We obtain some W^2,2 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.

1. Introduction

Let Ω be an open subset of ℝⁿ, 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 . First generalizations in this 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 are in the weak-Lⁿ 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 Lⁿ(Ω). A further extension, to a very general case, has been proved in [6, 7], supposing that the a_ij are in VMO, 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

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

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 ℝⁿ 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,

()

where s ∈ ℝ,

, and

are weighted Sobolev spaces where the weight ρ^s is power of a function

, of class

, 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.

2. A Class of Weighted Sobolev Spaces

Let Ω be an open subset of ℝⁿ, not necessarily bounded, n ≥ 2. We want to introduce a class of weight functions defined on .

To this aim, given k ∈ ℕ₀, we consider a function

such that

and

()

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.

Lemma 2.1. If assumption (2.1) is satisfied, then

()

Proof. The proof is obtained by induction. From (2.1), we get

()

with c₁ positive constant depending only on s. Thus (2.3) holds for |α | = 1.

Now, let us assume that (2.3) holds for any β such that |β | <|α| and any s ∈ ℝ, and fix a β such that |β | = |α | − 1. Then, using (2.1) and by the induction hypothesis written for s − 1, we have

()

with c₃ 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.

For k ∈ ℕ₀, p ∈ [1, +∞[, and s ∈ ℝ, and given a weight function ρ satisfying (2.1), we define the space

of distributions u on Ω such that

()

equipped with the norm given in (2.6). Moreover, we denote by

the closure of

and put

Lemma 2.2. Let k ∈ ℕ₀, p ∈ [1, +∞[, and s ∈ ℝ . If assumption (2.1) is satisfied, then there exist two constants c₁, c₂ ∈ ℝ₊ such that

()

with c₁ = c₁(t) and c₂ = c₂(t).

Proof. Observe that from (2.3), we have

()

with c₂ ∈ 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 c₃ ∈ ℝ₊ depending only on t.

We will prove (2.9) by induction. From (2.3), one has

()

for i = 1, …, n, with c₄ ∈ 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

()

with c₇ ∈ ℝ₊ depending only on t.

Let us specify a density result.

Lemma 2.3. Let k ∈ ℕ₀, p ∈ [1, +∞[, and s ∈ ℝ . If Ω has the segment property and assumption (2.1) is satisfied, then is dense in .

Proof. The proof follows by Lemma 2.2 in [12], since clearly both .

This allows us to prove the following inclusion.

Lemma 2.4. Let k ∈ ℕ₀, p ∈ [1, +∞[, and s ∈ ℝ . If Ω has the segment property and assumption (2.1) is satisfied, then

()

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 .

Lemma 2.5. Let k ∈ ℕ₀, p ∈ [1, +∞[, and s ∈ ℝ . If Ω has the segment property and assumption (2.1) is satisfied, then the map

()

defines a topological isomorphism from

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 if and only if .

If , there exists a sequence converging to u in . Therefore, fixed ε ∈ ℝ₊, there exists h₀ ∈ ℕ such that

()

Fix h₁ > h₀, clearly

, because of its compact support. Therefore, there exists a sequence

converging to

in W^k,p(Ω). Hence, there exists n₀ ∈ ℕ such that

()

Putting together (2.14) and (2.15), we get

()

for all n > n₀. Thus,

Vice versa, if we assume that , we find a sequence converging to ρ^su in W^k,p(Ω). Hence, there exists h₀ ∈ ℕ such that

()

Fix h₁ > h₀, since

, which is contained in

by Lemma 2.4, there exists a sequence

converging to

. Therefore, there exists n₀ ∈ ℕ such that

()

From (2.17) and (2.18), we get

()

for all n > n₀, so that

3. Preliminary Results

From now on, we consider a weight

, 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

()

Clearly σ verifies (2.1) and

()

Now, let us fix a cutoff function

such that

()

Then, set

()

By our definition, it follows that

and

()

Finally, we introduce the sequence

()

For any k ∈ ℕ , one has

()

where c_k ∈ ℝ₊ depends only on k. This entails that

()

Concerning the derivatives, easy calculations give that, for any k ∈ ℕ,

()

with c₁ and c₂ 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 ,

()

where

and

are positive constants independent of x and k.

Combining (3.13) and (3.16), we have also, for any k ∈ ℕ,

()

We conclude this section proving the following lemma.

Lemma 3.1. Let σ and Ω_k, k ∈ ℕ, be defined by (3.3) and (3.6), respectively. Then

()

Proof. Set

()

By the second relation in (3.4), the supremum of φ over

is actually a maximum; thus, for every k ∈ ℕ, there exists

such that ψ_k = φ(x_k).

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 ε₀ > 0 such that, for any k ∈ ℕ, there exists n_k > k such that .

If the sequence is bounded, there exists a subsequence converging to a limit , and by the continuity of σ, converges to σ(x). On the other hand, , thus , which is in contrast with the fact that is a convergent sequence.

Therefore, is unbounded, so that there exists a subsequence such that . Thus, by the second relation in (3.4), one has . This gives the contradiction since .

4. 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 ℝⁿ, 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 ∈ ℝⁿ, 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

such that

()

endowed with the norm defined in (4.1). It is easily seen that, for any t₁, t₂ ∈ ℝ₊, a function g belongs to M^p,λ(Ω, t₁) if and only if it belongs to M^p,λ(Ω, t₂); 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 VM^p,λ(Ω) is made up of the functions g ∈ M^p,λ(Ω) such that

()

while

and

denote the closures of L^∞(Ω) and

in M^p,λ(Ω), respectively. We point out that

()

We put M^p(Ω) = M^p,0(Ω), VM^p(Ω) = VM^p,0(Ω),

, and

We want to define the moduli of continuity of functions belonging to

. 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:

()

while

()

where ζ_h denotes a function of class

such that

()

Thus, if g is a function in

, a modulus of continuity of g in

is a map

such that

()

While, if g belongs to

, a modulus of continuity of g in

is an application

such that

()

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

such that

()

where

()

If g ∈ BMO(Ω) = BMO(Ω, τ_A), where

()

we say that g ∈ VMO(Ω) if [g] _BMO(Ω,τ) → 0 for τ → 0⁺.

