The Dirichlet Problem for elliptic equations in unbounded domains of the plane

In this paper we prove a uniqueness and existence theorem for the Dirichlet problem in W 2,p for second order linear elliptic equations in unbounded domains of the plane. Here the leading coefficients are locally of class VMO and satisfy a suitable condition at infinity.


Introduction
Consider the Dirichlet problem (1.1) u ∈ W 2,p (Ω)∩ where Ω is an open subset of R n , n ≥ 2 , with a suitable regularity property, p ∈ ]1, +∞[, and L is a second-order, linear, uniformly elliptic operator, i.e. (1.2) If n ≥ 3, p < n and Ω is bounded, it is well known that the sole assumption (1.3) a ij = a ji ∈ L ∞ (Ω) (together with suitable summability conditions for the coefficients a i and a) is not enough to ensure the uniqueness for the problem (1.1).It has been a challenging problem to determine suitable additional hypotheses, in terms of discontinuity, on the leading coefficients a ij in order to get the well-posedness of (1.1).In particular, Miranda [13] proved that it holds whenever p = 2 and the coefficients a ij belong to W 1,n (Ω); later generalizations required the derivatives of the leading coefficients to belong to certain wider spaces.Recently, in two profound papers [9] and [10], it has been pointed out that the solvability of (1.1) holds, for every p ∈ ]1, +∞[, under the assumption that the a ij 's are of class V MO , a i = a = 0 (note that V MO contains W 1,n (Ω)) and, later, the condition a i = a = 0 has been removed in [19] and [20].
If the set Ω is unbounded, problem (1.1) has for instance been studied in [16], where the leading coefficients satisfy restrictions similar to those of [13] and p = 2 , in [3], where the investigation has been extended to the case p ∈ ]1, +∞[, and in [4] and [5], where the a ij 's verify assumptions similar to those of [9] and [10].
In the planar case (n = 2 ) and whenever Ω is bounded, Talenti [15] proved that, if p = 2 , condition (1.3) is enough to establish certain estimates for the solutions of (1.1) which lead to an existence and uniqueness result.Further studies have shown that this result still holds even if p lies in a certain neighborhood of 2 (see for instance [12]), and this interval has recently been determined in [2].Observe that the lower critical exponent here is precisely the one conjectured by C. Pucci [14], who also proved that if p is smaller than this exponent, then the uniqueness of the solution of (1.1) cannot be proved.In our previous paper [7] we have studied problem (1.1) when p is an arbitrary real number > 1 and the a ij 's belong to V MO(Ω).
The aim of the present one is to extend the investigation of [7] to the case of unbounded open sets.In fact, under the hypotheses that the coefficients a ij are in V MO loc ( Ω) and satisfy a suitable condition at infinity, some W 2,p -estimates for the solutions of (1.1) and some regularity results will be obtained, and an existence and uniqueness theorem will be deduced.

Preliminaries
In this section we collect the notation as well as the definitions of function spaces being used throughout the paper.
Let Ω be an open subset of R n , and let Σ(Ω) be the collection of all Lebesgue measurable subsets of Ω.For each E ∈ Σ(Ω), |E| and χ E denote the Lebesgue measure and the characteristic function of E , respectively.Moreover, for any open ball B(x, ρ) in R n of center x and radius ρ.For convenience, we write If X(Ω) is a space of functions defined on Ω, we denote as X loc ( Ω) the space of all functions g : Ω → R such that ζg ∈ X(Ω) for every ζ ∈ D( Ω), where endowed with the norm defined by (2.1) and M p (Ω) (resp.
If Ω has the property where A is some positive constant independent of x and ρ, one can consider the space BM O(Ω, t), t ∈ R + , consisting of all functions g in L 1 loc ( Ω) such that For a more detailed account of properties of the above function spaces we refer to [16], [17] and [18].

