Elliptic Equations in Weighted Sobolev Spaces on Unbounded Domains

We study in this paper a class of second-order linear elliptic equations in weighted Sobolev spaces on unbounded domains of ℝ𝑛, 𝑛≥3. We obtain an a priori bound, and a regularity result from which we deduce a uniqueness theorem.


Introduction
Let Ω be an open subset of R n , n ≥ 3. Assign in Ω the uniformly elliptic second-order linear differential operator with coefficients a ij a ji ∈ L ∞ Ω , i, j 1, . . ., n, and consider the associate Dirichlet problem: where p ∈ 1, ∞ .
It is well known that if Ω is a bounded and sufficiently regular set, the above problem has been widely investigated by several authors under various hypotheses of discontinuity on the leading coefficients, in the case p 2 or p sufficiently close to 2. In particular, some W 2,p -bounds for the solutions of the problem 1.2 and related existence and uniqueness results have been obtained.Among the other results on this subject, we quote here those International Journal of Mathematics and Mathematical Sciences proved in 1 , where the author assumed that a ij 's belong to W 1,n Ω and considered the case p 2 and in 2-4 where the coefficients belong to some classes wider than W 1,n Ω .More recently, a relevant contribution has been given in [5][6][7][8] , where the coefficients a ij are assumed to be in the class VMO and p ∈ 1, ∞ ; observe here that VMO contains the space W 1,n Ω .
If the set Ω is unbounded and regular enough, under assumptions similar to those required in 1 , problem 1.2 has for instance been studied in 9-11 with p 2, and in 12 with p ∈ 1, ∞ .Instead, in 13, 14 the leading coefficients satisfy restrictions similar to those in 5, 6 .
In this paper, we extend some results of 13, 14 to a weighted case.More precisely, we denote by ρ a weight function belonging to a suitable class such that inf and consider the Dirichlet problem: s Ω , and L p s Ω are some weighted Sobolev spaces and the weight functions are a suitable power of ρ.We obtain an a priori bound for the solutions of 1.4 .Moreover, we state a regularity result that allows us to deduce a uniqueness theorem for the problem 1.4 .A similar weighted case was studied in 15 with the leading coefficients satisfying hypotheses of Miranda's type and when p 2.

Weight functions and weighted 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.If F ∈ Σ G , denote by |F| the Lebesgue measure of F, by χ F the characteristic function of F, by F x, r the intersection F ∩ B x, r x ∈ R n , r ∈ R -where B x, r is the open ball of radius r centered at x-and by D F the class of restrictions to is a space of functions defined on F, we denote by X loc F the class of all functions g : F → R, such that ζg ∈ X F for any ζ ∈ D F .
We introduce a class of weight functions defined on an open subset Ω of R n .Denote by A Ω the set of all measurable functions ρ : Ω → R , such that where γ ∈ R is independent of x and y.Examples of functions in A Ω are the function and, if Ω / R n and S is a nonempty subset of ∂Ω, the function x ∈ Ω −→ a dist x, S , a ∈ 0, 1 .

2.3
For ρ ∈ A Ω , we put It is known that We assign an unbounded open subset Ω of R n .Let ρ 1 be a function, such that We put For any a ∈ 0, 1 and x ∈ R n , we set Moreover, denote by A more detailed account of properties of the above defined spaces can be found, for instance, in 18 .
From 15, Lemmas 1.1 and 2.1 , we deduce the following two lemmas, respectively.

Lemma 2.1. For any
where c 1 and c 2 depend on n, p, s, a, and ρ.

Lemma 2.2. If Ω has the segment property, then for any
International Journal of Mathematics and Mathematical Sciences

Some embedding lemmas
We now recall the definitions of the function spaces in which the coefficients of the operator will be chosen.If Ω has the property where A is a positive constant independent of x and r, it is possible to consider the space BMO Ω, τ τ ∈ R of functions g ∈ L 1 loc Ω such that 3.5 For t ∈ 1, ∞ and λ ∈ 0, n , we denote by M t,λ Ω the set of all functions g in L t loc Ω such that endowed with the norm defined by 3.6 .Then, we define • Ω .In order to define the modulus of continuity of a function g in M t,λ • Ω , recall first that for a function g ∈ M t,λ Ω the following characterization holds: where with the position B r B 0, r .Thus, the modulus of continuity of g ∈ M t,λ • Ω is a function A more detailed account of properties of the above defined function spaces can be found in 9, 19, 20 .
We consider the following condition: From 21, Theorem 3.1 we have the following.