If g belongs to VMO(Ω), a modulus of continuity of g in VMO(Ω) is function η[g]:]0,1] → ℝ₊ such that

()

For more details on the above-defined function spaces, we refer to [8, 13–15].

Let us start proving a useful lemma.

Lemma 4.1. If Ω has the uniform C^1,1-regularity property and

()

then g ∈ VMO(Ω).

Proof. For n > 2, the result can be found in [16], combining Lemma 4.1 and the argument in the proof of Lemma 4.2.

Concerning n = 2, we firstly apply a known extension result, see [9, Corollary 2.2], stating that any function g such that g, g_x ∈ VM^r(Ω) admits an extension p(g) such that p(g), (p(g)) _x ∈ VM^r(ℝ²).

Then, we prove that for all x₀ ∈ ℝ² and t ∈ ℝ₊, there exists a constant c ∈ ℝ₊ such that

()

Indeed, in view of the above considerations, if (4.16) holds true, one has that p(g) ∈ VMO(ℝ²), so g ∈ VMO(Ω).

Consider the function

()

By Poincaré-Wirtinger inequality and Hölder inequality, one gets

()

this gives (4.16).

For reader’s convenience, we recall here some results proved in [17], adapted to our needs.

Lemma 4.2. If Ω is an open subset of ℝⁿ having the cone property and g ∈ M^r,λ(Ω), with r > 2 and λ = 0 if n = 2, and r∈]2, n] and λ = n − r if n > 2, then

()

is a bounded operator from W^1,2(Ω) to L²(Ω). Moreover, there exists a constant c ∈ ℝ₊, such that

()

with c = c (Ω, n, r).

Furthermore, if , then for any ε > 0 there exists a constant c_ε ∈ ℝ₊, such that

()

with

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 L²(Ω). Moreover, there exists a constant c^′ ∈ ℝ₊, such that

()

with c^′ = c^′(Ω, n, t, μ).

Furthermore, if , then for any ε > 0 there exists a constant , such that

()

with

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 ℝⁿ, 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 h1–h3 the operator L : W^2,2(Ω) → L²(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 h1–h3, there exists a constant c ∈ ℝ₊ such that

()

with

Proof. Let us put

()

and fix

. Lemma 4.1 being true, Lemma 3.1 of [18] (for n = 2) and Theorem 5.1 of [17] (for n > 2) apply, so that there exists a constant c₁ ∈ ℝ₊ such that

()

with

. Therefore,

()

On the other hand, from Lemma 4.2, one has

()

with

and

Furthermore, classical interpolation results give that there exists a constant K ∈ ℝ₊ such that

()

with K = K(Ω). Combining (4.28), (4.29) and (4.30) we conclude that there exists c₂ ∈ ℝ₊ such that

()

with

To show (4.25), it remains to estimate . To this aim let us rewrite our operator in divergence form

()

in order to adapt to our framework some known results concerning operators in variational form. 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 of Lemma 4.2—we obtain that

()

with

. Putting together (4.31) and (4.33), we obtain (4.25).

5. 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 . 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 h1–h3, there exists a constant c ∈ ℝ₊ such that

()

with

Proof. Fix . 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 applies giving that . Therefore, in view of Theorem 4.3, there exists c₀ ∈ ℝ₊, such that

()

with

. Easy computations give

()

Putting together (5.2) and (5.3), we deduce that

()

where c₁ ∈ ℝ₊ depends on the same parameters as c₀ and on s.

On the other hand, from Lemma 4.2 and (3.17), we get

()

with c₂ = c₂(Ω, n, r).

Combining (3.17), (3.18), (5.4), and (5.5), with simple calculations we obtain the bound

()

where c₃ depends on the same parameters as c₁ and on

By Lemma 3.1, it follows that there exists k_o ∈ ℕ such that

()

Now, if we still denote by η the function , from (5.6) and (5.7), we deduce that

()

Then, by Lemma 2.2 and by (3.12), written for k = k_o, we have

()

with c₄ depending on the same parameters as c₃ and on k_o.

This last estimate being true for every s ∈ ℝ, we also have

()

The bounds in (5.9) and (5.10) together with the definition (3.3) of σ give estimate (5.1).

Lemma 5.2. The Dirichlet problem

()

where

()

is uniquely solvable.

Proof. Observe that u is a solution of problem (5.11) if and only if w = σ^su is a solution of the problem

()

Clearly, for any i ∈ {1, …, n},

()

then (5.13) is equivalent to the problem

()

where

()

Using Theorem 5.2 in [18] (for n = 2), 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.

Theorem 5.3. Let L be defined in (4.24). Under hypotheses h1–h3, the problem

()

is uniquely solvable.

Proof. For each τ ∈ [0,1], we put

()

In view of Theorem 5.1,

()

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.

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

Abstract

1. Introduction

2. A Class of Weighted Sobolev Spaces

3. Preliminary Results

4. A No Weighted A Priori Bound

5. Uniqueness and Existence Results

References

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W2,2 A Priori Bounds for a Class of Elliptic Operators

Abstract

1. Introduction

2. A Class of Weighted Sobolev Spaces

3. Preliminary Results

4. A No Weighted A Priori Bound

5. Uniqueness and Existence Results

References

Citing Literature

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