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

Abstract

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 ℝ^𝑛,𝑛≥2. The uniformly elliptic second-order linear differential operator 𝐿=−^𝑛_𝑖_,𝑗=1𝑎_𝑖𝑗𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗+^𝑛_𝑖=1𝑎_𝑖𝜕𝜕𝑥_𝑖+𝑎,(1.1) with leading coefficients 𝑎_𝑖𝑗=𝑎_𝑗𝑖∈𝐿^∞(Ω),𝑖,𝑗=1,…,𝑛, and the associated Dirichlet problem 𝑢∈𝐿𝑢𝑊=𝑓^2,2,(Ω)𝑓∩∈𝑊𝐿^∘1,2₂(Ω),(Ω),(1.2) 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)_𝑥𝑘∈𝐿^𝑛(Ω),𝑛≥3. 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 (𝑎_𝑖𝑗)_𝑥𝑘 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 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 (𝑎_𝑖𝑗)_𝑥𝑘 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‖_𝐿²_(Ω) in the right-hand side, ‖𝑢‖_𝑊^2,2_(Ω)≤𝑐‖𝐿𝑢‖_𝐿²_(Ω),∀𝑢∈𝑊^2,2(Ω)∩𝑊^∘1,2(Ω),(1.3) 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, 𝑢∈𝐿𝑢𝑊=𝑓^𝑠2,2,(Ω)𝑓∩∈𝑊𝐿^∘𝑠1,2_2𝑠(Ω),(Ω),(1.4) where 𝑠∈ℝ, 𝑊^𝑠2,2(Ω),𝑊^∘𝑠1,2(Ω), and 𝐿^2𝑠(Ω) are weighted Sobolev spaces where the weight 𝜌^𝑠 is power of a function 𝜌∶Ω→ℝ₊, of class 𝐶²(Ω), and such that sup_𝑥∈Ω||𝜕^𝛼𝜌(𝑥)||lim_{|𝑥|→+∞}𝜌(𝑥)<+∞,∀|𝛼|≤2,𝜌(𝑥)+𝜌(1𝑥)=+∞,lim^{|𝑥|→+∞}𝜌_𝑥(𝑥)+𝜌_𝑥𝑥(𝑥)𝜌(𝑥)=0,(1.5) see Sections 2 and 3 for more details. Also in this weighted case, we obtain the bound ‖𝑢‖_𝑊^𝑠2,2_(Ω)≤𝑐‖𝐿𝑢‖_𝐿^2𝑠_(Ω),∀𝑢∈𝑊^𝑠2,2(Ω)∩𝑊^∘𝑠1,2(Ω),(1.6) 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 ℝ^𝑛, not necessarily bounded, 𝑛≥2. We want to introduce a class of weight functions defined on Ω.

To this aim, given 𝑘∈ℕ₀, we consider a function 𝜌∶Ω→ℝ₊ such that 𝜌∈𝐶^𝑘(Ω) and sup_𝑥∈Ω||𝜕^𝛼𝜌(𝑥)||𝜌(𝑥)<+∞,∀|𝛼|≤𝑘.(2.1) As an example, we can think of the function 𝜌(𝑥)=1+|𝑥|²^𝑡,𝑡∈ℝ.(2.2) 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 sup_𝑥∈Ω||𝜕^𝛼𝜌^𝑠(𝑥)||𝜌^𝑠(𝑥)<+∞∀𝑠∈ℝ,∀|𝛼|≤𝑘.(2.3)

Proof. The proof is obtained by induction. From (2.1), we get ||(𝜌^𝑠)_𝑥𝑖||=||𝑠𝜌^𝑠−1𝜌_𝑥𝑖||≤𝑐₁𝜌𝜌^𝑠−1=𝑐₁𝜌^𝑠,𝑖=1,…,𝑛,(2.4) 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 ||𝜕^𝛼𝜌^𝑠||=≤𝑐||𝜕^𝛽₂(𝜌^𝑠)_𝑥𝑖||=||𝜕^𝛽𝑠𝜌^𝑠−1𝜌_𝑥𝑖||_𝛾≤𝛽||𝜕^𝛽−𝛾𝜌_𝑥𝑖𝜕^𝛾𝜌^𝑠−1||≤𝑐₃𝜌𝜌^𝑠−1=𝑐₃𝜌^𝑠,for𝑖=1,…,𝑛,(2.5) 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 𝑊^{𝑠𝑘,𝑝}(Ω) of distributions 𝑢 on Ω such that ‖𝑢‖_𝑊^{𝑠𝑘,𝑝}_(Ω)=_{|𝛼|≤𝑘}‖𝜌^𝑠𝜕^𝛼𝑢‖_𝐿^𝑝_(Ω)<+∞,(2.6) equipped with the norm given in (2.6). Moreover, we denote by 𝑊^{∘𝑠𝑘,𝑝}(Ω) the closure of 𝐶^∞∘(Ω) in 𝑊^{𝑠𝑘,𝑝}(Ω) and put 𝑊^𝑠0,𝑝(Ω)=𝐿^𝑝𝑠(Ω).

