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International Journal of Differential Equations
VolumeΒ 2011Β (2011), Article IDΒ 572824, 17 pages
Research Article

π‘ŠπŸ,𝟐 A Priori Bounds for a Class of Elliptic Operators

Dipartimento di Matematica, UniversitΓ  di Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy

Received 17 June 2011; Accepted 5 August 2011

Academic Editor: Charles E.Β Chidume

Copyright Β© 2011 Sara MonsurrΓ² et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We obtain some π‘Š2,2 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 Ξ© be an open subset of ℝ𝑛,𝑛β‰₯2. The uniformly elliptic second-order linear differential operator 𝐿=βˆ’π‘›ξ“π‘–,𝑗=1π‘Žπ‘–π‘—πœ•2πœ•π‘₯π‘–πœ•π‘₯𝑗+𝑛𝑖=1π‘Žπ‘–πœ•πœ•π‘₯𝑖+π‘Ž,(1.1) with leading coefficients π‘Žπ‘–π‘—=π‘Žπ‘—π‘–βˆˆπΏβˆž(Ξ©),𝑖,𝑗=1,…,𝑛, and the associated Dirichlet problem π‘’βˆˆπ‘Š2,2(Ξ©)βˆ©βˆ˜π‘Š1,2(Ξ©),𝐿𝑒=𝑓,π‘“βˆˆπΏ2(Ξ©),(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 π‘Š2,2 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 (π‘Žπ‘–π‘—)π‘₯π‘˜βˆˆπΏπ‘›(Ξ©),𝑛β‰₯3. First generalizations in this direction have been carried on, always considering a bounded and sufficiently regular set Ξ©, assuming that the derivatives belong to some wider spaces. In particular, in [3], the (π‘Žπ‘–π‘—)π‘₯π‘˜ are in the weak-𝐿𝑛 space, while, in [4], they are supposed to be in an appropriate subspace of the classical Morrey space 𝐿2𝑝,π‘›βˆ’2𝑝(Ξ©), where π‘βˆˆ]1,𝑛/2[. In [5], the leading coefficients are supposed to be close to functions whose derivatives are in 𝐿𝑛(Ξ©). A further extension, to a very general case, has been proved in [6, 7], supposing that the π‘Žπ‘–π‘— 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 (π‘Žπ‘–π‘—)π‘₯π‘˜ 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 π‘Š2,2-bound, having the only term ‖𝐿𝑒‖𝐿2(Ξ©) in the right-hand side, β€–π‘’β€–π‘Š2,2(Ξ©)≀𝑐‖𝐿𝑒‖𝐿2(Ξ©),βˆ€π‘’βˆˆπ‘Š2,2(Ξ©)βˆ©βˆ˜π‘Š1,2(Ξ©),(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 π‘Žπ‘–π‘— 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 π‘Š2,2-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, π‘’βˆˆπ‘Šπ‘ 2,2(Ξ©)βˆ©βˆ˜π‘Šπ‘ 1,2(Ξ©),𝐿𝑒=𝑓,π‘“βˆˆπΏ2𝑠(Ξ©),(1.4) where π‘ βˆˆβ„, π‘Šπ‘ 2,2(Ξ©),βˆ˜π‘Šπ‘ 1,2(Ξ©), and 𝐿2𝑠(Ξ©) are weighted Sobolev spaces where the weight πœŒπ‘  is power of a function πœŒβˆΆΞ©β†’β„+, of class 𝐢2(Ξ©), and such that supπ‘₯∈Ω||πœ•π›Ό||𝜌(π‘₯)𝜌(π‘₯)<+∞,βˆ€|𝛼|≀2,lim|π‘₯|β†’+βˆžξ‚΅1𝜌(π‘₯)+ξ‚ΆπœŒ(π‘₯)=+∞,lim|π‘₯|β†’+∞𝜌π‘₯(π‘₯)+𝜌π‘₯π‘₯(π‘₯)𝜌(π‘₯)=0,(1.5) see Sections 2 and 3 for more details. Also in this weighted case, we obtain the bound β€–π‘’β€–π‘Šπ‘ 2,2(Ξ©)≀𝑐‖𝐿𝑒‖𝐿2𝑠(Ξ©),βˆ€π‘’βˆˆπ‘Šπ‘ 2,2(Ξ©)βˆ©βˆ˜π‘Šπ‘ 1,2(Ξ©),(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 Ξ© be an open subset of ℝ𝑛, not necessarily bounded, 𝑛β‰₯2. We want to introduce a class of weight functions defined on Ξ©.

To this aim, given π‘˜βˆˆβ„•0, we consider a function πœŒβˆΆΞ©β†’β„+ such that πœŒβˆˆπΆπ‘˜(Ξ©) and supπ‘₯∈Ω||πœ•π›Ό||𝜌(π‘₯)𝜌(π‘₯)<+∞,βˆ€|𝛼|β‰€π‘˜.(2.1) As an example, we can think of the function ξ€·πœŒ(π‘₯)=1+|π‘₯|2𝑑,π‘‘βˆˆβ„.(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 supπ‘₯∈Ω||πœ•π›ΌπœŒπ‘ ||(π‘₯)πœŒπ‘ (π‘₯)<+βˆžβˆ€π‘ βˆˆβ„,βˆ€|𝛼|β‰€π‘˜.(2.3)

Proof. The proof is obtained by induction. From (2.1), we get ||(πœŒπ‘ )π‘₯𝑖||=||π‘ πœŒπ‘ βˆ’1𝜌π‘₯𝑖||≀𝑐1πœŒπœŒπ‘ βˆ’1=𝑐1πœŒπ‘ ,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 π‘ βˆˆβ„, and fix a 𝛽 such that |𝛽|=|𝛼|βˆ’1. Then, using (2.1) and by the induction hypothesis written for π‘ βˆ’1, we have ||πœ•π›ΌπœŒπ‘ ||=||πœ•π›½(πœŒπ‘ )π‘₯𝑖||=||πœ•π›½ξ€·π‘ πœŒπ‘ βˆ’1𝜌π‘₯𝑖||≀𝑐2𝛾≀𝛽||πœ•π›½βˆ’π›ΎπœŒπ‘₯π‘–πœ•π›ΎπœŒπ‘ βˆ’1||≀𝑐3πœŒπœŒπ‘ βˆ’1=𝑐3πœŒπ‘ ,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 π‘˜βˆˆβ„•0, π‘βˆˆ[1,+∞[, and π‘ βˆˆβ„, and given a weight function 𝜌 satisfying (2.1), we define the space π‘Šπ‘ π‘˜,𝑝(Ξ©) of distributions 𝑒 on Ξ© such that β€–π‘’β€–π‘Šπ‘ π‘˜,𝑝(Ξ©)=|𝛼|β‰€π‘˜β€–πœŒπ‘ πœ•π›Όπ‘’β€–πΏπ‘(Ξ©)<+∞,(2.6) equipped with the norm given in (2.6). Moreover, we denote by βˆ˜π‘Šπ‘ π‘˜,𝑝(Ξ©) the closure of 𝐢∞∘(Ξ©) in π‘Šπ‘ π‘˜,𝑝(Ξ©) and put π‘Šπ‘ 0,𝑝(Ξ©)=𝐿𝑝𝑠(Ξ©).

