W 2 , 2 A Priori Bounds for a Class of Elliptic Operators

We obtain some 𝑊2,2 a priori bounds for a class of uniformly elliptic second-order differential operators, both in a no-weighted and in a weighted case. We deduce a uniqueness and existence theorem for the related Dirichlet problem in some weighted Sobolev spaces on unbounded domains.


Introduction
Let Ω be an open subset of R n , n ≥ 2. The uniformly elliptic second-order linear differential operator with leading coefficients a ij a ji ∈ L ∞ Ω , i, j 1, . . ., n, and the associated Dirichlet problem have been extensively studied under different hypotheses of discontinuity on the coefficients of L we refer to 1 for a general survey on the subject .In particular, some W 2,2 bounds and the related existence and uniqueness results have been obtained.Among the various hypotheses, in the framework of discontinuous coefficients, we are interested here in those of Miranda's type, having in mind the classical result of 2 where the leading coefficients have derivatives a ij x k ∈ L n Ω , n ≥ 3. First generalizations in this International Journal of Differential Equations direction have been carried on, always considering a bounded and sufficiently regular set Ω, assuming that the derivatives belong to some wider spaces.In particular, in 3 , the a ij x k are in the weak-L n space, while, in 4 , they are supposed to be in an appropriate subspace of the classical Morrey space L 2p,n−2p Ω , where p ∈ 1, n/2 .In 5 , the leading coefficients are supposed to be close to functions whose derivatives are in L n Ω .A further extension, to a very general case, has been proved in 6, 7 , supposing that the a ij are in V MO, which means a kind of continuity in the average sense and not in the pointwise sense.
In this paper, we deal with unbounded domains and we impose hypotheses of Miranda's type on the leading coefficients, assuming that their derivatives a ij x k belong to a suitable Morrey type space, which is a generalization to unbounded domains of the classical Morrey space.The existence of the derivatives is of crucial relevance in our analysis, since it allows us to rewrite the operator L in divergence form and puts us in position to use some known results concerning variational operators.A straightforward consequence of our argument is the following W 2,2 -bound, having the only term Lu L 2 Ω in the right-hand side, where the dependence of the constant c is explicitly described see Section 4 .This kind of estimate often cannot be obtained when dealing with unbounded domains and clearly immediately takes to the uniqueness of the solution of problem 1.2 .
In the framework of unbounded domains, under more regular boundary conditions, an analogous a priori bound can be found in 8 , where different assumptions on the a ij are taken into account.We quote here also the results of 9 , where, in the spirit of 5 , the leading coefficients are supposed to be close, in as specific sense, to functions whose derivatives are in spaces of Morrey type and have a suitable behavior at infinity.
The W 2,2 -bound obtained in 1.3 allows us to extend our result to a weighted case.The relevance of Sobolev spaces with weight in the study of the theory of PDEs with prescribed boundary conditions on unbounded open subsets of R n is well known.Indeed, in this framework, it is necessary to require not only conditions on the boundary of the set, but also conditions controlling the behaviour of the solution at infinity.In this order of ideas, we also consider the Dirichlet problem, s Ω , and L 2 s Ω are weighted Sobolev spaces where the weight ρ s is power of a function ρ : Ω → R , of class C 2 Ω , and such that see Sections 2 and 3 for more details.Also in this weighted case, we obtain the bound where the dependence of the constant c is again completely determined.From this a priori estimate, in Section 5, we deduce the solvability of problem 1.4 .Existence and uniqueness results for similar problems in the weighted case, but with different weight functions and different assumptions on the coefficients, have been proved in 10 .Recent results concerning a priori estimates for solutions of the Poisson and heat equations in weighted spaces can be found in 11 , where weights of Kondrat'ev type are considered.

A Class of Weighted Sobolev Spaces
Let Ω be an open subset of R n , not necessarily bounded, n ≥ 2. We want to introduce a class of weight functions defined on Ω.
To this aim, given k ∈ N 0 , we consider a function ρ : As an example, we can think of the function In the following lemma, we show a property, needed in the sequel, concerning this class of weight functions.
with c 3 positive constant depending only on s.Hence, 2.3 holds true also for α.
Now, let us study some properties of a new class of weighted Sobolev spaces, with weight function of the above-mentioned type.

