UNIQUENESS RESULTS FOR ELLIPTIC PROBLEMS 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 set Ω 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 Let Ω be an open subset of R n , n ≥ 3. Consider in Ω the uniformly elliptic differential operator with measurable coefficients L =− n i, j=1 a ij ∂ 2 ∂x i ∂x j + n i=1 a i ∂ ∂x i + a, (1.1) and the Dirichlet problem Lu = 0, u ∈ W 2,p (Ω) ∩ o W 1,p (Ω), (D) with p ∈]1,+∞[. Suppose that Ω verifies suitable regularity assumptions. If p ≥ n, a ij ∈ L ∞ (Ω)(i, j = 1, ,n), and the coefficients a i (i = 1, ,n), a satisfy cer- tain local summability conditions (with a>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). If p<n, some more assumptions on the a ij ’s are necessary to get uniqueness results for the problem (D). If Ω is bounded, problem (D) has been widely studied by several authors under various hypotheses on the leading coefficients. In particular, if the coefficients a ij Hindawi Publishing Cor poration Boundary Value Problems Volume 2006, Article ID 98923, Pages 1–9 DOI 10.1155/BVP/2006/98923 2 Uniqueness results for elliptic problems with singular data belong to the space C o (Ω), then uniqueness results for problem (D)havebeenobtained (see [12–15]). On the other hand, when the coefficients a ij are required to be discon- tinuous, the classical result by Miranda [16] must be quoted, where the author assumed that the a ij ’s belong to W 1,n (Ω) (and consider the case p = 2). More recently, a relevant contribution has been given in [11, 22], where the coefficients a ij are supposed to be in the class VMO and p ∈]1,∞[; observe here that VMO contains both classes C o (Ω)and W 1,n (Ω) (see [10]). If Ω 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] with p = 2andin[5]withp ∈]1,∞[. Moreover, f uther uniqueness results for (D), when the a ij ’s are in VMO and p ∈]1,∞[, can be found in [6, 9]. Suppose now that Ω has singular boundary. In [8], a problem of type (D)hasbeen investigated, with (a ij ) x k , a i and a singular near a nonempty subset S ρ of ∂Ω,andp = 2. In particular, the data are supposed to be depending on an appropriate weight function ρ related to the distance function from S 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]onthea ij ’s, and with p>1. More precisely, if there exist extensions a o ij of the coefficients a ij (i, j = 1, , n)inVMO(Ω o ) ∩ L ∞ (Ω o ), where Ω o is a regular open set containing Ω, and the functions ρa i (i = 1, ,n), ρ 2 a are assumed to be bounded with essinf Ω ρ 2 a>0, we can prove a uniqueness result for the problem Lu = 0, u ∈ W 2,p loc Ω \ S ρ ∩ o W 1,p loc Ω \ S ρ ∩ L p t (Ω), (D 1 ) where L p t (Ω), t ∈ R, is a weighted Sobolev space. Observe that if S ρ = ∂Ω and Ω has the segment property, we are able to deduce from the above result that the problem u ∈ W 2,p loc (Ω) ∩ L p (Ω), Lu = 0, (D 2 ) admits only the trivial solution. 