Lemma 2.2. Let π‘˜βˆˆβ„•0, π‘βˆˆ[1,+∞[, and π‘ βˆˆβ„. If assumption (2.1) is satisfied, then there exist two constants 𝑐1,𝑐2βˆˆβ„+ such that 𝑐1β€–π‘’β€–π‘Šπ‘ π‘˜,𝑝(Ξ©)β‰€β€–β€–πœŒπ‘‘π‘’β€–β€–π‘Šπ‘˜,π‘π‘ βˆ’π‘‘(Ξ©)≀𝑐2β€–π‘’β€–π‘Šπ‘ π‘˜,𝑝(Ξ©),βˆ€π‘‘βˆˆβ„,βˆ€π‘’βˆˆπ‘Šπ‘ π‘˜,𝑝(Ξ©),(2.7) with 𝑐1=𝑐1(𝑑) and 𝑐2=𝑐2(𝑑).

Proof. Observe that from (2.3), we have ||πœ•π›Όξ€·πœŒπ‘‘π‘’ξ€Έ||≀𝑐1𝛽≀𝛼||πœ•π›Όβˆ’π›½πœŒπ‘‘πœ•π›½π‘’||≀𝑐2||πœŒπ‘‘πœ•π›½π‘’||,βˆ€|𝛼|β‰€π‘˜,(2.8) with 𝑐2βˆˆβ„+ depending only on 𝑑. 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 ||πœŒπ‘‘πœ•π›Όπ‘’||≀𝑐3𝛽≀𝛼||πœ•π›½ξ€·πœŒπ‘‘π‘’ξ€Έ||,βˆ€|𝛼|β‰€π‘˜,(2.9) with 𝑐3βˆˆβ„+ depending only on 𝑑.
We will prove (2.9) by induction. From (2.3), one has ||πœŒπ‘‘π‘’π‘₯𝑖||=|||ξ€·πœŒπ‘‘π‘’ξ€Έπ‘₯π‘–βˆ’ξ€·πœŒπ‘‘ξ€Έπ‘₯𝑖𝑒|||≀𝑐4πœŒξ€·ξ€·π‘‘π‘’ξ€Έπ‘₯+πœŒπ‘‘ξ€Έ,|𝑒|(2.10) for 𝑖=1,…,𝑛, with 𝑐4βˆˆβ„+ depending only on 𝑑. 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 ||πœŒπ‘‘πœ•π›Όπ‘’||≀||πœ•π›Όξ€·πœŒπ‘‘π‘’ξ€Έ||+𝑐5𝛽<𝛼||πœ•π›Όβˆ’π›½πœŒπ‘‘||||πœ•π›½π‘’||≀||πœ•π›Όξ€·πœŒπ‘‘π‘’ξ€Έ||+𝑐6𝛽<𝛼||πœŒπ‘‘πœ•π›½π‘’||≀𝑐7𝛽≀𝛼||πœ•π›½ξ€·πœŒπ‘‘π‘’ξ€Έ||,(2.11) with 𝑐7βˆˆβ„+ depending only on 𝑑.

Let us specify a density result.

Lemma 2.3. Let π‘˜βˆˆβ„•0, π‘βˆˆ[1,+∞[, and π‘ βˆˆβ„. If Ξ© has the segment property and assumption (2.1) is satisfied, then π’Ÿ(Ξ©) is dense in π‘Šπ‘ π‘˜,𝑝(Ξ©).

Proof. The proof follows by Lemma 2.2 in [12], since clearly both 𝜌,πœŒβˆ’1∈𝐿∞loc(Ξ©).

This allows us to prove the following inclusion.

Lemma 2.4. Let π‘˜βˆˆβ„•0, π‘βˆˆ[1,+∞[, and π‘ βˆˆβ„. If Ξ© has the segment property and assumption (2.1) is satisfied, then π‘Šπ‘˜,𝑝(Ξ©)βˆ©βˆ˜π‘Šπ‘˜,𝑝(Ξ©)βŠ‚βˆ˜π‘Šπ‘ π‘˜,𝑝(Ξ©).(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 Ξ© has the segment property, also πΆπ‘˜π‘œ(Ξ©)βŠ‚βˆ˜π‘Šπ‘ π‘˜,𝑝(Ξ©).

Lemma 2.5. Let π‘˜βˆˆβ„•0, π‘βˆˆ[1,+∞[, and π‘ βˆˆβ„. If Ξ© has the segment property and assumption (2.1) is satisfied, then the map π‘’βŸΆπœŒπ‘ π‘’(2.13) defines a topological isomorphism from π‘Šπ‘ π‘˜,𝑝(Ξ©) to π‘Šπ‘˜,𝑝(Ξ©) and from βˆ˜π‘Šπ‘ π‘˜,𝑝(Ξ©) to βˆ˜π‘Šπ‘˜,𝑝(Ξ©).

