Regularity results for singular elliptic problems

Some local and global regularity results for solutions of linear elliptic equations in weighted spaces are proved. Here the leading coefficients are VMO functions, while the hypotheses on the other coefficients and the boundary conditions involve a suitable weight function.


Introduction
Consider the second order linear differential equation a. e. in Ω , (1.1) where L o is a uniformly elliptic operator in a bounded open subset Ω of R n , n ≥ 3, and f ∈ L p (Ω), p > 1 .A classical problem in the theory of linear elliptic equations in non-divergence form is the study of local and global regularity properties of solutions of (1.1).
It is well known that, when the a ij 's are uniformly continuous in Ω, any solution u of (1.1) in W 2,q  loc (Ω), 1 < q ≤ p, belongs to W 2,p loc (Ω) (see [11], Chapter 9).On the other hand, if the coefficients a ij are required to be discontinuous, some other kind of assumptions are necessary to get regularity results for the solutions of (1.1).For instance, if either the a ij 's belong to W 1,n (Ω) or they satisfy the "Cordes condition", then local W 2,pregularity results (with p belonging to a suitable neighborhood of p = 2) for the solutions of (1.1) have been obtained (see [10] and [2]).
More recently, some local regularity results in W 2,p , for any p ∈]1, +∞[, have been proved under the assumption that the coefficients a ij are bounded and of class V MO (see [8]).Observe that this latter condition is always satisfied if the a ij 's either are uniformly continuous or lie in W 1,n .The results of [8] have been extended to the case of operators whose lower order terms occur and belong to suitable spaces L r (Ω) (see [16]).
If Ω is an arbitrary open subset of R n , n ≥ 3, a W 2,p (Ω)-regularity result has been proved in [6] for the solutions in W 2,q loc ( Ω)∩ o W 1,q loc ( Ω)∩L p (Ω), 1 < q ≤ p, of the linear elliptic equation , under the conditions that the coefficients a ij are bounded and belong to the space V MO(Ω), while the coefficients a i and a lie in suitable spaces of Morrey type.Moreover, in [4] this result has been improved, obtaining local and global regularity results under weaker assumptions; in particular, the a ij 's are assumed locally V MO and satisfying a suitable condition at infinity.
The aim of this paper is to obtain some local and global regularity results for the solutions of the equation (1.2) in certain weighted Sobolev spaces.More precisely, let ρ be a suitable weight function and denote by S ρ the subset of ∂Ω where ρ goes to zero.Suppose that there exist extensions , where Ω o is a regular open set containing Ω; assume also that the coefficients a i and a satisfy certain local summability hypotheses and are singular near S ρ .If f ∈ L p loc (Ω), we will prove that all solutions of (1.2) in W 2,q loc ( Ω \ S ρ ) ∩ o W 1,q loc ( Ω \ S ρ ), 1 < q ≤ p, belong to the Sobolev space W 2,p loc ( Ω \ S ρ ).Moreover, if f and the solutions of (1.2) belong to certain weighted L p -spaces, where the weight functions are suitable powers of ρ, then such solutions belong to the weighted Sobolev space W 2,p s (Ω).

Notation and function spaces
In this paper we use the following basic notation: E , a generic Lebesgue measurable subset of R n ; Σ(E), the Lebesgue σ -algebra on E ; |A|, the Lebesgue measure of A ∈ Σ(E); χ A , the caratteristic function of A; D(A), the class of restrictions to ), the class of functions g , defined on A, such that ζ g ∈ L p (A) for all ζ ∈ D(A); B(x, r), the open ball of radius r centered at x.
Let Ω be an open subset of R n .For each E ∈ Σ(Ω) we put Denote by A(Ω) the class of measurable functions ρ : Ω → R + such that where γ ∈ R + is independent of x and y .For ρ ∈ A(Ω), we put (see [5], [14]).
If r ∈ N, 1 ≤ p ≤ +∞, s ∈ R and ρ ∈ A(Ω), we consider the space W r,p s (Ω) of distributions u on Ω such that ρ s+|α|−r ∂ α u ∈ L p (Ω) for |α| ≤ r , equipped with the norm Moreover, we denote by o W r,p s (Ω) the closure of C ∞ o (Ω) in W r,p s (Ω) and put W 0,p s (Ω) = L p s (Ω).A more detailed account of properties of the above defined spaces can be found in [9], [1] and [15].
For some properties of the spaces K p s (Ω) and Kp s (Ω) we refer to [3] and [5].If Ω has the property where A is a positive constant independent of x and r , it is possible to consider the space BM O(Ω, t) where
If (α) holds, we consider the sets Ẽλ (x) and Ĩλ (x), x ∈ R n \ S ρ , defined by (3.1), (3.2), respectively, corresponding to ρ = ρ.From Remark 3.1 it follows that where ω n is the volume of the unit ball in R n .

