-Solvability of the Dirichlet Problem for Elliptic Equations with Singular Data

and Applied Analysis 3 It is easy to prove that each ψ k belongs toD(Ω \ S ρ ) and 0 ≤ ψ k ≤ 1, (ψ k ) |Ω k = 1, suppψ k ⊂ Ω 2k , (15) whereΩ k = {x ∈ Ω | |x| < k, α(x) > 1/k}. Given g inK s (Ω), it is known (see [10]) that


Introduction
Let Ω be an open subset of R , ≥ 2. Consider in Ω the Dirichlet problem where ∈]1, +∞[ and is the second-order linear elliptic differential operator in nondivergence form defined by If ≥ 3 and Ω is a bounded set with a suitable regularity property, the well-posedness of the Dirichlet problem (1) has been largely studied by several authors under various hypotheses of discontinuity on the coefficients. It must be mentioned the classical contribution by Miranda [1], where the author assumed that the belong to 1, (Ω) and considered the case = 2. This result was later on generalized in [2,3] by considering the coefficients belonging to the wider class of spaces VMO and ∈]1, +∞[.
In the framework of open sets, nonnecessarily bounded, whose boundary has various singularities, for example, corners or edges, in accordance with the linear theory, it is natural to assume that the lower order coefficients and the right-hand side of the equation of problem (1) belong to some weighted Sobolev spaces, where the weight is usually a power of the distance function from the "singular set" of the boundary of domain. In these cases, near to the "singular set, " the solution of the boundary value problem may have a singularity which can be often characterized by a weight of mentioned type. For instance, if is a bounded weight function related to the distance function from nonempty subset of the boundary of an arbitrary domain Ω, not necessarily bounded and regular (see Section 2 for the definition of such weight function), a problem similar to (1) has been studied, in the weighted case, by several authors under suitable hypotheses on the weight function and when the coefficients of lower order terms are singular near to . This kind of Dirichlet problem has been dealt, for example, in [4][5][6] under hypotheses as those in [2,3] with ∈]1, +∞[. To be more precise, in [4][5][6], the authors consider the problem case we needed to employ some variational results (see [6] and its bibliography).
The aim of this work is to give an overview on the abovementioned results concerning the solvability of (3) when ∈ ]1, +∞[ as well as to extend such results to the planar case.
Here we note also that, for = 2, if ≥ 3 the required summability on ( ) ℎ can be equal to while if = 2 we have to take a summability greater than (see hypothesis (ℎ 2 )).
At last, we observe that the existence and uniqueness result for the problem (3) are based on the unique solvability of the problem (3) for = 2 and of a problem similar to (3) whose associated operator −1 differs from that in (2) for a compact operator (see Section 4).
For further results concerning elliptic boundary value problems similar to (3), involving different classes of weighted Sobolev spaces, we refer the reader also to [7,8].

Notation and Function Spaces
In this section we recall the definitions and the main properties of the class of weights we are interested in and of certain classes of function spaces where the coefficients of our operator belong. Thus, from now on, let Ω be an unbounded open subset of R , ≥ 2. By Σ ( ) we denote the -algebra of all Lebesgue measurable subsets of Ω. For ∈ Σ ( ), is its characteristic function, | | is its Lebesgue measure is the open ball with center in and radius . The class of restrictions to of functions ∈ ∞ 0 (R ) (resp., ∈ 0 0 (R )) with ∩ supp ⊆ is D ( ) (resp., D 0 ( )). For ∈ [1, +∞], loc ( ) is the class of all functions , defined on , such that ∈ ( ) for all ∈ D ( ). We denote by A(Ω) the class of all measurable weight functions : Ω → R + such that −1 ( ) ≤ ( ) ≤ ( ) ∀ ∈ Ω, ∀ ∈ Ω ( , ( )) , where ∈ R + is independent on and . Given ∈ A(Ω), we put It is known that and if ̸ = 0, (see [9,10] for further details). For ∈ N 0 , 1 ≤ ≤ +∞, ∈ R and ∈ A(Ω), the related weighted Sobolev space , (Ω) is made up of all the distributions on Ω such that +| |− ∈ (Ω) for | | ≤ . We observe that , (Ω) is a Banach space with the norm defined by Moreover, it is separable if 1 ≤ < +∞, reflexive if 1 < < +∞, and, in particular, ,2 (Ω) is an Hilbert space. We also denote by ∘ , (Ω) the closure of ∞ 0 (Ω) in , (Ω) and we put 0, (Ω) = (Ω). Clearly the following embeddings hold: A more detailed account of the properties of the abovedefined weighted Sobolev spaces can be found in [11][12][13].
In the end, we recall the definitions of two other classes of function spaces where the leading coefficients of our operator belong.
If Ω has the property where is a positive constant independent of and , it is possible to consider the space BMO(Ω, where − ∫ A more detailed account of properties of the abovedefined spaces BMO(Ω, ) and VMO(Ω) can be found in [16] or in some general reference books, for example, [17].