Proof. The first part of the proof easily follows from Lemma 2.2 with 𝑑=𝑠. Let us show that π‘’βˆˆβˆ˜π‘Šπ‘ π‘˜,𝑝(Ξ©) if and only if πœŒπ‘ π‘’βˆˆβˆ˜π‘Šπ‘˜,𝑝(Ξ©).
If π‘’βˆˆβˆ˜π‘Šπ‘ π‘˜,𝑝(Ξ©), there exists a sequence (πœ™β„Ž)β„Žβˆˆβ„•βŠ‚πΆβˆžπ‘œ(Ξ©) converging to 𝑒 in π‘Šπ‘ π‘˜,𝑝(Ξ©). Therefore, fixed πœ€βˆˆβ„+, there exists β„Ž0βˆˆβ„• such that β€–β€–πœŒπ‘ ξ€·πœ™β„Žξ€Έβ€–β€–βˆ’π‘’π‘Šπ‘˜,𝑝(Ξ©)<πœ€2,βˆ€β„Ž>β„Ž0.(2.14) Fix β„Ž1>β„Ž0, clearly πœŒπ‘ πœ™β„Ž1βˆˆβˆ˜π‘Šπ‘˜,𝑝(Ξ©), because of its compact support. Therefore, there exists a sequence (πœ“π‘›)π‘›βˆˆβ„•βŠ‚πΆβˆžπ‘œ(Ξ©) converging to πœŒπ‘ πœ™β„Ž1 in π‘Šπ‘˜,𝑝(Ξ©). Hence, there exists 𝑛0βˆˆβ„• such that β€–β€–πœ“π‘›βˆ’πœŒπ‘ πœ™β„Ž1β€–β€–π‘Šπ‘˜,𝑝(Ξ©)<πœ€2,βˆ€π‘›>𝑛0.(2.15) Putting together (2.14) and (2.15), we get β€–β€–πœ“π‘›βˆ’πœŒπ‘ π‘’β€–β€–π‘Šπ‘˜,𝑝(Ξ©)β‰€β€–β€–πœ“π‘›βˆ’πœŒπ‘ πœ™β„Ž1β€–β€–π‘Šπ‘˜,𝑝(Ξ©)+β€–β€–πœŒπ‘ πœ™β„Ž1βˆ’πœŒπ‘ π‘’β€–β€–π‘Šπ‘˜,𝑝(Ξ©)<πœ€,(2.16)  forall𝑛>𝑛0. Thus, πœŒπ‘ π‘’βˆˆβˆ˜π‘Šπ‘˜,𝑝(Ξ©).
Vice versa, if we assume that πœŒπ‘ π‘’βˆˆβˆ˜π‘Šπ‘˜,𝑝(Ξ©), we find a sequence (πœ™β„Ž)β„Žβˆˆβ„•βŠ‚πΆβˆžπ‘œ(Ξ©) converging to πœŒπ‘ π‘’ in π‘Šπ‘˜,𝑝(Ξ©). Hence, there exists β„Ž0βˆˆβ„• such that β€–β€–πœŒβˆ’π‘ πœ™β„Žβ€–β€–βˆ’π‘’π‘Šπ‘ π‘˜,𝑝(Ξ©)<πœ€2,βˆ€β„Ž>β„Ž0.(2.17) Fix β„Ž1>β„Ž0, since πœŒβˆ’π‘ πœ™β„Ž1βˆˆπΆπ‘˜π‘œ(Ξ©), which is contained in βˆ˜π‘Šπ‘ π‘˜,𝑝(Ξ©) by Lemma 2.4, there exists a sequence (πœ“π‘›)π‘›βˆˆβ„•βŠ‚πΆβˆžπ‘œ(Ξ©) converging to πœŒβˆ’π‘ πœ™β„Ž1 in π‘Šπ‘ π‘˜,𝑝(Ξ©). Therefore, there exists 𝑛0βˆˆβ„• such that β€–β€–πœ“π‘›βˆ’πœŒβˆ’π‘ πœ™β„Ž1β€–β€–π‘Šπ‘ π‘˜,𝑝(Ξ©)<πœ€2,βˆ€π‘›>𝑛0.(2.18) From (2.17) and (2.18), we get β€–β€–πœ“π‘›β€–β€–βˆ’π‘’π‘Šπ‘ π‘˜,𝑝(Ξ©)β‰€β€–β€–πœ“π‘›βˆ’πœŒβˆ’π‘ πœ™β„Ž1β€–β€–π‘Šπ‘ π‘˜,𝑝(Ξ©)+β€–β€–πœŒβˆ’π‘ πœ™β„Ž1β€–β€–βˆ’π‘’π‘Šπ‘ π‘˜,𝑝(Ξ©)<πœ€,(2.19)   forall𝑛>𝑛0, so that π‘’βˆˆβˆ˜π‘Šπ‘ π‘˜,𝑝(Ξ©).

3. Preliminary Results

From now on, we consider a weight πœŒβˆΆΞ©β†’β„+, 𝜌∈𝐢2(Ξ©), and such that (2.1) is satisfied (for π‘˜=2). Moreover, we assume that lim|π‘₯|β†’+βˆžξ‚΅1𝜌(π‘₯)+ξ‚ΆπœŒ(π‘₯)=+∞,lim|π‘₯|β†’+∞𝜌π‘₯(π‘₯)+𝜌π‘₯π‘₯(π‘₯)𝜌(π‘₯)=0.(3.1) An example of a function verifying our hypotheses is given by ξ€·πœŒ(π‘₯)=1+|π‘₯|2𝑑,π‘‘βˆˆβ„β§΅{0}.(3.2) We associate to 𝜌 a function 𝜎 defined by 𝜎=𝜌if𝜌⟢+∞,for1|π‘₯|⟢+∞,𝜎=𝜌if𝜌⟢0,for|π‘₯|⟢+∞.(3.3) Clearly 𝜎 verifies (2.1) and lim|π‘₯|β†’+∞𝜎(π‘₯)=+∞,lim|π‘₯|β†’+∞𝜎π‘₯(π‘₯)+𝜎π‘₯π‘₯(π‘₯)𝜎(π‘₯)=0.(3.4)

Now, let us fix a cutoff function π‘“βˆˆπΆβˆžβˆ˜(ℝ+) such that 0≀𝑓≀1,𝑓(𝑑)=1if[]π‘‘βˆˆ0,1,𝑓(𝑑)=0if[[.π‘‘βˆˆ2,+∞(3.5) Then, set πœπ‘˜βˆΆπ‘₯βˆˆξ‚΅Ξ©βŸΆπ‘“πœŽ(π‘₯)π‘˜ξ‚ΆΞ©,π‘˜βˆˆβ„•,π‘˜={π‘₯∈Ω∢𝜎(π‘₯)<π‘˜},π‘˜βˆˆβ„•.(3.6) By our definition, it follows that πœπ‘˜βˆˆπΆβˆžβˆ˜(Ξ©) and 0β‰€πœπ‘˜β‰€1,πœπ‘˜=1onΞ©π‘˜,πœπ‘˜=0onΩ⧡Ω2π‘˜,π‘˜βˆˆβ„•.(3.7)

Finally, we introduce the sequence πœ‚π‘˜βˆΆπ‘₯∈Ω⟢2π‘˜πœπ‘˜ξ€·(π‘₯)+1βˆ’πœπ‘˜ξ€Έ(π‘₯)𝜎(π‘₯),π‘˜βˆˆβ„•.(3.8)

For any π‘˜βˆˆβ„•, one has πœ‚π‘˜=πœπ‘˜(2π‘˜βˆ’πœŽ)+𝜎β‰₯𝜎,inΞ©2π‘˜,(3.9)πœ‚π‘˜ξƒ©β‰€2π‘˜+πœŽβ‰€2π‘˜infΞ©2π‘˜πœŽξƒͺ𝑐+1𝜎=π‘˜ξ€Έ+1𝜎,inΞ©2π‘˜,(3.10)πœ‚π‘˜=𝜎,inΩ⧡Ω2π‘˜,(3.11) where π‘π‘˜βˆˆβ„+ depends only on π‘˜. This entails that πœŽβˆΌπœ‚π‘˜,βˆ€π‘˜βˆˆβ„•.(3.12) Concerning the derivatives, easy calculations give that, for any π‘˜βˆˆβ„•, ξ€·πœ‚π‘˜ξ€Έπ‘₯=ξ€·πœ‚π‘˜ξ€Έπ‘₯π‘₯=0,inΞ©π‘˜,(3.13)ξ€·πœ‚π‘˜ξ€Έπ‘₯=𝜎π‘₯,ξ€·πœ‚π‘˜ξ€Έπ‘₯π‘₯=𝜎π‘₯π‘₯,inΩ⧡Ω2π‘˜,(3.14)ξ€·πœ‚π‘˜ξ€Έπ‘₯≀𝑐1𝜎π‘₯,ξ€·πœ‚π‘˜ξ€Έπ‘₯π‘₯≀𝑐2ξ‚΅πœŽ2π‘₯𝜎+𝜎π‘₯π‘₯ξ‚Ά,inΩ⧡Ω2π‘˜,(3.15) with 𝑐1 and 𝑐2 positive constants independent of π‘₯ and π‘˜.