2. Notation and function spaces Let G be any Lebesgue measurable subset of R n and let Σ(G) be the collection of all Lebesgue measurable subsets of G.IfF ∈ Σ(G), denote by |F| the Lebesgue measure of F and by Ᏸ(F) the class of restrictions to F of functions ζ ∈ C ∞ o (R n )withF ∩ supp ζ ⊆ F. Moreover, for p ∈ [1,+∞], let L p loc (F) be the class of functions g such that ζg ∈ L p (F)for all ζ ∈ Ᏸ(F). Let Ω be an open subset of R n .Weput Ω(x, r) = Ω ∩ B(x,r) ∀x ∈ R n , ∀r ∈ R + , (2.1) where B(x,r)istheopenballofradiusr centered at x. Denote by Ꮽ(Ω) the class of all measurable functions ρ : Ω → R + such that γ −1 ρ(y) ≤ ρ(x) ≤ γρ(y) ∀y ∈ Ω, ∀x ∈ Ω y,ρ(y) , (2.2) Loredana Ca so 3 where γ ∈ R + is independent of x and y.Forρ ∈ Ꮽ(Ω), we put S ρ = z ∈ ∂Ω :lim x→z ρ(x) = 0 . (2.3) It is known that ρ ∈ L ∞ loc (Ω), ρ −1 ∈ L ∞ loc Ω \ S ρ , (2.4) and, if S ρ =∅, ρ(x) ≤ dist x, S ρ ∀ x ∈ Ω (2.5) (see [7, 20]). If r ∈ N,1≤ p ≤ +∞, s ∈ R,andρ ∈ Ꮽ(Ω), we consider the space W r,p s (Ω) of distri- butions u 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 L p (Ω) . (2.6) Moreover, we denote by o W r,p s (Ω)theclosureofC ∞ o (Ω)inW r,p s (Ω)andputW 0,p s (Ω) = L p s (Ω). A detailed account of properties of the above-defined function spaces can be found in [21]. If Ω has the property Ω(x, r) ≥ Ar n ∀x ∈ Ω, ∀r ∈]0,1], (2.7) where A is a positive constant independent of x and r, it i s possible to consider the space BMO(Ω,t)(t ∈ R + ) of functions g ∈ L 1 loc (Ω)suchthat [g] BMO(Ω,t) = sup x∈Ω r ∈]0,t] − Ω(x,r) g − − Ω(x,r) g dy <+∞, (2.8) where − Ω(x,r) gdy=(1/|Ω(x,r)|) Ω(x,r) gdy.Wewillsaythatg ∈ VMO( Ω)ifg ∈ BMO(Ω)= BMO(Ω,t A ), where t A = sup t∈R + ⎛ ⎜ ⎜ ⎝ sup x∈Ω r ∈]0,t] r n Ω(x, r) ≤ 1 A ⎞ ⎟ ⎟ ⎠ , (2.9) and [g] BMO(Ω,t) → 0fort → 0 + . 4 Uniqueness results for elliptic problems with singular data 3. Some density results Let ρ ∈ Ꮽ(Ω). We consider the following conditions on ρ. (i 1 ) There exists an open subset Ω o of R n with the segment property such that Ω ⊂ Ω o , ∂Ω \ S ρ ⊂ ∂Ω o . (3.1) (i 2 ) H = inf Ω ρ −n (x)|Ω( x,ρ(x))|∈R + . Remark 3.1. If condition (i 2 ) holds, then it is possible to find a function σ ∈ Ꮽ(Ω) ∩ C ∞ (Ω) ∩ C 0,1 (Ω)whichisequivalenttoρ and such that ∂ α σ(x) ≤ c α σ 1−|α| (x) ∀x ∈ Ω, ∀α ∈ N n o , (3.2) where c α is independent of x (see [20]). Fix r ∈ N and p ∈ [1, +∞[. We denote by o W r,p (Ω \ S ρ ) the space of distributions u on Ω such that u ∈ W r,p (Ω), suppu ⊂ Ω \ S ρ . (3.3) Lemma 3.2. Assume that condition (i 1 ) holds. Then Ᏸ(Ω \ S ρ ) is dense in o W r,p (Ω \ S ρ ). Proof. Fix u ∈ o W r,p (Ω \ S ρ ) and denote by u o the zero extension of u to Ω o .Itiseasy to prove that u o belongs to W r,p (Ω o ). It follows from (i 1 ) that there exists a sequence {u k } k∈N ⊂ Ᏸ(Ω o )suchthat u k −→ u o in W r,p Ω o (3.4) (see [1, Theorem 3.18]). Let ψ ∈ Ᏸ(Ω \ S ρ )suchthatψ = 1onsuppu.Observethat{ψu k } k∈N ⊂ Ᏸ(Ω \ S ρ )and ψu k − u W r,p (Ω) ≤ ψ u k − u o W r,p (Ω o ) ≤ c 1 u k − u o W r,p (Ω o ) , (3.5) where c 1 depends on n,ψ. Thus the statement