International Journal of Differential Equations
For k ∈ N 0 , p ∈ 1, ∞ , and s ∈ R, and given a weight function ρ satisfying 2.1 , we define the space W k,p s Ω of distributions u on Ω such that equipped with the norm given in 2.6 .Moreover, we denote by Proof.Observe that from 2.3 , we have with c 2 ∈ R depending only on t.This entails the inequality on the right-hand side of 2.7 .
To get the left-hand side inequality, it is enough to show that with c 3 ∈ R depending only on t.We will prove 2.9 by induction.From 2.3 , one has for i 1, . . ., n, with c 4 ∈ R depending only on t.Hence, 2.9 holds for |α| 1.If 2.3 holds for any β such that |β| < |α|, then, using again 2.3 and by the induction hypothesis, we have Let us specify a density result.
Proof.The density result stated in Lemma 2.3 being true, we can argue as in the proof of Lemma 2.1 of 10 to obtain the claimed inclusion.
From this last lemma, we easily deduce that, if Ω has the segment property, also Ω to W k,p Ω and from Proof.The first part of the proof easily follows from Lemma 2.2 with t s.Let us show that Putting together 2.14 and 2.15 , we get for all n > n 0 .Thus, Vice versa, if we assume that ρ s u ∈ , ∀h > h 0 .

2.17
Fix Ω .Therefore, there exists n 0 ∈ N such that From 2.17 and 2.18 , we get

Preliminary Results
From now on, we consider a weight ρ : Ω → R , ρ ∈ C 2 Ω , and such that 2.1 is satisfied for k 2 .Moreover, we assume that An example of a function verifying our hypotheses is given by We associate to ρ a function σ defined by

3.3
Clearly σ verifies 2.1 and International Journal of Differential Equations 7 Then, set

3.6
By our definition, it follows that Finally, we introduce the sequence For any k ∈ N , one has where c k ∈ R depends only on k.This entails that σ ∼ η k , ∀k ∈ N.

3.12
Concerning the derivatives, easy calculations give that, for any k ∈ N, with c 1 and c 2 positive constants independent of x and k.From 3.9 , 3.11 , 3.13 , 3.14 , and 3.15 , we obtain, for any k ∈ N ,

3.16
where c 1 and c 2 are positive constants independent of x and k.

International Journal of Differential Equations
Combining 3.13 and 3.16 , we have also, for any k ∈ N, , in Ω.

3.18
We conclude this section proving the following lemma.

3.20
By the second relation in 3.4 , the supremum of ϕ over Ω \ Ω k is actually a maximum; thus, for every k ∈ N, there exists To prove 3.19 , we have to show that lim k → ∞ ψ k 0. We proceed by contradiction.Hence, let us assume that there exists ε 0 > 0 such that, for any k ∈ N, there exists n k > k such that ψ n k ϕ x n k ≥ ε 0 .
If the sequence x n k k∈N is bounded, there exists a subsequence x n k k∈N converging to a limit x ∈ Ω, and by the continuity of σ, σ x n k k∈N converges to σ x .On the other hand, x n k ∈ Ω \ Ω k , thus σ x n k ≥ n k , which is in contrast with the fact that σ x n k k∈N is a convergent sequence.
Therefore, x n k k∈N is unbounded, so that there exists a subsequence x n k k∈N such that lim k → ∞ |x n k | ∞.Thus, by the second relation in 3.4 , one has lim k → ∞ ϕ x n k 0. This gives the contradiction since ϕ x n k ≥ ε 0 .

A No Weighted A Priori Bound
We want to prove a W 2,2 bound for an uniformly elliptic second-order linear differential operator.Let us start recalling the definitions of the function spaces in which the coefficients of our operator will be chosen.
For any Lebesgue measurable subset G of R n , let Σ G be the σ-algebra of all Lebesgue measurable subsets of G. Given E ∈ Σ G , we denote by |E| the Lebesgue measure of E, by χ E its characteristic function, and by E x, r the intersection E ∩ B x, r x ∈ R n , r ∈ R , where B x, r is the open ball with center x and radius r.
For n ≥ 2, λ ∈ 0, n , p ∈ 1, ∞ , and fixed t in R , the space of Morrey type M p,λ Ω, t is the set of all functions g in L p loc Ω such that endowed with the norm defined in 4.1 .It is easily seen that, for any t 1 , t 2 ∈ R , a function g belongs to M p,λ Ω, t 1 if and only if it belongs to M p,λ Ω, t 2 ; moreover, the norms of g in these two spaces are equivalent.This allows us to restrict our attention to the space M p,λ Ω M p,λ Ω, 1 .
We now introduce three subspaces of M p,λ Ω needed in the sequel.The set V M p,λ Ω is made up of the functions g ∈ M p,λ Ω such that lim • Ω .We want to define the moduli of continuity of functions belonging to M p,λ Ω or M p,λ • Ω .To this aim, let us put, for h ∈ R and g ∈ M p,λ Ω , Recall first that for a function g ∈ M p,λ Ω the following characterization holds: where Thus, if g is a function in M p,λ Ω , a modulus of continuity of g in M p,λ Ω is a map σ p,λ g : R → R such that 4.9 If Ω has the property where A is a positive constant independent of x and r, it is possible to consider the space where g dy.