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 𝑐₁‖𝑢‖_𝑊^{𝑠𝑘,𝑝}_(Ω)≤‖‖𝜌^𝑡𝑢‖‖_𝑊^{𝑘,𝑝𝑠−𝑡}_(Ω)≤𝑐₂‖𝑢‖_𝑊^{𝑠𝑘,𝑝}_(Ω),∀𝑡∈ℝ,∀𝑢∈𝑊^{𝑠𝑘,𝑝}(Ω),(2.7) with 𝑐₁=𝑐₁(𝑡) and 𝑐₂=𝑐₂(𝑡).

Proof. Observe that from (2.3), we have ||𝜕^𝛼𝜌^𝑡𝑢||≤𝑐₁_𝛽≤𝛼||𝜕^𝛼−𝛽𝜌^𝑡𝜕^𝛽𝑢||≤𝑐₂||𝜌^𝑡𝜕^𝛽𝑢||,∀|𝛼|≤𝑘,(2.8) 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 ||𝜌^𝑡𝜕^𝛼𝑢||≤𝑐₃_𝛽≤𝛼||𝜕^𝛽𝜌^𝑡𝑢||,∀|𝛼|≤𝑘,(2.9) with 𝑐₃∈ℝ₊ depending only on 𝑡.
We will prove (2.9) by induction. From (2.3), one has ||𝜌^𝑡𝑢_𝑥𝑖||=|||𝜌^𝑡𝑢_𝑥𝑖−𝜌^𝑡_𝑥𝑖𝑢|||≤𝑐₄𝜌^𝑡𝑢_𝑥+𝜌^𝑡|𝑢|,(2.10) 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 ||𝜌^𝑡𝜕^𝛼𝑢||≤||𝜕^𝛼𝜌^𝑡𝑢||+𝑐₅_𝛽<𝛼≤||𝜕_𝛼||𝜕^𝛼−𝛽𝜌^𝑡||||𝜕^𝛽𝑢||𝜌^𝑡𝑢||+𝑐₆_𝛽<𝛼||𝜌^𝑡𝜕^𝛽𝑢||≤𝑐₇_𝛽≤𝛼||𝜕^𝛽𝜌^𝑡𝑢||,(2.11) 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 𝒟(Ω) is dense in 𝑊^{𝑠𝑘,𝑝}(Ω).

Proof. The proof follows by Lemma 2.2 in [12], since clearly both 𝜌,𝜌⁻¹∈𝐿^∞loc(Ω).

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 𝑊^𝑘,𝑝(Ω)∩𝑊^∘𝑘,𝑝(Ω)⊂𝑊^{∘𝑠𝑘,𝑝}(Ω).(2.12)

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 𝐶^𝑘𝑜(Ω)⊂𝑊^{∘𝑠𝑘,𝑝}(Ω).

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 𝑢⟶𝜌^𝑠𝑢(2.13) defines a topological isomorphism from 𝑊^{𝑠𝑘,𝑝}(Ω) to 𝑊^𝑘,𝑝(Ω) and from 𝑊^{∘𝑠𝑘,𝑝}(Ω) to 𝑊^∘𝑘,𝑝(Ω).

Proof. The first part of the proof easily follows from Lemma 2.2 with 𝑡=𝑠. Let us show that 𝑢∈𝑊^{∘𝑠𝑘,𝑝}(Ω) if and only if 𝜌^𝑠𝑢∈𝑊^∘𝑘,𝑝(Ω).
If 𝑢∈𝑊^{∘𝑠𝑘,𝑝}(Ω), there exists a sequence (𝜙_ℎ)_ℎ∈ℕ⊂𝐶^∞𝑜(Ω) converging to 𝑢 in 𝑊^{𝑠𝑘,𝑝}(Ω). Therefore, fixed 𝜀∈ℝ₊, there exists ℎ₀∈ℕ such that ‖‖𝜌^𝑠𝜙_ℎ−𝑢‖‖_{𝑊𝑘,𝑝(Ω)}<𝜀2,∀ℎ>ℎ⁰.(2.14) Fix ℎ₁>ℎ₀, clearly 𝜌^𝑠𝜙_ℎ1∈𝑊^∘𝑘,𝑝(Ω), because of its compact support. Therefore, there exists a sequence (𝜓_𝑛)_𝑛∈ℕ⊂𝐶^∞𝑜(Ω) converging to 𝜌^𝑠𝜙_ℎ1 in 𝑊^𝑘,𝑝(Ω). Hence, there exists 𝑛₀∈ℕ such that ‖‖𝜓_𝑛−𝜌^𝑠𝜙_ℎ1‖‖_{𝑊𝑘,𝑝(Ω)}<𝜀2,∀𝑛>𝑛⁰.(2.15) Putting together (2.14) and (2.15), we get ‖‖𝜓_𝑛−𝜌^𝑠𝑢‖‖_{𝑊𝑘,𝑝(Ω)}≤‖‖𝜓_𝑛−𝜌^𝑠𝜙_ℎ1‖‖_{𝑊𝑘,𝑝(Ω)}+‖‖𝜌^𝑠𝜙_ℎ1−𝜌^𝑠𝑢‖‖_{𝑊𝑘,𝑝(Ω)}<𝜀,(2.16)  forall𝑛>𝑛₀. Thus, 𝜌^𝑠𝑢∈𝑊^∘𝑘,𝑝(Ω).
Vice versa, if we assume that 𝜌^𝑠𝑢∈𝑊^∘𝑘,𝑝(Ω), we find a sequence (𝜙_ℎ)_ℎ∈ℕ⊂𝐶^∞𝑜(Ω) converging to 𝜌^𝑠𝑢 in 𝑊^𝑘,𝑝(Ω). Hence, there exists ℎ₀∈ℕ such that ‖‖𝜌^−𝑠𝜙_ℎ−𝑢‖‖_𝑊^{𝑠𝑘,𝑝}_(Ω)<𝜀2,∀ℎ>ℎ⁰.(2.17) Fix ℎ₁>ℎ₀, since 𝜌^−𝑠𝜙_ℎ1∈𝐶^𝑘𝑜(Ω), which is contained in 𝑊^{∘𝑠𝑘,𝑝}(Ω) by Lemma 2.4, there exists a sequence (𝜓_𝑛)_𝑛∈ℕ⊂𝐶^∞𝑜(Ω) converging to 𝜌^−𝑠𝜙_ℎ1 in 𝑊^{𝑠𝑘,𝑝}(Ω). Therefore, there exists 𝑛₀∈ℕ such that ‖‖𝜓_𝑛−𝜌^−𝑠𝜙_ℎ1‖‖_{𝑊𝑠𝑘,𝑝(Ω)}<𝜀2,∀𝑛>𝑛⁰.(2.18) From (2.17) and (2.18), we get ‖‖𝜓_𝑛−𝑢‖‖_𝑊^{𝑠𝑘,𝑝}_(Ω)≤‖‖𝜓_𝑛−𝜌^−𝑠𝜙_ℎ1‖‖_{𝑊𝑠𝑘,𝑝(Ω)}+‖‖𝜌^−𝑠𝜙_ℎ1−𝑢‖‖_{𝑊𝑠𝑘,𝑝(Ω)}<𝜀,(2.19)   forall𝑛>𝑛₀, so that 𝑢∈𝑊^{∘𝑠𝑘,𝑝}(Ω).