From (3.9), (3.11), (3.13), (3.14), and (3.15), we obtain, for any π‘˜βˆˆπ‘, ξ€·πœ‚π‘˜ξ€Έπ‘₯πœ‚π‘˜β‰€π‘ξ…ž1𝜎π‘₯𝜎,inξ€·πœ‚Ξ©,π‘˜ξ€Έπ‘₯π‘₯πœ‚π‘˜β‰€π‘ξ…ž2𝜎2π‘₯+𝜎𝜎π‘₯π‘₯𝜎2,inΞ©,(3.16) where π‘ξ…ž1 and π‘ξ…ž2 are positive constants independent of π‘₯ and π‘˜.

Combining (3.13) and (3.16), we have also, for any π‘˜βˆˆβ„•, ξ€·πœ‚π‘˜ξ€Έπ‘₯πœ‚π‘˜β‰€π‘ξ…ž1supΞ©β§΅Ξ©π‘˜πœŽπ‘₯𝜎,inΞ©,(3.17)ξ€·πœ‚π‘˜ξ€Έπ‘₯π‘₯πœ‚π‘˜β‰€π‘ξ…ž2supΞ©β§΅Ξ©π‘˜πœŽ2π‘₯+𝜎𝜎π‘₯π‘₯𝜎2,inΞ©.(3.18)

We conclude this section proving the following lemma.

Lemma 3.1. Let 𝜎 and Ξ©π‘˜,π‘˜βˆˆβ„•, be defined by (3.3) and (3.6), respectively. Then limπ‘˜β†’+∞supΞ©β§΅Ξ©π‘˜πœŽπ‘₯(π‘₯)+𝜎π‘₯π‘₯(π‘₯)𝜎(π‘₯)=0.(3.19)

Proof. Set πœŽπœ‘(π‘₯)=π‘₯(π‘₯)+𝜎π‘₯π‘₯(π‘₯)𝜎(π‘₯),π‘₯βˆˆπœ“Ξ©,π‘˜=supΞ©β§΅Ξ©π‘˜πœ‘,π‘˜βˆˆβ„•.(3.20) By the second relation in (3.4), the supremum of πœ‘ over Ξ©β§΅Ξ©π‘˜ is actually a maximum; thus, for every π‘˜βˆˆβ„•, there exists π‘₯π‘˜βˆˆΞ©β§΅Ξ©π‘˜ such that πœ“π‘˜=πœ‘(π‘₯π‘˜).
To prove (3.19), we have to show that limπ‘˜β†’+βˆžπœ“π‘˜=0.
We proceed by contradiction. Hence, let us assume that there exists πœ€0>0 such that, for any π‘˜βˆˆβ„•, there exists π‘›π‘˜>π‘˜ such that πœ“π‘›π‘˜=πœ‘(π‘₯π‘›π‘˜)β‰₯πœ€0.
If the sequence (π‘₯π‘›π‘˜)π‘˜βˆˆβ„• is bounded, there exists a subsequence (π‘₯β€²π‘›π‘˜)π‘˜βˆˆβ„• converging to a limit π‘₯∈Ω, and by the continuity of 𝜎, (𝜎(π‘₯β€²π‘›π‘˜))π‘˜βˆˆβ„• converges to 𝜎(π‘₯). On the other hand, π‘₯β€²π‘›π‘˜βˆˆΞ©β§΅Ξ©π‘˜, thus 𝜎(π‘₯β€²π‘›π‘˜)β‰₯π‘›π‘˜, which is in contrast with the fact that (𝜎(π‘₯β€²π‘›π‘˜))π‘˜βˆˆβ„• is a convergent sequence.
Therefore, (π‘₯π‘›π‘˜)π‘˜βˆˆβ„• is unbounded, so that there exists a subsequence (π‘₯π‘›ξ…žξ…žπ‘˜)π‘˜βˆˆβ„• such that limπ‘˜β†’+∞|π‘₯π‘›ξ…žξ…žπ‘˜|=+∞. Thus, by the second relation in (3.4), one has limπ‘˜β†’+βˆžπœ‘(π‘₯π‘›ξ…žξ…žπ‘˜)=0. This gives the contradiction since πœ‘(π‘₯π‘›ξ…žξ…žπ‘˜)β‰₯πœ€0.

4. A No Weighted A Priori Bound

We want to prove a π‘Š2,2 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 Ξ£(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 πœ’πΈ its characteristic function, and by 𝐸(π‘₯,π‘Ÿ) the intersection 𝐸∩𝐡(π‘₯,π‘Ÿ) (π‘₯βˆˆβ„π‘›,π‘Ÿβˆˆβ„+), where 𝐡(π‘₯,π‘Ÿ) is the open ball with center π‘₯ and radius π‘Ÿ.

For 𝑛β‰₯2, πœ†βˆˆ[0,𝑛[, π‘βˆˆ[1,+∞[, and fixed 𝑑 in ℝ+, the space of Morrey type 𝑀𝑝,πœ†(Ξ©,𝑑) is the set of all functions 𝑔 in 𝐿𝑝loc(Ξ©) such that ‖𝑔‖𝑀𝑝,πœ†(Ξ©,𝑑)=sup𝜏∈]0,𝑑]π‘₯βˆˆΞ©πœβˆ’πœ†/𝑝‖𝑔‖𝐿𝑝(Ξ©(π‘₯,𝜏))<+∞,(4.1) endowed with the norm defined in (4.1). It is easily seen that, for any 𝑑1,𝑑2βˆˆβ„+, a function 𝑔 belongs to 𝑀𝑝,πœ†(Ξ©,𝑑1) if and only if it belongs to 𝑀𝑝,πœ†(Ξ©,𝑑2); moreover, the norms of 𝑔 in these two spaces are equivalent. This allows us to restrict our attention to the space 𝑀𝑝,πœ†(Ξ©)=𝑀𝑝,πœ†(Ξ©,1).