Preliminary Results
In this section we recall some embedding and compactness estimates for a multiplication operator as well as a regularity result which will be also extended to the planar case.
Let us assume that Ω has the segment property (for all definitions of regularity properties of open subsets of R we will refer to [18]) and fix ∈ A(Ω) ∩ ∞ (Ω) such that ̸ = 0. For our purposes, we suppose that the following condition on Ω holds.
(ℎ 0 ) There exists an open subset Ω of R with the uniform 1,1 -regularity property such that We note that since the required segment property gives that Ω lies on one side of the "singular" part of its boundary the hypothesis (ℎ ), roughly speaking, means that it is possible to "widen" suitably Ω on the other side of .
Let us start collecting two results of [10] which provide the boundedness and compactness of the multiplication operator where the function belongs to suitable spaces (Ω).

Theorem 2.
Let , , , be numbers such that If condition (ℎ 0 ) holds, then for all ∈ , (Ω) and for any ∈ − + (Ω) we have ∈ (Ω) and where the constant ∈ R + is independent of and . Furthermore, if ∈ ∘ − + (Ω) then for any ∈ R + there exist a constant ( ) ∈ R + and a bounded open set Ω ⊂⊂ Ω with the cone property such that Abstract and Applied Analysis and we have that the multiplication operator is compact.
We go on collecting a density result, an a priori estimate, and a regularity result, which will be some of the crucial analytic tools of our main results.

Lemma 3.
If Ω verifies condition (ℎ 0 ), then for every Now consider in Ω the second-order linear differential operator in nondivergence form Assume that the leading coefficients satisfy the hypothesis.
The following a priori estimate holds.
In the next Lemma we recall a regularity result of [5] (see Theorem 3.4) and we extend it to the case = 2.
where ∈ R + depends only on Ω, , , , 1 , 2 , , , ] , Let us give an overview of the proof of Lemma 5. The idea is to use a local regularity result which is based on an analogous result for solutions of boundary value problems in classical Sobolev spaces defined on regular domains. The mentioned regularity result has been proved in [19] when ≥ 3 (see, Theorem 5.1 of [19]) and it can be easily extended to the planar case, taking into account to apply, at right time, Lemma 4.2 in [20] in place of Lemma 4.2 in [22].