3. Preliminary Results

From now on, we consider a weight 𝜌∶Ω→ℝ₊, 𝜌∈𝐶²(Ω), and such that (2.1) is satisfied (for 𝑘=2). Moreover, we assume that lim_{|𝑥|→+∞}𝜌(𝑥)+𝜌(1𝑥)=+∞,lim^{|𝑥|→+∞}𝜌_𝑥(𝑥)+𝜌_𝑥𝑥(𝑥)𝜌(𝑥)=0.(3.1) An example of a function verifying our hypotheses is given by 𝜌(𝑥)=1+|𝑥|²^𝑡,𝑡∈ℝ⧵{0}.(3.2) We associate to 𝜌 a function 𝜎 defined by 𝜎𝜎==1𝜌if𝜌⟶+∞,for|𝑥|⟶+∞,𝜌if𝜌⟶0,for|𝑥|⟶+∞.(3.3) Clearly 𝜎 verifies (2.1) and lim_{|𝑥|→+∞}𝜎(𝑥)=+∞,lim_{|𝑥|→+∞}𝜎_𝑥(𝑥)+𝜎_𝑥𝑥(𝑥)𝜎(𝑥)=0.(3.4)

Now, let us fix a cutoff function 𝑓∈𝐶^∞∘(ℝ₊) such that 0≤𝑓≤1,𝑓(𝑡)=1if𝑡∈[0,1],𝑓(𝑡)=0if𝑡∈[2,+∞[.(3.5) Then, set 𝜁_𝑘∶𝑥∈Ω⟶𝑓𝜎(𝑥)Ω𝑘,𝑘∈ℕ,_𝑘={𝑥∈Ω∶𝜎(𝑥)<𝑘},𝑘∈ℕ.(3.6) By our definition, it follows that 𝜁_𝑘∈𝐶^∞∘(Ω) and 0≤𝜁_𝑘≤1,𝜁_𝑘=1onΩ_𝑘,𝜁_𝑘=0onΩ⧵Ω_2𝑘,𝑘∈ℕ.(3.7)

Finally, we introduce the sequence 𝜂_𝑘∶𝑥∈Ω⟶2𝑘𝜁_𝑘(𝑥)+1−𝜁_𝑘(𝑥)𝜎(𝑥),𝑘∈ℕ.(3.8)

For any 𝑘∈ℕ, one has 𝜂_𝑘=𝜁_𝑘(2𝑘−𝜎)+𝜎≥𝜎,inΩ_2𝑘,(3.9)𝜂_𝑘≤2𝑘+𝜎≤2𝑘inf_Ω2𝑘𝜎+1𝜎=𝑐^𝑘+1𝜎,inΩ_2𝑘,(3.10)𝜂_𝑘=𝜎,inΩ⧵Ω_2𝑘,(3.11) 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, 𝜂_𝑘_𝑥=𝜂_𝑘_𝑥𝑥=0,inΩ_𝑘,(3.13)𝜂_𝑘_𝑥=𝜎_𝑥,𝜂_𝑘_𝑥𝑥=𝜎_𝑥𝑥,inΩ⧵Ω_2𝑘,(3.14)𝜂_𝑘_𝑥≤𝑐₁𝜎_𝑥,𝜂_𝑘_𝑥𝑥≤𝑐₂𝜎𝜎^2𝑥+𝜎^𝑥𝑥,inΩ⧵Ω_2𝑘,(3.15) 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, 𝜂_𝑘_𝑥𝜂_𝑘≤𝑐^1𝜎_𝑥𝜎,𝜂inΩ,_𝑘_𝑥𝑥𝜂_𝑘≤𝑐^2𝜎^2𝑥+𝜎𝜎_𝑥𝑥𝜎²,inΩ,(3.16) where 𝑐^1 and 𝑐^2 are positive constants independent of 𝑥 and 𝑘.