Some a priori estimates
Let Ω be an unbounded open set in R 2 satisfying the C 1,1 -regularity property, namely ) for some δ > 0; • there is m 0 ∈ N such that the intersection of any m 0 + 1 distinct U i 's is empty; Notice that any open set Ω with the property (P Ω ) also satisfies the condition (2.3) above.
From this point on, we assume that p ∈]1, +∞[, L is an elliptic operator in Ω of the form We stress that, under the assumptions (h 1 ) and (h 2 ), the operator is bounded (see e.g.Theorem 3.1 in [11]).
The next lemma provides an a priori bound.
Lemma 3.1.Assume (h 1 ), (h 2 ) and (h 3 ).If u is a solution of the Dirichlet problem where the constant c depends on Ω, p, r, ν, The regularity hypothesis on Ω implies that there exists for any fixed x ∈ Ω, the function Then from Lemma 3.1 in [7] it follows that , [18] for the existence of such extension).Since with c 2 depending on the same parameters as c 1 .On the other hand, an application of Theorem 3.1 in [11] yields where c 3 depends only on p and r .Therefore, from (3.7) and (3.8), one has that with c 4 depending on the same parameters as c 2 and on r .By (3.9) and by Lemma 4.1 in [16] we obtain that u ∈ W 2,p (Ω) and where c 5 depends on the same parameters on which c 4 does.Finally, there exists for any ε ∈]0, 1] (see e.g.[1], Theorem 4.14).This combined with (3.10) gives (3.4).
Consider now the following hypotheses on the coefficients of L : there exist functions α ij , i, j = 1, 2, g and μ ∈ R + such that We point out that (h 3 ) -(h 4 ) are weaker than (h 3 ), as an example in [4] shows.The next result provides another a priori estimate, but under these weaker assumptions.In the claim, for any Theorem 3.2.Assume (h 1 ), (h 2 ), (h 3 ) and (h 4 ).Then there exist positive real numbers ρ 0 , c such that where c depends only on Ω, p, r, ν, μ, Proof.The proof is similar to the one given in [4] (Theorem 3.1), taking into account to apply our Lemma 3.1 in place of Theorem 5.1 in [6].

Some regularity results
The aim of this section is to prove two regularity results, the first one of local type and the other of global type.Note that for the local regularity case, we assume only the conditions (h 1 ) and (h 3 ) on the leading coefficients, while for the coefficients of lower order the condition (h 2 ) is replaced by the weaker assumption The following result is proved.
Lemma 4.1.Assume (h 1 ), (h 2 ) and (h 3 ).If u is a solution of the problem where q ∈ ]1, p], then u belongs to W 2,p loc ( Ω). Proof.We may assume that q < p.Let k be the positive integer such that (4.1) In order to prove the claim it is enough to show that since by (4.4) and by Lemma 3.3 in [7], it will follow that u ∈ W 2,q1 loc ( Ω), and so the statement can be deduced iterating the above argument k times. Fixed where q ∈ ]1, p] and q 0 ∈ [1, p], then u belongs to W 2,p (Ω).
Proof.The proof follows the lines of that one of Lemma 4.2 in [4], but taking care to apply our Lemma 4.1 and Theorem 3.2 in place of Lemma 4.1 and Theorem 3.1 in [4], respectively.

Uniqueness and existence results
Now we turn to the Dirichlet problem (1.1) and we first prove the following uniqueness result.Theorem 5.1.Assume (h 1 ), (h 2 ), (h 3 ) and that (5.1) ∃ a 0 ∈ R + such that a ≥ a 0 a.e. in Ω.
Moreover, if p ≤ 2 , suppose also (h 2 ) (with r > 2 ) and (h 4 ).Then the problem admits only the zero solution in Ω.
Proof.Note that Sobolev embedding yields u ∈ C 0 ( Ω)∩  we are now in position to state the following uniqueness and existence result.
Proof.One can proceed as in the proof of Theorem 4.3 in [5], but replacing the applications of Theorem 5.2 in [4] and Theorem 5.1 in [6] with the ones of previous Theorem 5.1 and Lemma 3.1, respectively.Remark 5.3.An example of functions a ij verifying the assumptions (h 1 ), (h 3 ), (h 4 ) and (h 5 ) of Theorem 5.2, but such that (a ii )
[11] , u xi ζ xj , a i u ζ xi ∈ L q1 (Ω) , i, j = 1, 2 .By Sobolev embedding, one obtains that u ζ xixj , u xi ζ xj belong to L q1 ; on the other hand, from Theorem 3.1 in[11]we can conclude that a i u ζ xi also lies in L q1 , and so (4.6) holds.We prove now the following regularity result of global type.Assume (h 1 ), (h 2 ) (with r > 2 ), (h 3 ) and (h 4 ).If u is a solution of the problem