Now we define the function ρ
For any x ∈ R n \ S ρ , put Lemma 3.2.Assume that either (α) or (β) holds.Then for any p, q ∈ [1, +∞[, q ≥ p, and for all s ∈ R there exist , where u o is the zero extension of u outside Ω, c 1 depends on γ, n, s, p and also on H if (β) holds, while c 2 depends on γ, n, s, p, q .
Consider the following hypothesis on Ω: where C θ (x) is an open indefinite cone with vertex in x and opening θ and Remark 3.2.If condition (i 1 ) holds, then also (β) holds (see [14]).
Fix k, r ∈ N, p, p o ∈ [1, +∞[ and consider the following condition Then for any function where Proof.The proof is similar to that of Theorem 2.1 in [13], using Lemma 3.2 instead of Lemmas 1.1 and 1.2 of [13].
We recall now two imbedding results proved in [5] that will be useful for our pourposes.Consider real numbers r, p, q, s such that Lemma 3.4.Suppose that conditions (i 1 ) and (i 3 ) hold.Then for all g ∈ K q −s+r (Ω) and for all u ∈ W r,p s (Ω), we have g u ∈ L p (Ω) and where c o depends on Ω, n, p, q, r, s, ρ.
Lemma 3.5.Suppose that the hypotheses of Lemma 3.4 are satisfied and let g ∈ Kq −s+r (Ω).Then for any ε ∈ R + there exists c(ε) ∈ R + such that where c(ε) depends on ε, Ω, n, p, q, r, s, ρ, g K q −s+r (Ω) and σq [g].Observe that in the original statements of the above lemmas given in [5], the dependence of the constants was not explicited.
with the following conditions on the coefficients: We put Theorem 4.1.Assume that conditions (h 1 ) − (h 3 ) hold.Then there exists a positive real number c such that For each x ∈ Ω, let We consider in Ω o the differential operator Then from hypotheses (h 1 ), (h 2 ) and from Theorem 5.1 in [6], we have where Thus, from (4.6) and (4.7) we deduce where c 2 depends on n, p, a o ij L ∞ (Ωo) .It follows from (4.4), (4.5) and (4.8) that Then by (4.9) and by Lemma 3.2 we have where c 4 depends on the same parameters as c 3 . Therefore where c 5 depends on the same parameters as c 4 .

Main results
In this section we will prove two regularity results, the first of local and the other of global type.In the case of local regularity, we require that the coefficients of lower order satisfy suitable local summability conditions: For p ∈]1, +∞[, we denote by W 2,p loc ( Ω \ S ρ ) and Theorem 5.1.Assume that conditions (h 1 ), (h 2 ) and (h 3 ) hold and let u be a solution of the problem (5.1) Proof.Assume that q < p and fix ψ ∈ D( Ω \ S ρ ).We claim that ψ u ∈ W 2,p (Ω).To this aim, let k ∈ N such that Thus it is enough to prove that Observe that Consider in Ω o the differential operator where a o i and a o are the zero extensions of a i and a, respectively, outside Ω.Observe that L(ψ u o ) L q h+1 (Ωo) = L(ψ u) L q h+1 (Ω) (5.5) and (5.6)where Using now the same argument of the proof of Lemma 4.1 in [4], it can be proved that L(ψ u) ∈ L q h+1 (Ω) and so, by (5.5), also that L(ψ u o ) ∈ L q h+1 (Ω o ).Application of Lemma 4.2 of [7] yields that ψ u o ∈ W 2,q h+1 (Ω o ) and hence (5.3) holds.
Proof.Fix a solution u of problem (5.7).By Theorem 5.1 we have that u For each x ∈ Ω, consider the function ψ defined by (4.3).In our situation we obtain and so it follows from Theorem 4.1 that and so, by (5.9), (5.10) where c 2 depends on the same parameters as c 1 .From Lemma 3.3 we have where c 3 depends on Ω, n, p, s, ρ.

4 )
where u o is the zero extension of u outside Ω.Let A be an open subset of R n such thatsupp ψ ⊂ A , A ∩ Ω ⊂ Ω \ S ρ ,and fix χ ∈ C ∞ o (A) such that χ = 1 on supp ψ .
.4) If u o denotes the zero extension of u outside Ω, it follows from (2.2), (2.3) and (3.2) that