Combining (3.13) and (3.16), we have also, for any 𝑘∈ℕ, 𝜂_𝑘_𝑥𝜂_𝑘≤𝑐^1sup_Ω⧵Ω𝑘𝜎_𝑥𝜎,inΩ,(3.17)𝜂_𝑘_𝑥𝑥𝜂_𝑘≤𝑐^2sup_Ω⧵Ω𝑘𝜎^2𝑥+𝜎𝜎_𝑥𝑥𝜎²,inΩ.(3.18)

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 lim_𝑘→+∞sup_Ω⧵Ω𝑘𝜎_𝑥(𝑥)+𝜎_𝑥𝑥(𝑥)𝜎(𝑥)=0.(3.19)

Proof. Set 𝜑(𝑥)=𝜎^𝑥(𝑥)+𝜎_𝑥𝑥(𝑥)𝜎(𝑥),𝑥∈𝜓Ω,_𝑘=sup_Ω⧵Ω𝑘𝜑,𝑘∈ℕ.(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 𝑥_𝑘∈Ω⧵Ω_𝑘 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 𝜓_𝑛𝑘=𝜑(𝑥_𝑛𝑘)≥𝜀₀.
If the sequence (𝑥_𝑛𝑘)_𝑘∈ℕ is bounded, there exists a subsequence (𝑥′_𝑛𝑘)_𝑘∈ℕ converging to a limit 𝑥∈Ω, and by the continuity of 𝜎, (𝜎(𝑥′_𝑛𝑘))_𝑘∈ℕ converges to 𝜎(𝑥). On the other hand, 𝑥′_𝑛𝑘∈Ω⧵Ω_𝑘, thus 𝜎(𝑥′_𝑛𝑘)≥𝑛_𝑘, which is in contrast with the fact that (𝜎(𝑥′_𝑛𝑘))_𝑘∈ℕ is a convergent sequence.
Therefore, (𝑥_𝑛𝑘)_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)≥𝜀₀.