is a consequence of (3.4). Lemma 3.3. Assume that conditions (i 1 )and(i 2 )hold.ThenᏰ(Ω \ S ρ ) is dense in W r,p s (Ω). Proof. It follows from (i 1 ), (i 2 ), and [20, Theorem 4.1] that there exists a sequence {δ k } k∈N ⊂ Ᏸ(Ω \ S ρ )suchthat lim k→+∞ ∂ α 1 − δ k (x) = 0 ∀x ∈ Ω, ∀α ∈ N n o , (3.6) sup k∈N ∂ α δ k (x) ≤ c α ρ −|α| (x) ∀x ∈ Ω, ∀α ∈ N n o , (3.7) where c α is independent of x. Fix u ∈ W r,p s (Ω). Observe that condition (3.7) implies that δ k u ∈ W r,p s (Ω)forallk ∈ N .Moreover,by(3.6)wehavethat δ k u −→ u in W r,p s (Ω). (3.8) Loredana Ca so 5 On the other hand, using (2.4), it is easy to show that δ k u ∈ W r,p (Ω), and so δ k u ∈ o W r,p (Ω \ S ρ ). For each k ∈ N, Lemma 3.2 yields that there exists a sequence {u k h } h∈N ⊂ Ᏸ(Ω \ S ρ )suchthat u k h −→ δ k u in W r,p (Ω). (3.9) Moreover, let ψ k ∈ C ∞ o (R n )suchthatψ k = 1onsupp(δ k u). Thus by (2.4), we have ψ k u k h − δ k u W r,p s (Ω) ≤ c 1 u k h − δ k u W r,p (Ω) , (3.10) where c 1 ∈ R + depends on ρ, r, s, k.Itfollowsfrom(3.9) that there exists h k ∈ N such that ψ k u k h k − δ k u W r,p s (Ω) ≤ 1 k . (3.11) If ϕ k = ψ k u k h k , k ∈ N,weobtainfrom(3.8)and(3.11)that ϕ k −→ u in W r,p s (Ω), (3.12) and the lemma is proved. If r ∈ N,1≤ p<+∞, we will denote by o W r,p loc (Ω \ S ρ ) the set of distri butions u on Ω such that ζu ∈ o W r,p (Ω)foranyζ ∈ Ᏸ(Ω \ S ρ ). Lemma 3.4. Assume that conditions (i 1 )and(i 2 )hold.Then o W r,p loc Ω \ S ρ ∩ W r,p s (Ω) = o W r,p s (Ω). (3.13) Proof. It is clearly enough to show that o W r,p loc Ω \ S ρ ∩ W r,p s (Ω) ⊆ o W r,p s (Ω). (3.14) Let u ∈ o W r,p loc (Ω \ S ρ ) ∩ W r,p s (Ω) and consider a sequence {δ k } k∈N ⊂ Ᏸ(Ω \ S ρ ) satis- fying (3.6)and(3.7). Since each δ k u belongs to o W r,p (Ω), for any k ∈ N, there exists a sequence {u k h } h∈N ⊂ C ∞ o (Ω)suchthat u k h −→ δ k u in W r,p (Ω). (3.15) Let ψ k ∈ C ∞ o (R n )suchthatψ k = 1onsupp(δ k u). Since ψ k u k h ∈ C ∞ o (Ω), the same argument used in Lemma 3.3 allows to deduce from (3.15)thatforeveryk ∈ N, there exists h k ∈ N such that ψ k u k h k − δ k u W r,p s (Ω) ≤ 1 k . (3.16) We put ϕ k = ψ k u k h k for each k. Therefore it follows from (3.16)that ϕ k − u W r,p s (Ω) ≤ 1 k + δ k u − u W r,p s (Ω) . (3.17) 6 Uniqueness results for elliptic problems with singular data As the sequence {δ k } k∈N satisfies (3.8), (3.17) yields that the sequence {ϕ k } k∈N converges to u in W r,p s (Ω), and hence (3.14)holds. 4. Main results Let Ω be an open subset of R n , n ≥ 3, with the segment property. Fix ρ ∈ Ꮽ(Ω) ∩ L ∞ (Ω) and consider the following condition on Ω. (h 1 ) There exists an open subset Ω o of R n with the uniform C 1,1 -regularity property, such that Ω ⊂ Ω o , ∂Ω \ S ρ ⊂ ∂Ω o . (4.1) Remark 4.1. If condition (h 1 )holdsandρ ∈ Ꮽ(Ω) ∩ L ∞ (Ω), then Ω satisfies (i 2 ) (see [20]). Let p ∈]1,+∞[, and let L be the differential operator in Ω defined by L =− n i, j=1 a ij ∂ 2 ∂x i ∂x j + n i=1 a i ∂ ∂x i + a. (4.2) Consider the