WITH SINGULAR DATA
LOREDANA CASOReceived 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 setΩwithout 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
LetΩbe an open subset ofRn,n≥3. Consider inΩthe uniformly elliptic differential
operator with measurable coefficients
L= −
n
i,j=1 ai j ∂
2
∂xi∂xj
+
n
i=1 ai∂x∂
i
+a, (1.1)
and the Dirichlet problem
Lu=0, u∈W2,p(Ω)∩Wo1,p(Ω), (D)
withp∈]1, +∞[.
Suppose thatΩverifies suitable regularity assumptions.
Ifp≥n,ai j∈L∞(Ω) (i,j=1,. . .,n), and the coefficientsai(i=1,. . .,n),asatisfy
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 theai j’s are necessary to get uniqueness results for
the problem (D). IfΩis bounded, problem (D) has been widely studied by several authors under various hypotheses on the leading coefficients. In particular, if the coefficientsai j
Hindawi Publishing Corporation Boundary Value Problems
Volume 2006, Article ID 98923, Pages1–9
belong to the spaceCo(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 theai j’s belong toW1,n(Ω) (and consider the casep=2). More recently, a relevant
contribution has been given in [11,22], where the coefficientsai j are supposed to be in
the class VMO and p∈]1,∞[; observe here that VMO contains both classesCo(Ω) and
W1,n(Ω) (see [10]). IfΩis 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 theai j’s are in VMO andp∈]1,∞[, can be found in [6,9].
Suppose now thatΩhas singular boundary. In [8], a problem of type (D) has been investigated, with (ai j)xk,aiandasingular near a nonempty subsetSρof∂Ω, andp=2. In particular, the data are supposed to be depending on an appropriate weight functionρ related to the distance function fromSρ.
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 theai j’s, and withp >1. More precisely,
if there exist extensionsao
i j of the coefficientsai j (i,j=1,. . .,n) in VMO(Ωo)∩L∞(Ωo),
whereΩois a regular open set containingΩ, and the functionsρai(i=1,. . .,n),ρ2aare
assumed to be bounded with ess infΩρ2a >0, we can prove a uniqueness result for the problem
Lu=0, u∈Wloc2,p
Ω\Sρ∩ o
W1,locp
Ω\Sρ∩Ltp(Ω), (D1)
whereLpt(Ω),t∈R, is a weighted Sobolev space.
Observe that ifSρ=∂ΩandΩhas the segment property, we are able to deduce from
the above result that the problem
u∈Wloc2,p(Ω)∩Lp(Ω), Lu=0, (D2)
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. IfF∈Σ(G), denote by|F|the Lebesgue measure ofF and byᏰ(F) the class of restrictions toFof functionsζ∈Co∞(Rn) withF∩suppζ⊆F.
Moreover, forp∈[1, +∞], letLlocp (F) be the class of functionsgsuch thatζg∈Lp(F) for allζ∈Ᏸ(F).
LetΩbe an open subset ofRn. We put
Ω(x,r)=Ω∩B(x,r) ∀x∈Rn,∀r∈R
+, (2.1)
whereB(x,r) is the open ball of radiusrcentered atx.
Denote byᏭ(Ω) the class of all measurable functionsρ:Ω→R+such that
whereγ∈R+is independent ofxandy. Forρ∈Ꮽ(Ω), we put
Sρ=
z∈∂Ω: lim
x→zρ(x)=0
. (2.3)
It is known that
ρ∈L∞loc(Ω), ρ−1∈L∞
loc
Ω\Sρ
, (2.4)
and, ifSρ= ∅,
ρ(x)≤distx,Sρ
∀x∈Ω (2.5)
(see [7,20]).
Ifr∈N, 1≤p≤+∞,s∈R, andρ∈Ꮽ(Ω), we consider the spaceWsr,p(Ω) of
distri-butionsuonΩsuch thatρs+|α|−r∂αu∈Lp(Ω) for|α| ≤r, equipped with the norm
uWsr,p(Ω)=
|α|≤r
ρs+|α|−r∂αu
Lp(Ω). (2.6)
Moreover, we denote byWo rs,p(Ω) the closure ofC∞o(Ω) inWsr,p(Ω) and putWs0,p(Ω)=
Lsp(Ω). A detailed account of properties of the above-defined function spaces can be
found in [21].