Main Results
Let be the operator defined in (32). Suppose that the coefficients of operator satisfy the assumptions (ℎ 1 ), where 1 , 2 are as in ( 2 ) and is the function defined in Remark 1. Moreover, suppose that the following condition on holds: Abstract and Applied Analysis 5 We are interested in the study of the Dirichlet problem: with ∈ ]1, +∞[ and ∈ R. We point out that the unique solvability of (41) has been firstly proved in [4,5] for ≥ 2 and ≥ 3. Later on, the results of [4,5] have been used in [6] to get the existence and uniqueness of solution also for 1 < < 2. Here, we collect the main results of the above-mentioned papers and we also extend them to the case = 2.
For the case where the assumptions (ℎ 0 )-(ℎ 4 ) are taken into account, with 1 > , 2 > /2 and = 0, and for = 2, the unique solvability of (41) has been proved in Lemma 4.1 of [5], which can be easily extended to the planar case, applying our Lemmas 4 and 5 at the right time. If ̸ = 2, the idea is to exploit the previous case and the unique solvability of a problem similar to (41), whose associated differential operator −1 differs from that in (32) for a compact operator.
Here, we only give an overview of the proof, pointing out to the crucial aspects. Since the operator −1 verifies the same assumptions of the operator , we can write for it the estimate (35). Now a crucial point is to provide a bound for ‖ ‖ −2 (Ω) in terms of −1 (Ω) , when the function is more regular; that is, ∈ 2, (Ω) ∩ ∘ 1,2 −1 (Ω) ∩ D 0 (Ω \ ). To this aim, we build the following bilinear form −1 related, in appropriate way, to the operator −1 : where , ∈ ∘ 1,2 −1 (Ω). By simple computations, we get .
Now we study separately the cases ≥ 2 and < 2. If ≥ 2, applying Lemma 4.1 of [4] in (46), with a suitable choice of function , we easily deduce the claimed bound on ‖ ‖ −2 (Ω) . Thus, from Lemma 3 we get the estimate (43). Furthermore, in this case, the uniqueness of the solution of problem (44) easily follows from (43). For the existence of solution, we refer to Theorem 4.2 in [5], whose analytic technique exploits the bound (43), which also gives the closure of the range of operator −1 , the unique solvability of our problem when = 2, and the regularity result of Lemma 5, which allow us to go up in summability.
Suppose now 1 < < 2. In this case, in order to get a bound on ‖ ‖ −2 (Ω) , when is more regular, we use a variational result (see the proof of Lemma 3.3 in [6]) whose main analytic tools exploit the existence and uniqueness result of previous case and Lemma 5. The mentioned variational result gives us a bound for ‖ ‖ −2 (Ω) , where is the unique solution of the Dirichlet problem associated with −1 ( , ) and is the conjugate exponent of . From such estimate and (46), with a suitable choice of function , we obtain our claim. Using again Lemma 3, we get (43) also in this case. Finally, for the existence and uniqueness of problem (44) we can refer to Lemma 3.4 of [6], whose proof can be easily extended also to the planar case, in view of previous considerations.
In the following Theorem we collect two results of [4,6], giving an a priori bound for the operator when ≥ 2.

Theorem 8. Under the same hypotheses of Theorem 7 with
1 > and 2 > /2, for any ∈]1, +∞[ and ∈ R, the problem (41) is an index problem with index equal to zero. Moreover, if = 0, then the problem is uniquely solvable.
Proof. First we prove that the problem (41) is an index problem with index equal to zero. In fact, from Lemma 6 we deduce that the operator −1 is a Fredholm operator with index zero. On the other hand, by Theorem 2 it follows that the operator is a compact operator. Thus, from (48), (49), and well known results of the classical Fredholm index theory, we deduce that the problem (41) is an index problem with index equal to zero. Assume now = 0. We prove only the uniqueness of solution of problem (41); the existence will easily follow from uniqueness and from what we have proved in the case ̸ = 0. At first, suppose ≥ 2. Then, by the unique solvability of problem (41) for = 2 there exists a unique ∈ 2,2 (Ω) ∩ ∘ 1,2 −1 (Ω) such that = 0. Moreover, by our Lemma 5 and by Lemma 3.2 of [5] it follows that belongs to 2, (Ω) ∩ ∘ 1, −1 (Ω) and thus we deduce that = 0. Now let 1 < < 2 and ∈ 2, (Ω) ∩ ∘ 1, −1 (Ω) such that = 0. Using again our Lemma 5 and Lemma 3.2 of [5] we get ∈ 2,2 (Ω) ∩ ∘ 1,2 −1 (Ω) with = + (1/ − 1/2) and = 0. Exploiting again the existence and uniqueness of (41) for = 2, we obtain that = 0. This concludes the proof.