We now introduce three subspaces of 𝑀𝑝,πœ†(Ξ©) needed in the sequel. The set 𝑉𝑀𝑝,πœ†(Ξ©) is made up of the functions π‘”βˆˆπ‘€π‘,πœ†(Ξ©) such that lim𝑑→0‖𝑔‖𝑀𝑝,πœ†(Ξ©,𝑑)=0,(4.2) while 𝑀𝑝,πœ†(Ξ©) and π‘€βˆ˜π‘,πœ†(Ξ©) denote the closures of 𝐿∞(Ξ©) and 𝐢∞∘(Ξ©) in 𝑀𝑝,πœ†(Ξ©), respectively. We point out that π‘€βˆ˜π‘,πœ†ξ‚‹π‘€(Ξ©)βŠ‚π‘,πœ†(Ξ©)βŠ‚π‘‰π‘€π‘,πœ†(Ξ©).(4.3) We put 𝑀𝑝(Ξ©)=𝑀𝑝,0(Ξ©), 𝑉𝑀𝑝(Ξ©)=𝑉𝑀𝑝,0(Ξ©), 𝑀𝑝𝑀(Ξ©)=𝑝,0(Ξ©), and π‘€π‘βˆ˜(Ξ©)=π‘€βˆ˜π‘,0(Ξ©).

We want to define the moduli of continuity of functions belonging to 𝑀𝑝,πœ†(Ξ©) or π‘€βˆ˜π‘,πœ†(Ξ©). To this aim, let us put, for β„Žβˆˆβ„+ and π‘”βˆˆπ‘€π‘,πœ†(Ξ©), 𝐹[𝑔](β„Ž)=sup𝐸∈Σ(Ξ©)supπ‘₯∈Ω|𝐸(π‘₯,1)|≀1/β„Žβ€–β€–π‘”πœ’πΈβ€–β€–π‘€π‘,πœ†(Ξ©).(4.4) Recall first that for a function π‘”βˆˆπ‘€π‘,πœ†(Ξ©) the following characterization holds: ξ‚‹π‘€π‘”βˆˆπ‘,πœ†(Ξ©)⟺limβ„Žβ†’+∞𝐹[𝑔](β„Ž)=0,(4.5) while π‘”βˆˆπ‘€βˆ˜π‘,πœ†(Ξ©)⟺limβ„Žβ†’+βˆžξ‚€πΉ[𝑔]β€–β€–ξ€·(β„Ž)+1βˆ’πœβ„Žξ€Έπ‘”β€–β€–π‘€π‘,πœ†(Ξ©)=0,(4.6) where πœβ„Ž denotes a function of class πΆβˆžπ‘œ(𝑅𝑛) such that 0β‰€πœβ„Žβ‰€1,πœβ„Ž|𝐡(0,β„Ž)=1,suppπœβ„ŽβŠ‚π΅(0,2β„Ž).(4.7) Thus, if 𝑔 is a function in 𝑀𝑝,πœ†(Ξ©), a modulus of continuity of 𝑔 in 𝑀𝑝,πœ†(Ξ©) is a map ξ‚πœŽπ‘,πœ†[𝑔]βˆΆβ„+→ℝ+ such that 𝐹[𝑔](β„Ž)β‰€ξ‚πœŽπ‘,πœ†[𝑔](β„Ž),limβ„Žβ†’+βˆžξ‚πœŽπ‘,πœ†[𝑔](β„Ž)=0.(4.8) While, if 𝑔 belongs to π‘€π‘œπ‘,πœ†(Ξ©), a modulus of continuity of 𝑔 in π‘€π‘œπ‘,πœ†(Ξ©) is an application πœŽπ‘œπ‘,πœ†[𝑔]βˆΆβ„+→ℝ+ such that 𝐹[𝑔]β€–β€–ξ€·(β„Ž)+1βˆ’πœβ„Žξ€Έπ‘”β€–β€–π‘€π‘,πœ†(Ξ©)β‰€πœŽπ‘œπ‘,πœ†[𝑔](β„Ž),limβ„Žβ†’+βˆžπœŽπ‘œπ‘,πœ†[𝑔](β„Ž)=0.(4.9) If Ξ© has the property ||||Ξ©(π‘₯,π‘Ÿ)β‰₯π΄π‘Ÿπ‘›]],βˆ€π‘₯∈Ω,βˆ€π‘Ÿβˆˆ0,1(4.10) where 𝐴 is a positive constant independent of π‘₯ and π‘Ÿ, it is possible to consider the space 𝐡𝑀𝑂(Ξ©,𝜏) (πœβˆˆβ„+) of functions π‘”βˆˆπΏ1loc(Ξ©) such that [𝑔]𝐡𝑀𝑂(Ξ©,𝜏)=supπ‘₯βˆˆΞ©π‘Ÿβˆˆ]0,𝜏]Ω(π‘₯,π‘Ÿ)||||ξ€§π‘”βˆ’Ξ©(π‘₯,π‘Ÿ)||||𝑔𝑑𝑦𝑑𝑦<+∞,(4.11) where Ω(π‘₯,π‘Ÿ)||||𝑔𝑑𝑦=Ξ©(π‘₯,π‘Ÿ)βˆ’1ξ€œΞ©(π‘₯,π‘Ÿ)𝑔𝑑𝑦.(4.12) If π‘”βˆˆπ΅π‘€π‘‚(Ξ©)=𝐡𝑀𝑂(Ξ©,𝜏𝐴), where 𝜏𝐴⎧βŽͺ⎨βŽͺ⎩=supπœβˆˆβ„+∢supπ‘₯βˆˆΞ©π‘Ÿβˆˆ]0,𝜏]π‘Ÿπ‘›||||≀1Ξ©(π‘₯,π‘Ÿ)𝐴⎫βŽͺ⎬βŽͺ⎭,(4.13) we say that π‘”βˆˆπ‘‰π‘€π‘‚(Ξ©) if [𝑔]𝐡𝑀𝑂(Ξ©,𝜏)β†’0 for πœβ†’0+.

If 𝑔 belongs to 𝑉𝑀𝑂(Ξ©), a modulus of continuity of 𝑔 in 𝑉𝑀𝑂(Ξ©) is function πœ‚[𝑔]∢]0,1]→ℝ+ such that [𝑔]𝐡𝑀𝑂(Ξ©,𝜏)[𝑔]]]β‰€πœ‚(𝜏)βˆ€πœβˆˆ0,1,limπœβ†’0+πœ‚[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 Ξ© has the uniform C1,1-regularity property and 𝑔,𝑔π‘₯βˆˆξ‚»VMr(Ξ©),r>2,for𝑛=2,VMr,nβˆ’r(Ξ©),r∈]2,n],forn>2,(4.15) then g∈VMO(Ξ©).