following conditions on the coefficients of L: (h 2 ) there exist extensions a o ij of a ij to Ω o such that a o ij = a o ji ∈ L ∞ Ω o ∩ VMO Ω o , i, j = 1, ,n, ∃ν ∈ R + : n i, j=1 a o ij ξ i ξ j ≥ ν|ξ| 2 a.e. in Ω o , ∀ξ ∈ R n , (4.3) (h 3 ) a i ∈ L ∞ 1 (Ω), i = 1, ,n, a ∈ L ∞ 2 (Ω), a o = essinf Ω σ 2 (x) a(x) > 0, (4.4) where σ is the function defined in Remark 3.1. Moreover, we suppose that the following hypothesis on ρ holds: (h 4 ) lim k→+∞ sup Ω\Ω k σ(x) x + σ(x) σ(x) xx = 0, (4.5) where Ω k = x ∈ Ω : σ(x) > 1 k , k ∈ N. (4.6) Loredana Ca so 7 In the proof of our main theorem, we need the following uniqueness result. Lemma 4.2. Assume that conditions (h 1 )–(h 4 ) hold and also that p>n/2. Then the problem Lu = 0, u ∈ W 2,p loc (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 condition S ρ = ∂Ω is replaced by the assumption (h 1 ). Theorem 4.3. Suppose that conditions (h 1 )–(h 4 ) are satisfied. Then for any t ∈ R, the prob- lem u ∈ W 2,p loc Ω \ S ρ ∩ o W 1,p loc Ω \ S ρ ∩ L p t (Ω), Lu = 0, (4.8) 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 2,p t+2 (Ω). Moreover, u belongs to W 1,p t+1 (Ω), and hence Lemma 3.4 yields that u ∈ W 2,p t+2 (Ω) ∩ o W 1,p t+1 (Ω). Using Remark 3.1,itiseasytoprovethat σ t+2 u ∈ W 2,p (Ω) ∩ o W 1,p (Ω). (4.9) Put v = σ t+2 u and denote by v o the zero extension of v to Ω o .Then v o ∈ W 2,p Ω o ∩ o W 1,p Ω o (4.10) by Lemma 3.3. Suppose first that p>n/2. By the Sobolev embedding theorem, v o belongs to C 0 (Ω o ) ∩ o W 1,p (Ω o ), and hence v o | ∂Ω o = 0. On the other hand, v o ∈ W 2,p (Ω o ), so that another application of the Sobolev embedding theorem gives that lim |x|→+∞ v o (x) = 0. Thus by (h 1 ), we have that lim |x|→+∞ σ t+2 u (x) = 0, σ t+2 u (x) | ∂Ω = 0. (4.11) In this case the statement follows now from Lemma 4.2. Assume now that p ∈]1,n/2]. Then by the Sobolev embedding theorem, we have that v o ∈ L q (Ω o ), where 1/q ≥ 1/p− 2/n.Itfollowsfrom[3, Theorem 5.2] that v o ∈ W 2,q 2 (Ω o ), and hence v o belongs to W 2,q (Ω o )by(2.4). If q>n/2, the previous case can be used to complete the proof. If finally q ≤ n/2, an iterated application of [3, Theorem 5.2] yields that v o ∈ W 2,q (Ω o )withq >n/2. Thus the first case applies again to complete the proof. As an application of Theorem 4.3, we consider the case S ρ = ∂Ω (examples of such situation can for instance be found in [20]). 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UNIQUENESS RESULTS FOR ELLIPTIC PROBLEMS 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. 1–9 DOI 10.1155/BVP/2006/98923 2 Uniqueness results for elliptic problems with singular data belong to the space C o (Ω), then uniqueness results for problem (D)havebeenobtained (see [12–15]) sup t∈R + ⎛ ⎜ ⎜ ⎝ sup x∈Ω r ∈]0,t] r n Ω(x, r) ≤ 1 A ⎞ ⎟ ⎟ ⎠ , (2.9) and [g] BMO(Ω,t) → 0fort → 0 + . 4 Uniqueness results for elliptic problems with singular data 3. Some density results Let ρ ∈ Ꮽ(Ω). We consider the following conditions