IfΩhas the property
Ω(x,r)≥Arn ∀x∈Ω,∀r∈]0, 1], (2.7)
whereAis a positive constant independent ofxandr, it is possible to consider the space BMO(Ω,t) (t∈R+) of functionsg∈L1loc(Ω) such that
[g]BMO(Ω,t)= sup
x∈Ω
r∈]0,t] −
Ω(x,r)
g−−
Ω(x,r)
gd y <+∞, (2.8)
where−Ω(x,r)gd y=(1/|Ω(x,r)|)Ω(x,r)g d y. We will say thatg∈VMO(Ω) ifg∈BMO(Ω)= BMO(Ω,tA), where
tA=sup t∈R+
⎛ ⎜ ⎜ ⎝sup
x∈Ω
r∈]0,t] rn
Ω(x,r)≤ 1 A
⎞ ⎟ ⎟
⎠, (2.9)
3. Some density results
Letρ∈Ꮽ(Ω). We consider the following conditions onρ.
(i1) There exists an open subsetΩoofRnwith the segment property such that
Ω⊂Ωo, ∂Ω\Sρ⊂∂Ωo. (3.1)
(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 toρand such that
∂ασ(x)≤c
ασ1−|α|(x) ∀x∈Ω,∀α∈Nno, (3.2)
wherecαis independent ofx(see [20]).
Fixr∈Nandp∈[1, +∞[. We denote byWor,p(Ω\S
ρ) the space of distributionsuon
Ωsuch that
u∈Wr,p(Ω), suppu⊂Ω\S
ρ. (3.3)
Lemma3.2. Assume that condition (i1) holds. ThenᏰ(Ω\Sρ)is dense in o
Wr,p(Ω\S ρ).
Proof. Fixu∈Wor,p(Ω\S
ρ) and denote byuo the zero extension ofutoΩo. It is easy
to prove that uo belongs toWr,p(Ωo). It follows from (i1) that there exists a sequence {uk}k∈N⊂Ᏸ(Ωo) such that
uk−→uo inWr,p
Ωo
(3.4)
(see [1, Theorem 3.18]).
Letψ∈Ᏸ(Ω\Sρ) such thatψ=1 on suppu. Observe that{ψuk}k∈N⊂Ᏸ(Ω\Sρ) and ψuk−u
Wr,p(Ω)≤ψ
uk−uoWr,p(Ω o)≤c1
uk−uo Wr,p(Ω
o), (3.5)
wherec1depends onn,ψ. Thus the statement is a consequence of (3.4).
Lemma3.3. Assume that conditions (i1) and (i2) hold. ThenᏰ(Ω\Sρ)is dense inWr
,p s (Ω).
Proof. It follows from (i1), (i2), and [20, Theorem 4.1] that there exists a sequence{δk}k∈N
⊂Ᏸ(Ω\Sρ) such that
lim
k→+∞∂ α1−δ
k(x)=0 ∀x∈Ω,∀α∈Nno, (3.6)
sup
k∈N
∂αδ
k(x)≤cαρ−|α|(x) ∀x∈Ω,∀α∈Nno, (3.7)
wherecαis independent ofx.
Fixu∈Wsr,p(Ω). Observe that condition (3.7) implies thatδku∈Wsr,p(Ω) for allk∈
N. Moreover, by (3.6) we have that
δku−→u inWr
,p
On the other hand, using (o 2.4), it is easy to show that δku∈Wr,p(Ω), and so δku∈
Wr,p(Ω\S
ρ). For eachk∈N,Lemma 3.2yields that there exists a sequence{ukh}h∈N⊂
Ᏸ(Ω\Sρ) such that
ukh−→δku inWr,p(Ω). (3.9)
Moreover, letψk∈C∞o(Rn) such thatψk=1 on supp(δku). Thus by (2.4), we have ψkuk
h−δkuWsr,p(Ω)≤c1u
k
h−δkuWr,p(Ω), (3.10) wherec1∈R+depends onρ,r,s,k. It follows from (3.9) that there existshk∈Nsuch
that
ψkuk
hk−δkuWsr,p(Ω)≤ 1
k. (3.11)
Ifϕk=ψkukhk,k∈N, we obtain from (3.8) and (3.11) that
ϕk−→u inWsr,p(Ω), (3.12)
and the lemma is proved.
Ifr∈N, 1≤p <+∞, we will denote byWo locr,p(Ω\Sρ) the set of distributionsuonΩ
such thatζu∈Wo r,p(Ω) for anyζ∈Ᏸ(Ω\S ρ).