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,𝑔π‘₯βˆˆπ‘‰π‘€π‘Ÿ(Ξ©) admits an extension 𝑝(𝑔) such that 𝑝(𝑔),(𝑝(𝑔))π‘₯βˆˆπ‘‰π‘€π‘Ÿ(ℝ2).
Then, we prove that for all π‘₯0βˆˆβ„2 and π‘‘βˆˆβ„+, there exists a constant π‘βˆˆβ„+ such that 𝐡(π‘₯0,𝑑)||||𝑝(𝑔)βˆ’π΅(π‘₯0,𝑑)𝑝||||𝑑(𝑔)𝑑π‘₯𝑑π‘₯≀𝑐(π‘Ÿβˆ’2/π‘Ÿ)β€–β€–(𝑝(𝑔))π‘₯β€–β€–πΏπ‘Ÿ(𝐡(π‘₯0,𝑑)).(4.16) Indeed, in view of the above considerations, if (4.16) holds true, one has that 𝑝(𝑔)βˆˆπ‘‰π‘€π‘‚(ℝ2), so π‘”βˆˆπ‘‰π‘€π‘‚(Ξ©).
Consider the function π‘”βˆ—βˆΆπ‘§βˆˆβ„2ξ€·π‘₯βŸΆπ‘(𝑔)0ξ€Έ+π‘‘π‘§βˆˆβ„.(4.17) By PoincarΓ©-Wirtinger inequality and HΓΆlder inequality, one gets 𝐡(π‘₯0,𝑑)||||𝑝(𝑔)(π‘₯)βˆ’π΅(π‘₯0,𝑑)𝑝||||(𝑔)(π‘₯)𝑑π‘₯𝑑π‘₯=πœ‹βˆ’1ξ€œπ΅(0,1)||||π‘”βˆ—(𝑧)βˆ’π΅(0,1)π‘”βˆ—(||||𝑧)𝑑𝑧𝑑𝑧≀𝑐1ξ€œπ΅(0,1)||ξ€·π‘”βˆ—ξ€Έπ‘§(||𝑧)𝑑𝑧=𝑐1π‘‘βˆ’1ξ€œπ΅(π‘₯0,𝑑)||(𝑝(𝑔))π‘₯(||π‘₯)𝑑π‘₯≀𝑐1π‘‘βˆ’1||𝐡π‘₯0ξ€Έ||,𝑑(π‘Ÿβˆ’1/π‘Ÿ)β€–β€–(𝑝(𝑔))π‘₯β€–β€–πΏπ‘Ÿ(𝐡(π‘₯0,𝑑)),(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 Ξ© is an open subset of ℝ𝑛 having the cone property and π‘”βˆˆπ‘€π‘Ÿ,πœ†(Ξ©), with π‘Ÿ>2 and πœ†=0 if 𝑛=2, and π‘Ÿβˆˆ]2,𝑛] and πœ†=π‘›βˆ’π‘Ÿ if 𝑛>2, then π‘’βŸΆπ‘”π‘’(4.19) is a bounded operator from π‘Š1,2(Ξ©) to 𝐿2(Ξ©). Moreover, there exists a constant π‘βˆˆβ„+, such that ‖𝑔𝑒‖𝐿2(Ξ©)β‰€π‘β€–π‘”β€–π‘€π‘Ÿ,πœ†(Ξ©)β€–π‘’β€–π‘Š1,2(Ξ©),(4.20) with 𝑐=𝑐(Ξ©,𝑛,π‘Ÿ).
Furthermore, if ξ‚‹π‘€π‘”βˆˆπ‘Ÿ,πœ†(Ξ©), then for any πœ€>0 there exists a constant π‘πœ€βˆˆβ„+, such that ‖𝑔𝑒‖𝐿2(Ξ©)β‰€πœ€β€–π‘’β€–π‘Š1,2(Ξ©)+π‘πœ€β€–π‘’β€–πΏ2(Ξ©),(4.21) with π‘πœ€=π‘πœ€(πœ€,Ξ©,𝑛,π‘Ÿ,ξ‚πœŽπ‘Ÿ,πœ†[𝑔]).
If π‘”βˆˆπ‘€π‘‘,πœ‡(Ξ©), with 𝑑β‰₯2 and πœ‡>π‘›βˆ’2𝑑, then the operator in (4.19) is bounded from π‘Š2,2(Ξ©) to 𝐿2(Ξ©). Moreover, there exists a constant π‘ξ…žβˆˆβ„+, such that ‖𝑔𝑒‖𝐿2(Ξ©)β‰€π‘ξ…žβ€–π‘”β€–π‘€π‘‘,πœ‡(Ξ©)β€–π‘’β€–π‘Š2,2(Ξ©),(4.22) with π‘ξ…ž=π‘ξ…ž(Ξ©,𝑛,𝑑,πœ‡).
Furthermore, if ξ‚‹π‘€π‘”βˆˆπ‘‘,πœ‡(Ξ©), then for any πœ€>0 there exists a constant π‘ξ…žπœ€βˆˆβ„+, such that ‖𝑔𝑒‖𝐿2(Ξ©)β‰€πœ€β€–π‘’β€–π‘Š2,2(Ξ©)+π‘ξ…žπœ€β€–π‘’β€–πΏ2(Ξ©),(4.23) with π‘ξ…žπœ€=π‘ξ…žπœ€(πœ€,Ξ©,𝑛,𝑑,πœ‡,ξ‚πœŽπ‘‘,πœ‡[𝑔]).

Proof. The proof easily follows from Theorem 3.2 and Corollary 3.3 of [17].

From now on, we assume that Ξ© is an unbounded open subset of ℝ𝑛,𝑛β‰₯2, with the uniform 𝐢1,1-regularity property.

We consider the differential operator 𝐿=βˆ’π‘›ξ“π‘–,𝑗=1π‘Žπ‘–π‘—πœ•2πœ•π‘₯π‘–πœ•π‘₯𝑗+𝑛𝑖=1π‘Žπ‘–πœ•πœ•π‘₯𝑖+π‘Ž,(4.24) with the following conditions on the coefficients: π‘Žπ‘–π‘—=π‘Žπ‘—π‘–βˆˆπΏβˆž(Ξ©),𝑖,𝑗=1,…,𝑛,βˆƒπœˆ>0βˆΆπ‘›ξ“π‘–,𝑗=1π‘Žπ‘–π‘—πœ‰π‘–πœ‰π‘—||πœ‰||β‰₯𝜈2,a.e.inΞ©,βˆ€πœ‰βˆˆπ‘…π‘›,(β„Ž1)ξ€·π‘Žπ‘–π‘—ξ€Έπ‘₯𝑗,π‘Žπ‘–βˆˆπ‘€π‘œπ‘Ÿ,πœ†(Ξ©),𝑖,𝑗=1,…,𝑛,withπ‘Ÿ>2,πœ†=0if𝑛=2,with]]π‘Ÿβˆˆ2,𝑛,πœ†=π‘›βˆ’π‘Ÿif𝑛>2,(β„Ž2)ξ‚‹π‘€π‘Žβˆˆπ‘‘,πœ‡(Ξ©),with𝑑β‰₯2,πœ‡>π‘›βˆ’2𝑑,essinfΞ©π‘Ž=π‘Ž0>0.(β„Ž3) We explicitly observe that under the assumptions β„Ž1β€“β„Ž3 the operator πΏβˆΆπ‘Š2,2(Ξ©)→𝐿2(Ξ©) 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 β„Ž1β€“β„Ž3, there exists a constant π‘βˆˆβ„+ such that β€–π‘’β€–π‘Š2,2(Ξ©)≀𝑐‖𝐿𝑒‖𝐿2(Ξ©),βˆ€π‘’βˆˆπ‘Š2,2(Ξ©)βˆ©βˆ˜π‘Š1,2(Ξ©),(4.25) with 𝑐=𝑐(Ξ©,𝑛,𝜈,π‘Ÿ,𝑑,πœ‡,||π‘Žπ‘–π‘—||𝐿∞(Ξ©),πœŽπ‘œπ‘Ÿ,πœ†[(π‘Žπ‘–π‘—)π‘₯𝑗],πœŽπ‘œπ‘Ÿ,πœ†[π‘Žπ‘–],ξ‚πœŽπ‘‘,πœ‡[π‘Ž],π‘Ž0).