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 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^𝑝loc(Ω) such that ‖𝑔‖_𝑀^𝑝,𝜆_(Ω,𝑡)=sup_{𝜏∈]0,𝑡]𝑥∈Ω}𝜏^−𝜆/𝑝‖𝑔‖_𝐿^𝑝_{(Ω(𝑥,𝜏))}<+∞,(4.1) 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 lim_𝑡→0‖𝑔‖_𝑀^𝑝,𝜆_(Ω,𝑡)=0,(4.2) while 𝑀^𝑝,𝜆(Ω) and 𝑀^∘𝑝,𝜆(Ω) denote the closures of 𝐿^∞(Ω) and 𝐶^∞∘(Ω) in 𝑀^𝑝,𝜆(Ω), respectively. We point out that 𝑀^∘𝑝,𝜆(Ω)⊂𝑀^𝑝,𝜆(Ω)⊂𝑉𝑀^𝑝,𝜆(Ω).(4.3) We put 𝑀^𝑝(Ω)=𝑀^𝑝,0(Ω), 𝑉𝑀^𝑝(Ω)=𝑉𝑀^𝑝,0(Ω), 𝑀^𝑝(Ω)=𝑀^𝑝,0(Ω), and 𝑀^𝑝∘(Ω)=𝑀^∘𝑝,0(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), 𝐹[𝑔](ℎ)=sup_{𝐸∈Σ(Ω)sup𝑥∈Ω|𝐸(𝑥,1)|≤1/ℎ}‖‖𝑔𝜒_𝐸‖‖_{𝑀𝑝,𝜆(Ω)}.(4.4) Recall first that for a function 𝑔∈𝑀^𝑝,𝜆(Ω) the following characterization holds: 𝑔∈𝑀^𝑝,𝜆(Ω)⟺lim_ℎ→+∞𝐹[𝑔](ℎ)=0,(4.5) while 𝑔∈𝑀^∘𝑝,𝜆(Ω)⟺lim_ℎ→+∞𝐹[𝑔](ℎ)+‖‖1−𝜁^ℎ𝑔‖‖_𝑀^𝑝,𝜆_(Ω)=0,(4.6) where 𝜁_ℎ denotes a function of class 𝐶^∞𝑜(𝑅^𝑛) such that 0≤𝜁_ℎ≤1,𝜁_{ℎ|𝐵(0,ℎ)}=1,supp𝜁_ℎ⊂𝐵(0,2ℎ).(4.7) 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 𝐹[𝑔](ℎ)≤𝜎^𝑝,𝜆[𝑔](ℎ),lim_ℎ→+∞𝜎^𝑝,𝜆[𝑔](ℎ)=0.(4.8) While, if 𝑔 belongs to 𝑀^{𝑜𝑝,𝜆}(Ω), a modulus of continuity of 𝑔 in 𝑀^{𝑜𝑝,𝜆}(Ω) is an application 𝜎_{𝑜𝑝,𝜆}[𝑔]∶ℝ₊→ℝ₊ such that 𝐹[𝑔](ℎ)+‖‖li1m−_ℎ→+∞𝜁_ℎ𝑔‖‖_𝑀^𝑝,𝜆_(Ω)≤𝜎_{𝑜𝑝,𝜆}[𝑔](ℎ),𝜎_{𝑜𝑝,𝜆}[𝑔](ℎ)=0.(4.9) If Ω has the property ||Ω(𝑥,𝑟)||≥𝐴𝑟^𝑛∀𝑥∈Ω,∀𝑟∈]0,1],(4.10) 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 𝑔∈𝐿^1loc(Ω) such that [𝑔]_{𝐵𝑀𝑂(Ω,𝜏)}=sup_{𝑥∈Ω𝑟∈]0,𝜏]}_{Ω(𝑥,𝑟)}||||𝑔−_{Ω(𝑥,𝑟)}𝑔𝑑𝑦||||𝑑𝑦<+∞,(4.11) where _{Ω(𝑥,𝑟)}𝑔𝑑𝑦=||Ω(𝑥,𝑟)||⁻¹_{Ω(𝑥,𝑟)}𝑔𝑑𝑦.