4.12
If g ∈ BMO Ω BMO Ω, τ A , where For more details on the above-defined function spaces, we refer to 8, 13-15 .
Let us start proving a useful lemma.

International Journal of Differential Equations
If g ∈ M t,μ Ω , with t ≥ 2 and μ > n − 2t, then the operator in 4.19 is bounded from W 2,2 Ω to L 2 Ω .Moreover, there exists a constant c ∈ R , such that with c c Ω, n, t, μ .Furthermore, if g ∈ M t,μ Ω , then for any ε > 0 there exists a constant c ε ∈ R , such that Proof.The proof easily follows from Theorem 3.2 and Corollary 3.3 of 17 .
From now on, we assume that Ω is an unbounded open subset of R n , n ≥ 2, with the uniform C 1,1 -regularity property.
We consider the differential operator with the following conditions on the coefficients: We explicitly observe that under the assumptions h 1 -h 3 the operator L : W 2,2 Ω → L 2 Ω is bounded, as a consequence of Lemma 4.2.
We are now in position to prove the above-mentioned a priori estimate.
Theorem 4.3.Let L be defined in 4.24 .Under hypotheses h 1 -h 3 , there exists a constant c ∈ R such that On the other hand, from Lemma 4.2, one has with c ε c ε ε, Ω, n, r, σ o r,λ a i and c ε c ε ε, Ω, n, t, μ, σ t,μ a .Furthermore, classical interpolation results give that there exists a constant K ∈ R such that

Uniqueness and Existence Results
This section is devoted to the proof of the solvability of a Dirichlet problem for a class of second-order linear elliptic equations in the weighted space W 2,2 s Ω .The W 2,2 -bound obtained in Theorem 4.3 allows us to show the following a priori estimate in the weighted case.
Theorem 5.1.Let L be defined in 4.24 .Under hypotheses h 1 -h 3 , there exists a constant c ∈ R such that s Ω .In the sequel, for sake of simplicity, we will write η k η, for a fixed k ∈ N. Observe that η satisfies 2.1 , as a consequence of 3.16 , so that Lemma 2.5

5.3
Putting together 5.2 and 5.3 , we deduce that where c 1 ∈ R depends on the same parameters as c 0 and on s.

International Journal of Differential Equations 15
On the other hand, from Lemma 4.2 and 3.17 , we get with c 2 c 2 Ω, n, r .Combining 3.17 , 3.18 , 5.4 , and 5.5 , with simple calculations we obtain the bound where c 3 depends on the same parameters as c 1 and on a i M  s Ω , ∀τ ∈ 0, 1 .

5.19
Thus, taking into account the result of Lemma 5.2 and using the method of continuity along a parameter see, e.g., Theorem 5.2 of 21 , we obtain the claimed result.
Following along the lines, the proofs of Theorem 4.3 of 19 for n 2 and of Theorem 4.2 of 13 for n > 2 , with opportune modifications-we explicitly observe that the continuity of the bilinear form associated to 4.32 in our case is an immediate consequence International Journal of Differential Equations of Lemma 4.2-we obtain that u L 2 Ω ≤ c 3 Lu L 2 Ω , r,λ Ω .By Lemma 3.1, it follows that there exists k o ∈ N such that Now, if we still denote by η the function η k o , from 5.6 and 5.7 , we deduce thatη s u W 2,2 Ω ≤ 2c 3 η s Lu L 2 Ω .5.8Then, by Lemma 2.2 and by 3.12 , written for k k o , we have|α|≤2 σ s ∂ α u L 2 Ω ≤ c 4 σ s Lu L 2 Ω , 5.9with c 4 depending on the same parameters as c 3 and on k o .This last estimate being true for every s ∈ R, we also have|α|≤2 σ −s ∂ α u L 2 Ω ≤ c 5 σ −s Lu L 2 Ω .5.10The bounds in 5.9 and 5.10 together with the definition 3.3 of σ give estimate 5.1 .Observe that u is a solution of problem 5.11 if and only if w σ s u is a solution of the problemw ∈ W 2,2 Ω ∩ • W 1,2 Ω , −Δ σ −s w bσ −s w f, f ∈ L 2 s Ω .−swx i x i − 2sσ −s−1 σ x i w x i s s 1 σ −s−2 σ 2 x i w − sσ −s−1 σ x i x i w, 5.14then 5.13 is equivalent to the problem w ∈ W 2,2 Ω ∩ Theorem 4.3 of 20 for n > 2 , 1.6 of 8 , and the hypotheses on σ, we obtain that 5.15 is uniquely solvable and then problem 5.11 is uniquely solvable too.Let L be defined in 4.24 .Under hypotheses h 1 -h 3 , the problem u ∈ W2,2 • W 1,2 Ω ,