Proof. Let us put 𝐿0=βˆ’π‘›ξ“π‘–,𝑗=1π‘Žπ‘–π‘—πœ•2πœ•π‘₯π‘–πœ•π‘₯𝑗(4.26) and fix π‘’βˆˆπ‘Š2,2(Ξ©)βˆ©βˆ˜π‘Š1,2(Ξ©). 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 𝑐1βˆˆβ„+ such that β€–π‘’β€–π‘Š2,2(Ξ©)≀𝑐1‖‖𝐿0𝑒‖‖𝐿2(Ξ©)+‖𝑒‖𝐿2(Ξ©)ξ€Έ,(4.27) with 𝑐1=𝑐1(Ξ©,𝑛,𝜈,||π‘Žπ‘–π‘—||𝐿∞(Ξ©),πœŽπ‘œπ‘Ÿ,πœ†[(π‘Žπ‘–π‘—)π‘₯𝑗]). Therefore, β€–π‘’β€–π‘Š2,2(Ξ©)≀𝑐1‖𝐿𝑒‖𝐿2(Ξ©)+‖𝑒‖𝐿2(Ξ©)+𝑛𝑖=1β€–β€–π‘Žπ‘–π‘’π‘₯𝑖‖‖𝐿2(Ξ©)+β€–π‘Žπ‘’β€–πΏ2(Ξ©)ξƒͺ.(4.28) On the other hand, from Lemma 4.2, one has β€–β€–π‘Žπ‘–π‘’π‘₯𝑖‖‖𝐿2(Ξ©)β‰€πœ€β€–π‘’β€–π‘Š2,2(Ξ©)+π‘πœ€β€–β€–π‘’π‘₯𝑖‖‖𝐿2(Ξ©),β€–π‘Žπ‘’β€–πΏ2(Ξ©)β‰€πœ€β€–π‘’β€–π‘Š2,2(Ξ©)+π‘ξ…žπœ€β€–π‘’β€–πΏ2(Ξ©),(4.29) with π‘πœ€=π‘πœ€(πœ€,Ξ©,𝑛,π‘Ÿ,πœŽπ‘œπ‘Ÿ,πœ†[π‘Žπ‘–]) and π‘ξ…žπœ€=π‘ξ…žπœ€(πœ€,Ξ©,𝑛,𝑑,πœ‡,ξ‚πœŽπ‘‘,πœ‡[π‘Ž]).
Furthermore, classical interpolation results give that there exists a constant πΎβˆˆβ„+ such that ‖‖𝑒π‘₯‖‖𝐿2(Ξ©)β‰€πΎπœ€β€–π‘’β€–π‘Š2,2(Ξ©)+πΎπœ€β€–π‘’β€–πΏ2(Ξ©),(4.30) with 𝐾=𝐾(Ξ©). Combining (4.28), (4.29) and (4.30) we conclude that there exists 𝑐2βˆˆβ„+ such that β€–π‘’β€–π‘Š2,2(Ξ©)≀𝑐2‖𝐿𝑒‖𝐿2(Ξ©)+‖𝑒‖𝐿2(Ξ©)ξ€Έ,(4.31) with 𝑐2=𝑐2(Ξ©,𝑛,𝜈,π‘Ÿ,𝑑,πœ‡,||π‘Žπ‘–π‘—||𝐿∞(Ξ©),πœŽπ‘œπ‘Ÿ,πœ†[(π‘Žπ‘–π‘—)π‘₯𝑗],πœŽπ‘œπ‘Ÿ,πœ†[π‘Žπ‘–],ξ‚πœŽπ‘‘,πœ‡[π‘Ž]).
To show (4.25), it remains to estimate ‖𝑒‖𝐿2(Ξ©). To this aim let us rewrite our operator in divergence form 𝐿𝑒=βˆ’π‘›ξ“π‘–,𝑗=1ξ€·π‘Žπ‘–π‘—π‘’π‘₯𝑖π‘₯𝑗+𝑛𝑖=1𝑛𝑗=1ξ€·π‘Žπ‘–π‘—ξ€Έπ‘₯𝑗+π‘Žπ‘–ξƒͺ𝑒π‘₯𝑖+π‘Žπ‘’,(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 modificationsβ€”we explicitly observe that the continuity of the bilinear form associated to (4.32) in our case is an immediate consequence of Lemma 4.2β€”we obtain that ‖𝑒‖𝐿2(Ξ©)≀𝑐3‖𝐿𝑒‖𝐿2(Ξ©),(4.33) with 𝑐3=𝑐3(𝑛,𝜈,π‘Ÿ,πœŽπ‘œπ‘Ÿ,πœ†[(π‘Žπ‘–π‘—)π‘₯𝑗],πœŽπ‘œπ‘Ÿ,πœ†[π‘Žπ‘–],π‘Ž0). 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 π‘Šπ‘ 2,2(Ξ©). The π‘Š2,2-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 β„Ž1β€“β„Ž3, there exists a constant π‘βˆˆβ„+ such that β€–π‘’β€–π‘Šπ‘ 2,2(Ξ©)≀𝑐‖𝐿𝑒‖𝐿2𝑠(Ξ©),βˆ€π‘’βˆˆπ‘Šπ‘ 2,2(Ξ©)βˆ©βˆ˜π‘Šπ‘ 1,2(Ξ©),(5.1) with 𝑐=𝑐(Ξ©,𝑛,𝑠,𝜈,π‘Ÿ,𝑑,πœ‡,||π‘Žπ‘–π‘—||𝐿∞(Ξ©),||π‘Žπ‘–||π‘€π‘Ÿ,πœ†(Ξ©),πœŽπ‘œπ‘Ÿ,πœ†[(π‘Žπ‘–π‘—)π‘₯𝑗],πœŽπ‘œπ‘Ÿ,πœ†[π‘Žπ‘–],ξ‚πœŽπ‘‘,πœ‡[π‘Ž],π‘Ž0).