(4.12) If 𝑔∈𝐵𝑀𝑂(Ω)=𝐵𝑀𝑂(Ω,𝜏_𝐴), where 𝜏_𝐴=sup⎧⎪⎨⎪⎩𝜏∈ℝ₊∶sup_{𝑥∈Ω𝑟∈]0,𝜏]}𝑟^𝑛||Ω(𝑥,𝑟)||≤𝐴1⎫⎪⎬⎪⎭,(4.13) 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 [𝑔]_{𝐵𝑀𝑂(Ω,𝜏)}≤𝜂[𝑔](𝜏)∀𝜏∈]0,1],lim_𝜏→0+𝜂[𝑔](𝜏)=0.(4.14) 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 𝑔,𝑔_𝑥∈VM^r(Ω),r>2,for𝑛=2,VM^r,n−r(Ω),r∈]2,n],forn>2,(4.15) 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 _{𝐵(𝑥0,𝑡)}||||𝑝(𝑔)−_{𝐵(𝑥0,𝑡)}𝑝(𝑔)𝑑𝑥||||𝑑𝑥≤𝑐𝑡^{(𝑟−2/𝑟)}‖‖(𝑝(𝑔))_𝑥‖‖_{𝐿𝑟(𝐵(𝑥0,𝑡))}.(4.16) 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 𝑔^∗∶𝑧∈ℝ²⟶𝑝(𝑔)𝑥₀+𝑡𝑧∈ℝ.(4.17) By Poincaré-Wirtinger inequality and Hölder inequality, one gets _{𝐵(𝑥0,𝑡)}||||𝑝(𝑔)(𝑥)−_{𝐵(𝑥0,𝑡)}=𝜋₋₁𝑝(𝑔)(𝑥)𝑑𝑥||||𝑑𝑥_𝐵(0,1)||||𝑔_∗(𝑧)−_𝐵(0,1)𝑔^∗(𝑧)𝑑𝑧||||𝑑𝑧≤𝑐₁_𝐵(0,1)=𝑐₁||𝑔^∗_𝑧(𝑧)||𝑑𝑧𝑡⁻¹_{𝐵(𝑥0,𝑡)}||(𝑝(𝑔))_𝑥(𝑥)||𝑑𝑥≤𝑐₁𝑡⁻¹||𝐵𝑥₀,𝑡||^{(𝑟−1/𝑟)}‖‖(𝑝(𝑔))_𝑥‖‖_{𝐿𝑟(𝐵(𝑥0,𝑡))},(4.18) 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 ℝ^𝑛 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 𝑢⟶𝑔𝑢(4.19) is a bounded operator from 𝑊^1,2(Ω) to 𝐿²(Ω). Moreover, there exists a constant 𝑐∈ℝ₊, such that ‖𝑔𝑢‖_𝐿²_(Ω)≤𝑐‖𝑔‖_𝑀^𝑟,𝜆_(Ω)‖𝑢‖_𝑊^1,2_(Ω),(4.20) with 𝑐=𝑐(Ω,𝑛,𝑟).
Furthermore, if 𝑔∈𝑀^𝑟,𝜆(Ω), then for any 𝜀>0 there exists a constant 𝑐_𝜀∈ℝ₊, such that ‖𝑔𝑢‖_𝐿²_(Ω)≤𝜀‖𝑢‖_𝑊^1,2_(Ω)+𝑐_𝜀‖𝑢‖_𝐿²_(Ω),(4.21) with 𝑐_𝜀=𝑐_𝜀(𝜀,Ω,𝑛,𝑟,𝜎^𝑟,𝜆[𝑔]).
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 ‖𝑔𝑢‖_𝐿²_(Ω)≤𝑐^‖𝑔‖_𝑀^𝑡,𝜇_(Ω)‖𝑢‖_𝑊^2,2_(Ω),(4.22) 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 ‖𝑔𝑢‖_𝐿²_(Ω)≤𝜀‖𝑢‖_𝑊^2,2_(Ω)+𝑐^𝜀‖𝑢‖_𝐿²_(Ω),(4.23) with 𝑐^𝜀=𝑐^𝜀(𝜀,Ω,𝑛,𝑡,𝜇,𝜎^𝑡,𝜇[𝑔]).