An a priori bound
Assume that Ω is an unbounded open subset of R n , n ≥ 3, with the uniform C 1,1 -regularity property, and let ρ be the function defined by 2.7 .Moreover, let p ∈ 1, ∞ and s ∈ R.
Consider in Ω the differential operator: International Journal of Mathematics and Mathematical Sciences with the following conditions on the coefficients: there exist functions e ij , i, j 1, . . ., n, g and μ ∈ R such that where

4.5
Observe that under the assumptions h 1 -h 3 , it follows that the operator L : W with c and Ω 0 depending on n, p, ρ, s, Ω, ν, μ, g 0 , a 0 , t, t Proof.We consider a function φ ∈ C ∞ • R n , such that

7
where c α ∈ R depends only on α, fix y ∈ R n and put Clearly we have 4.9 Now, we put Ω , from 14, Theorem 3.3 , it follows that there exist c 1 ∈ R and a bounded open subset Ω 1 ⊂⊂ Ω, with the cone property, such that a ij ψ x i x j u, 4.12 from 4.11 and 4.12 , we have with c 2 dependent on the same parameters of c 1 .
On the other hand, since ρ ∈ L ∞ loc Ω , we have that where c 3 ∈ R depends only on ρ.Therefore, by 4.13 and 4.14 , we deduce the bound: with c 6 ∈ R dependent on the same parameters of c 5 and also on s.Moreover, from Lemma 3.2 it follows that for any ε ∈ R , there exist c ε , c ε ∈ R , and two bounded open sets Ω ε , Ω ε ⊂⊂ Ω, both with the cone property, such that where c ε , Ω ε depend on ε, Ω, n, p, ρ, s, and c ε , Ω ε depend on ε, Ω, n, p, t 1 , t 2 , ρ, s, σ • a i , and σ • a .From 4.18 and 4.19 it follows 4.6 and then we have the result.

A uniqueness result
In this section, we will prove our uniqueness theorem.We begin to prove a regularity result.
Lemma 5.1.Suppose that the assumptions h 1 , h 2 , and h 3 (with t 1 > n and t 2 > n/2) hold, and let u be a solution of the problem We choose r, r ∈ R , with r < r < 1, and a function φ where c α ∈ R depends only on α.
We fix y ∈ R n and put Clearly we have Ω , from 14, Theorem 3.3 it follows that there exist c 1 ∈ R and a bounded open subset Ω 1 ⊂⊂ Ω, with the cone property, such that with c 1 and Ω 1 depending on n, p, Ω, ν, μ, g 0 , a 0 , t, t a i ψ x i u, 5.7 from 5.6 and 5.7 , we have

5.8
with c 2 dependent on the same parameters of c 1 .

International Journal of Mathematics and Mathematical Sciences
From Lemma 3.1 with s 0, we have that with c 3 dependent on Ω, n, p, and t 1 .Using 22, Corollary 4.5 , we can obtain the following interpolation estimate: where the constant c 4 depends on Ω, n, p.Thus, by 5.8 -5.10 , with easy computations, we deduce the bound: By a well-known lemma of monotonicity of Miranda see 23, Lemma 3.1 , it follows from 5.11 that 12 and then, using Young's inequality, we deduce from 5.12 that u W Ω .Thus, from 13, Theorem 5.2 we deduce that u 0.

Theorem 4 . 1 .
bounded from Lemma 3.1.If the hypotheses h 1 , h 2 , and h 3 are verified, then there exist a constant c ∈ R and a bounded open subset Ω 0 ⊂⊂ Ω, with the cone property, such that

4 . 15 where c 4 ∈From 4 where c 5 ∈
R depends on the same parameters of c 2 and on ρ.International Journal of Mathematics and Mathematical Sciences R depends on the same parameters of c 4

from 4 .
16 and from Lemma 2.1 we have that

5 . 1
where q ∈ 1, p and m ∈ R.Then, u belongs to W 14p I 1/2 y ≤ c 7 Lu L p I 1 y ∈ R depends on the same parameters of c 7 .If m ≥ s − 1, since ∈ R dependent on the same parameters of c 8 and on s.Therefore, u belongs to W If m < s − 1, we denote by k the positive integer, such thats − m − 1 ≤ k < s − m.5.17Then, for i 1, ..., k, we have that Therefore, using 5.14and 5.16 with m i, i 1, . . ., k, instead of s, we deduce that u ∈ W Ω , 5.14 holds.Thus, u satisfies 5.16 and then u ∈ W If conditions h 1 , h 2 , and h 3 (with t 1 > n and t 2 > n/2) hold, and a ≥ a 0 > 0 a.e. in Ω, then the problem • W 1,p s Ω , such that Lu 0. From Lemma 5.1 it follows that u ∈ W 2,p Ω .On the other hand, since u ∈ W 1,p Ω ∩ • W