Lemma3.4. Assume that conditions (i1) and (i2) hold. Then
o
Wrloc,pΩ\Sρ
∩Wsr,p(Ω)= o
Wrs,p(Ω). (3.13)
Proof. It is clearly enough to show that
o
Wrloc,pΩ\Sρ
∩Wsr,p(Ω)⊆ o
Wrs,p(Ω). (3.14)
Letu∈Wo rloc,p(Ω\Sρ)∩Wr
,p
s (Ω) and consider a sequence{δk}k∈N⊂Ᏸ(Ω\Sρ)
satis-fying (3.6) and (3.7). Since eachδkubelongs to o
Wr,p(Ω), for anyk∈N, there exists a
sequence{uk
h}h∈N⊂Co∞(Ω) such that
uk
h−→δku inWr,p(Ω). (3.15)
Letψk∈C∞o(Rn) such thatψk=1 on supp(δku). Sinceψkukh∈C∞o(Ω), the same argument
used inLemma 3.3allows to deduce from (3.15) that for everyk∈N, there existshk∈N
such that
ψkuk
hk−δkuWsr,p(Ω)≤ 1
k. (3.16)
We putϕk=ψkukhkfor eachk. Therefore it follows from (3.16) that
ϕk−u
Wsr,p(Ω)≤ 1
As the sequence{δk}k∈Nsatisfies (3.8), (3.17) yields that the sequence{ϕk}k∈Nconverges
touinWsr,p(Ω), and hence (3.14) holds.
4. Main results
LetΩbe 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
Ω⊂Ωo, ∂Ω\Sρ⊂∂Ωo. (4.1)
Remark 4.1. If condition (h1) holds andρ∈Ꮽ(Ω)∩L∞(Ω), then Ωsatisfies (i2) (see [20]).
Letp∈]1, +∞[, and letLbe the differential operator inΩdefined by
L= −
n
i,j=1 ai j ∂
2
∂xi∂xj + n
i=1 ai ∂
∂xi+a. (4.2)
Consider the following conditions on the coefficients ofL: (h2) there exist extensionsaoi jofai jtoΩosuch that
aoi j=aoji∈L∞Ωo
∩VMOΩo
, i,j=1,. . .,n,
∃ν∈R+:
n
i,j=1
aoi jξiξj≥ν|ξ|2 a.e. inΩo,∀ξ∈Rn,
(4.3)
(h3)
ai∈L∞1(Ω), i=1,. . .,n,a∈L∞2(Ω),
ao=ess inf
Ω
σ2(x)a(x)>0, (4.4)
whereσ is the function defined inRemark 3.1.
Moreover, we suppose that the following hypothesis onρholds: (h4)
lim
k→+∞
sup Ω\Ωk
σ(x)x+σ(x)σ(x)xx=0, (4.5)
where
Ωk=
x∈Ω:σ(x)>1 k
In the proof of our main theorem, we need the following uniqueness result.
Lemma4.2. Assume that conditions (h1)–(h4) hold and also thatp > n/2. Then the problem
Lu=0, u∈Wloc2,p(Ω),
lim
x→xo
σsu(x)=0, ∀x o∈∂Ω,
lim
|x|→+∞
σsu(x)=0, ifΩis 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 conditionSρ=∂Ωis replaced by the assumption (h1).
Theorem4.3. Suppose that conditions (h1)–(h4) are satisfied. Then for anyt∈R, the prob-lem
u∈Wloc2,pΩ\Sρ
∩Wo1,locpΩ\Sρ
∩Ltp(Ω), Lu=0, (4.8)
admits only the zero solution.
Proof. Letube a solution of the problem (4.8). It follows from [3, Theorem 5.2] that u∈Wt2,+2p(Ω). Moreover,u belongs toW
1,p
t+1(Ω), and henceLemma 3.4 yields thatu∈ Wt2,+2p(Ω)∩
o
W1,t+1p(Ω). UsingRemark 3.1, it is easy to prove that
σt+2u∈W2,p(Ω)∩Wo 1,p(Ω). (4.9)
Putv=σt+2uand denote byv
othe zero extension ofvtoΩo. Then
vo∈W2,p
Ωo
∩Wo 1,pΩo
(4.10)
byLemma 3.3. Suppose first thatp > n/2. By the Sobolev embedding theorem,vobelongs
toC0(Ω
o)∩ o
W1,p(Ω
o), and hencevo|∂Ωo=0. On the other hand,vo∈W 2,p(Ω
o), so that
another application of the Sobolev embedding theorem gives that lim|x|→+∞vo(x)=0.
Thus by (h1), we have that
lim
|x|→+∞
σt+2u(x)=0, σt+2u(x)|∂Ω=0. (4.11)
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 vo∈Lq(Ωo), where 1/q≥1/ p−2/n. It follows from [3, Theorem 5.2] thatvo∈W
2,q
2 (Ωo),
and hencevo 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 thatvo∈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 Sρ=∂Ω(examples of such
eachΩo⊃Ωwith the uniformC1,1-regularity property; in this case, condition (h2) means that the coefficientsai jadmit extensions outsideΩin the classL∞(Ωo)∩VMO(Ωo).
Corollary4.4. Assume that (h2), (h3), (h4) hold and thatSρ=∂Ω. Then the problem
u∈Wloc2,p(Ω)∩Lp(Ω), Lu=0 (4.12)
admits only the zero solution.
Proof. The statement follows fromTheorem 4.3observing that, in this case,ubelongs to
o
W1,locp(Ω).
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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