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 𝐿=−^𝑛_𝑖_,𝑗=1𝑎_𝑖𝑗𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗+^𝑛_𝑖=1𝑎_𝑖𝜕𝜕𝑥_𝑖+𝑎,(4.24) with the following conditions on the coefficients: ∃𝜈>0𝑎∶_𝑖𝑗=_𝑛𝑎_𝑗𝑖∈𝐿^∞(Ω),𝑖,𝑗=1,…,𝑛,_𝑖_,𝑗=1𝑎_𝑖𝑗𝜉_𝑖𝜉_𝑗≥𝜈||𝜉||²,a.e.inΩ,∀𝜉∈𝑅^𝑛,(ℎ₁)𝑎_𝑖𝑗_𝑥𝑗,𝑎_𝑖∈𝑀^{𝑜𝑟,𝜆}(Ω),𝑖,𝑗=1,…,𝑛,with𝑟>2,𝜆=0if𝑛=2,with𝑟∈]2,𝑛],𝜆=𝑛−𝑟if𝑛>2,(ℎ₂)𝑎∈𝑀^𝑡,𝜇(Ω),with𝑡≥2,𝜇>𝑛−2𝑡,essinf_Ω𝑎=𝑎₀>0.(ℎ₃) 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 ‖𝑢‖_𝑊^2,2_(Ω)≤𝑐‖𝐿𝑢‖_𝐿²_(Ω),∀𝑢∈𝑊^2,2(Ω)∩𝑊^∘1,2(Ω),(4.25) with 𝑐=𝑐(Ω,𝑛,𝜈,𝑟,𝑡,𝜇,||𝑎_𝑖𝑗||_𝐿^∞_(Ω),𝜎_{𝑜𝑟,𝜆}[(𝑎_𝑖𝑗)_𝑥𝑗],𝜎_{𝑜𝑟,𝜆}[𝑎_𝑖],𝜎^𝑡,𝜇[𝑎],𝑎₀).

Proof. Let us put 𝐿₀=−^𝑛_𝑖_,𝑗=1𝑎_𝑖𝑗𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗(4.26) and fix 𝑢∈𝑊^2,2(Ω)∩𝑊^∘1,2(Ω). 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 ‖𝑢‖_𝑊^2,2_(Ω)≤𝑐₁‖‖𝐿₀𝑢‖‖_𝐿2(Ω)+‖𝑢‖_𝐿²_(Ω),(4.27) with 𝑐₁=𝑐₁(Ω,𝑛,𝜈,||𝑎_𝑖𝑗||_𝐿^∞_(Ω),𝜎_{𝑜𝑟,𝜆}[(𝑎_𝑖𝑗)_𝑥𝑗]). Therefore, ‖𝑢‖_𝑊^2,2_(Ω)≤𝑐₁‖𝐿𝑢‖^𝐿2(Ω)+‖𝑢‖_𝐿²_(Ω)+^𝑛_𝑖=1‖‖𝑎_𝑖𝑢_𝑥𝑖‖‖_𝐿2(Ω)+‖𝑎𝑢‖_𝐿²_(Ω).(4.28) On the other hand, from Lemma 4.2, one has ‖‖𝑎‖_𝑖𝑎𝑢𝑢_𝑥𝑖‖‖‖_{𝐿2(Ω)𝐿2(Ω)}≤𝜀‖𝑢‖_𝑊^2,2_(Ω)+𝑐_𝜀‖‖𝑢_𝑥𝑖‖‖_𝐿2(Ω),≤𝜀‖𝑢‖_𝑊^2,2_(Ω)+𝑐^𝜀‖𝑢‖_𝐿²_(Ω),(4.29) with 𝑐_𝜀=𝑐_𝜀(𝜀,Ω,𝑛,𝑟,𝜎_{𝑜𝑟,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 ‖‖𝑢_𝑥‖‖_𝐿2(Ω)≤𝐾𝜀‖𝑢‖_𝑊^2,2_(Ω)+𝐾𝜀‖𝑢‖^𝐿2(Ω),(4.30) with 𝐾=𝐾(Ω). Combining (4.28), (4.29) and (4.30) we conclude that there exists 𝑐₂∈ℝ₊ such that ‖𝑢‖_𝑊^2,2_(Ω)≤𝑐₂‖𝐿𝑢‖_𝐿²_(Ω)+‖𝑢‖_𝐿²_(Ω),(4.31) with 𝑐₂=𝑐₂(Ω,𝑛,𝜈,𝑟,𝑡,𝜇,||𝑎_𝑖𝑗||_𝐿^∞_(Ω),𝜎_{𝑜𝑟,𝜆}[(𝑎_𝑖𝑗)_𝑥𝑗],𝜎_{𝑜𝑟,𝜆}[𝑎_𝑖],𝜎^𝑡,𝜇[𝑎]).
To show (4.25), it remains to estimate ‖𝑢‖_𝐿²_(Ω). To this aim let us rewrite our operator in divergence form 𝐿𝑢=−^𝑛_𝑖_,𝑗=1𝑎_𝑖𝑗𝑢_𝑥𝑖_𝑥𝑗+^𝑛_𝑖=1^𝑛_𝑗=1𝑎_𝑖𝑗_𝑥𝑗+𝑎_𝑖𝑢^𝑥𝑖+𝑎𝑢,(4.32) 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 ‖𝑢‖_𝐿²_(Ω)≤𝑐₃‖𝐿𝑢‖_𝐿²_(Ω),(4.33) with 𝑐₃=𝑐₃(𝑛,𝜈,𝑟,𝜎_{𝑜𝑟,𝜆}[(𝑎_𝑖𝑗)_𝑥𝑗],𝜎_{𝑜𝑟,𝜆}[𝑎_𝑖],𝑎₀). 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 𝑊^𝑠2,2(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 ℎ₁–ℎ₃, there exists a constant 𝑐∈ℝ₊ such that ‖𝑢‖_𝑊^𝑠2,2_(Ω)≤𝑐‖𝐿𝑢‖_𝐿^2𝑠_(Ω),∀𝑢∈𝑊^𝑠2,2(Ω)∩𝑊^∘𝑠1,2(Ω),(5.1) with 𝑐=𝑐(Ω,𝑛,𝑠,𝜈,𝑟,𝑡,𝜇,||𝑎_𝑖𝑗||_𝐿^∞_(Ω),||𝑎_𝑖||_𝑀^𝑟,𝜆_(Ω),𝜎_{𝑜𝑟,𝜆}[(𝑎_𝑖𝑗)_𝑥𝑗],𝜎_{𝑜𝑟,𝜆}[𝑎_𝑖],𝜎^𝑡,𝜇[𝑎],𝑎₀).

