le Matematiche (Catania) 57 (2002), no 2, 287–302 (2005).

[6] , Uniqueness results for elliptic equations VMO-coe fficients, International Journal of Pure

and Applied Mathematics 13 (2004), no 4, 499–512.

[7] L Caso and M Transirico, Some remarks on a class of weight functions, Commentationes

Math-ematicae Universitatis Carolinae 37 (1996), no 3, 469–477.

[8] , The Dirichlet problem for second order elliptic equations with singular data, Acta

Mathe-matica Hungarica 76 (1997), no 1-2, 1–16.

[9] P Cavaliere, M Transirico, and M Troisi, Uniqueness result for elliptic equations in unbounded

domains, Le Matematiche (Catania) 54 (1999), no 1, 139–146 (2000).

[10] F Chiarenza, M Frasca, and P Longo, Interior W2,p estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche di Matematica 40 (1991), no 1, 149–168.

[11] ,W2,p -solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Transactions of the American Mathematical Society 336 (1993), no 2, 841–853.

[12] M Chicco, Solvability of the Dirichlet problem in H2,p(Ω) for a class of linear second order elliptic

partial differential equations, Bollettino della Unione Matematica Italiana 4 (1971), 374–387.

[13] D Gilbarg and N S Trudinger, Elliptic Partial Di fferential Equations of Second Order, 2nd ed.,

Fundamental Principles of Mathematical Sciences, vol 224, Springer, Berlin, 1983.

[14] D Greco, Nuove formole integrali di maggiorazione per le soluzioni di un’equazione lineare di tipo

ellittico ed applicazioni alla teoria del potenziale, Ricerche di Matematica 5 (1956), 126–149.

[15] A I Koˇselev, On boundedness of L p of derivatives of solutions of elliptic differential equations,

Matematicheskij Sbornik Novaya Seriya 38(80) (1956), 359–372.

[16] C Miranda, Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coe fficienti

discontinui, Annali di Matematica Pura ed Applicata Serie Quarta 63 (1963), no 1, 353–386.

[17] C Pucci, Limitazioni per soluzioni di equazioni ellittiche, Annali di Matematica Pura ed Applicata.

Serie Quarta 74 (1966), no 1, 15–30.

[18] M Transirico and M Troisi, Second-order nonvariational elliptic equations in unbounded open

sets, Annali di Matematica Pura ed Applicata Serie Quarta 152 (1988), 209–226 (Italian).

[19] , Further contributions to the study of second-order elliptic equations in unbounded open

sets, Bollettino della Unione Matem`atica Italiana Serie VII B 4 (1990), no 3, 679–691 (Italian).

[20] M Troisi, On a class of weight functions, Rendiconti della Accademia Nazionale delle Scienze

detta dei XL Memorie di Matematica e Applicazioni Serie V Parte I 10 (1986), no 1, 141–152

(Italian).

[21] , On a class of weighted Sobolev spaces, Rendiconti della Accademia Nazionale delle

Scienze detta dei XL Memorie di Matematica e Applicazioni Serie V Parte I 10 (1986), no 1,

177–189 (Italian).

[22] C Vitanza, A new contribution to the W2,p regularity for a class of elliptic second order equations with discontinuous coefficients, Le Matematiche (Catania) 48 (1993), no 2, 287–296 (1994).

Loredana Caso: Dipartimento di Matematica e Informatica, Facolt`a di Scienze Matematiche, Fisiche e Naturali (MM FF NN.), Universit`a degli Studi di Salerno, Via Ponte don Melillo,

Fisciano 84084, Italy

E-mail address:lorcaso@unisa.it

[3] , Regularity results for singular elliptic problems, Journal of Function Spaces and

Applica-tions... 469–477.

[8] , The Dirichlet problem for second order elliptic equations with singular data, Acta

Mathe-matica Hungarica 76... , Solvability of the Dirichlet problem in W2,p for elliptic equations with discontinuous

coef-ficients in unbounded domains,