Proof. Fix 𝑢∈𝑊^𝑠2,2(Ω)∩𝑊^∘𝑠1,2(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 applies giving that 𝜂^𝑠𝑢∈𝑊^2,2(Ω)∩𝑊^∘1,2(Ω). Therefore, in view of Theorem 4.3, there exists 𝑐₀∈ℝ₊, such that ‖𝜂^𝑠𝑢‖_𝑊^2,2_(Ω)≤𝑐₀‖𝐿(𝜂^𝑠𝑢)‖_𝐿²_(Ω),(5.2) with 𝑐₀=𝑐₀(Ω,𝑛,𝜈,𝑟,𝑡,𝜇,||𝑎_𝑖𝑗||_𝐿^∞_(Ω),𝜎_{𝑜𝑟,𝜆}[(𝑎_𝑖𝑗)_𝑥𝑗],𝜎_{𝑜𝑟,𝜆}[𝑎_𝑖],𝜎^𝑡,𝜇[𝑎],𝑎₀). Easy computations give 𝐿(𝜂^𝑠𝑢)=𝜂^𝑠𝐿𝑢−𝑠^𝑛_𝑖_,𝑗=1+𝑠_𝑛𝑎_𝑖𝑗(𝑠−1)𝜂^𝑠−2𝜂_𝑥𝑖𝜂_𝑥𝑗𝑢+𝜂^𝑠−1𝜂_{𝑥𝑖𝑥𝑗}𝑢+2𝜂^𝑠−1𝜂_𝑥𝑖𝑢_𝑥𝑗_𝑖=1𝑎_𝑖𝜂^𝑠−1𝜂_𝑥𝑖𝑢.(5.3) Putting together (5.2) and (5.3), we deduce that ‖𝜂^𝑠𝑢‖_𝑊^2,2_(Ω)≤𝑐₁‖𝜂^𝑠𝐿𝑢‖_𝐿²_(Ω)+^𝑛_𝑖_,𝑗=1‖‖𝜂^𝑠−2𝜂_𝑥𝑖𝜂_𝑥𝑗𝑢‖‖_𝐿2(Ω)+‖‖𝜂^𝑠−1𝜂_{𝑥𝑖𝑥𝑗}𝑢‖‖_𝐿2(Ω)+‖‖𝜂^𝑠−1𝜂_𝑥𝑖𝑢_𝑥𝑗‖‖_𝐿2(Ω)+^𝑛_𝑖=1‖‖𝑎_𝑖𝜂^𝑠−1𝜂_𝑥𝑖𝑢‖‖_𝐿2(Ω),(5.4) 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 ‖‖𝑎_𝑖𝜂^𝑠−1𝜂_𝑥𝑖𝑢‖‖_𝐿2(Ω)≤𝑐₂sup_Ω⧵Ω𝑘𝜎_𝑥𝜎‖‖𝑎^𝑖‖‖_{𝑀𝑟,𝜆(Ω)}‖𝜂^𝑠𝑢‖_𝑊^1,2_(Ω),(5.5) with 𝑐₂=𝑐₂(Ω,𝑛,𝑟).
Combining (3.17), (3.18), (5.4), and (5.5), with simple calculations we obtain the bound ‖𝜂^𝑠𝑢‖_𝑊^2,2_(Ω)≤𝑐₃⎡⎢⎢⎣‖𝜂_𝑠𝐿𝑢‖_𝐿²_(Ω)+⎛⎜⎜⎝sup_Ω⧵Ω𝑘𝜎^2𝑥+𝜎𝜎_𝑥𝑥𝜎²+sup_Ω⧵Ω𝑘𝜎_𝑥𝜎⎞⎟⎟⎠‖𝜂^𝑠𝑢‖_𝑊^2,2_(Ω)⎤⎥⎥⎦,(5.6) where 𝑐₃ depends on the same parameters as 𝑐₁ and on ‖𝑎_𝑖‖_𝑀^𝑟,𝜆_(Ω).
By Lemma 3.1, it follows that there exists 𝑘_𝑜∈ℕ such that ⎛⎜⎜⎝sup_{Ω⧵Ω𝑘𝑜}𝜎^2𝑥+𝜎𝜎_𝑥𝑥𝜎²+sup_{Ω⧵Ω𝑘𝑜}𝜎_𝑥𝜎⎞⎟⎟⎠≤21𝑐₃.(5.7)
Now, if we still denote by 𝜂 the function 𝜂_𝑘𝑜, from (5.6) and (5.7), we deduce that ‖𝜂^𝑠𝑢‖_𝑊^2,2_(Ω)≤2𝑐₃‖𝜂^𝑠𝐿𝑢‖_𝐿²_(Ω).(5.8) Then, by Lemma 2.2 and by (3.12), written for 𝑘=𝑘_𝑜, we have _|_𝛼|≤2‖𝜎^𝑠𝜕^𝛼𝑢‖_𝐿²_(Ω)≤𝑐₄‖𝜎^𝑠𝐿𝑢‖_𝐿²_(Ω),(5.9) 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 _|_𝛼|≤2‖𝜎^−𝑠𝜕^𝛼𝑢‖_𝐿²_(Ω)≤𝑐₅‖𝜎^−𝑠𝐿𝑢‖_𝐿²_(Ω).(5.10)
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 −𝑢Δ∈𝑢𝑊+𝑏𝑢^𝑠2,2(=Ω𝑓),∩𝑓𝑊^∘∈𝐿^𝑠1,2_2𝑠(Ω),(Ω),(5.11) where 𝑏=1+|||||−𝑠(𝑠+1)^𝑛_𝑖=1𝜎^2𝑥_𝑖𝜎²+𝑠^𝑛_𝑖=1𝜎_{𝑥𝑖𝑥𝑖}𝜎|||||,(5.12) 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 −Δ(𝜎𝑤_−𝑠∈𝑊^2,2(Ω)∩𝑊^∘1,2(Ω),𝑤)+𝑏𝜎^−𝑠𝑤=𝑓,𝑓∈𝐿^2𝑠(Ω).(5.13) Clearly, for any 𝑖∈{1,…,𝑛}, 𝜕²𝜕𝑥^2𝑖(𝜎^−𝑠𝑤)=𝜎^−𝑠𝑤_{𝑥𝑖𝑥𝑖}−2𝑠𝜎^−𝑠−1𝜎_𝑥𝑖𝑤_𝑥𝑖+𝑠(𝑠+1)𝜎^−𝑠−2𝜎^2𝑥_𝑖𝑤−𝑠𝜎^−𝑠−1𝜎_{𝑥𝑖𝑥𝑖}𝑤,(5.14) then (5.13) is equivalent to the problem −Δ𝑤+𝑤_𝑛∈𝑊^2,2(Ω)∩𝑊^∘1,2(Ω),_𝑖∑₌₁𝛼_𝑖𝑤_𝑥𝑖+𝛼𝑤=𝑔,𝑔∈𝐿²(Ω),(5.15) where 𝛼_𝑖=2𝑠𝜎^𝑥𝑖𝜎,𝑖=1,…,𝑛,𝛼=𝑏−𝑠(𝑠+1)^𝑛_𝑖=1𝜎^2𝑥_𝑖𝜎²+𝑠^𝑛_𝑖=1𝜎_{𝑥𝑖𝑥𝑖}𝜎,𝑔=𝜎^𝑠𝑓.(5.16) 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 𝑢∈𝐿𝑢𝑊=𝑓^𝑠2,2,(Ω)𝑓∩∈𝑊𝐿^∘𝑠1,2_2𝑠(Ω),(Ω),(5.17) is uniquely solvable.

Proof. For each 𝜏∈[0,1], we put 𝐿_𝜏=𝜏(𝐿)+(1−𝜏)(−Δ+𝑏).(5.18) In view of Theorem 5.1, ‖𝑢‖_𝑊^𝑠2,2_(Ω)≤𝑐‖‖𝐿_𝜏𝑢‖‖_𝐿^𝑝𝑠_(Ω),∀𝑢∈𝑊^𝑠2,2(Ω)∩𝑊^∘𝑠1,2(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.