Proof. Fix π‘’βˆˆπ‘Šπ‘ 2,2(Ξ©)βˆ©βˆ˜π‘Šπ‘ 1,2(Ξ©). In the sequel, for sake of simplicity, we will write πœ‚π‘˜=πœ‚, for a fixed π‘˜βˆˆβ„•. Observe that πœ‚ satisfies (2.1), as a consequence of (3.16), so that Lemma 2.5 applies giving that πœ‚π‘ π‘’βˆˆπ‘Š2,2(Ξ©)βˆ©βˆ˜π‘Š1,2(Ξ©). Therefore, in view of Theorem 4.3, there exists 𝑐0βˆˆβ„+, such that β€–πœ‚π‘ π‘’β€–π‘Š2,2(Ξ©)≀𝑐0(πœ‚β€–πΏπ‘ π‘’)‖𝐿2(Ξ©),(5.2) with 𝑐0=𝑐0(Ξ©,𝑛,𝜈,π‘Ÿ,𝑑,πœ‡,||π‘Žπ‘–π‘—||𝐿∞(Ξ©),πœŽπ‘œπ‘Ÿ,πœ†[(π‘Žπ‘–π‘—)π‘₯𝑗],πœŽπ‘œπ‘Ÿ,πœ†[π‘Žπ‘–],ξ‚πœŽπ‘‘,πœ‡[π‘Ž],π‘Ž0). Easy computations give 𝐿(πœ‚π‘ π‘’)=πœ‚π‘ πΏπ‘’βˆ’π‘ π‘›ξ“π‘–,𝑗=1π‘Žπ‘–π‘—ξ‚€(π‘ βˆ’1)πœ‚π‘ βˆ’2πœ‚π‘₯π‘–πœ‚π‘₯𝑗𝑒+πœ‚π‘ βˆ’1πœ‚π‘₯𝑖π‘₯𝑗𝑒+2πœ‚π‘ βˆ’1πœ‚π‘₯𝑖𝑒π‘₯𝑗+𝑠𝑛𝑖=1π‘Žπ‘–πœ‚π‘ βˆ’1πœ‚π‘₯𝑖𝑒.(5.3) Putting together (5.2) and (5.3), we deduce that β€–πœ‚π‘ π‘’β€–π‘Š2,2(Ξ©)≀𝑐1ξƒ©β€–πœ‚π‘ πΏπ‘’β€–πΏ2(Ξ©)+𝑛𝑖,𝑗=1ξ‚€β€–β€–πœ‚π‘ βˆ’2πœ‚π‘₯π‘–πœ‚π‘₯𝑗𝑒‖‖𝐿2(Ξ©)+β€–β€–πœ‚π‘ βˆ’1πœ‚π‘₯𝑖π‘₯𝑗𝑒‖‖𝐿2(Ξ©)+β€–β€–πœ‚π‘ βˆ’1πœ‚π‘₯𝑖𝑒π‘₯𝑗‖‖𝐿2(Ξ©)+𝑛𝑖=1β€–β€–π‘Žπ‘–πœ‚π‘ βˆ’1πœ‚π‘₯𝑖𝑒‖‖𝐿2(Ξ©)ξƒͺ,(5.4) where 𝑐1βˆˆβ„+ depends on the same parameters as 𝑐0 and on 𝑠.
On the other hand, from Lemma 4.2 and (3.17), we get β€–β€–π‘Žπ‘–πœ‚π‘ βˆ’1πœ‚π‘₯𝑖𝑒‖‖𝐿2(Ξ©)≀𝑐2supΞ©β§΅Ξ©π‘˜πœŽπ‘₯πœŽβ€–β€–π‘Žπ‘–β€–β€–π‘€π‘Ÿ,πœ†(Ξ©)β€–πœ‚π‘ π‘’β€–π‘Š1,2(Ξ©),(5.5) with 𝑐2=𝑐2(Ξ©,𝑛,π‘Ÿ).
Combining (3.17), (3.18), (5.4), and (5.5), with simple calculations we obtain the bound β€–πœ‚π‘ π‘’β€–π‘Š2,2(Ξ©)≀𝑐3βŽ‘βŽ’βŽ’βŽ£β€–πœ‚π‘ πΏπ‘’β€–πΏ2(Ξ©)+βŽ›βŽœβŽœβŽsupΞ©β§΅Ξ©π‘˜πœŽ2π‘₯+𝜎𝜎π‘₯π‘₯𝜎2+supΞ©β§΅Ξ©π‘˜πœŽπ‘₯πœŽβŽžβŽŸβŽŸβŽ β€–πœ‚π‘ π‘’β€–π‘Š2,2(Ξ©)⎀βŽ₯βŽ₯⎦,(5.6) where 𝑐3 depends on the same parameters as 𝑐1 and on β€–π‘Žπ‘–β€–π‘€π‘Ÿ,πœ†(Ξ©).
By Lemma 3.1, it follows that there exists π‘˜π‘œβˆˆβ„• such that βŽ›βŽœβŽœβŽsupΞ©β§΅Ξ©π‘˜π‘œπœŽ2π‘₯+𝜎𝜎π‘₯π‘₯𝜎2+supΞ©β§΅Ξ©π‘˜π‘œπœŽπ‘₯πœŽβŽžβŽŸβŽŸβŽ β‰€12𝑐3.(5.7)
Now, if we still denote by πœ‚ the function πœ‚π‘˜π‘œ, from (5.6) and (5.7), we deduce that β€–πœ‚π‘ π‘’β€–π‘Š2,2(Ξ©)≀2𝑐3β€–πœ‚π‘ πΏπ‘’β€–πΏ2(Ξ©).(5.8) Then, by Lemma 2.2 and by (3.12), written for π‘˜=π‘˜π‘œ, we have |𝛼|≀2β€–πœŽπ‘ πœ•π›Όπ‘’β€–πΏ2(Ξ©)≀𝑐4β€–πœŽπ‘ πΏπ‘’β€–πΏ2(Ξ©),(5.9) with 𝑐4 depending on the same parameters as 𝑐3 and on π‘˜π‘œ.
This last estimate being true for every π‘ βˆˆβ„, we also have |𝛼|≀2β€–πœŽβˆ’π‘ πœ•π›Όπ‘’β€–πΏ2(Ξ©)≀𝑐5β€–πœŽβˆ’π‘ πΏπ‘’β€–πΏ2(Ξ©).(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 π‘’βˆˆπ‘Šπ‘ 2,2(Ξ©)βˆ©βˆ˜π‘Šπ‘ 1,2(Ξ©),βˆ’Ξ”π‘’+𝑏𝑒=𝑓,π‘“βˆˆπΏ2𝑠(Ξ©),(5.11) where |||||𝑏=1+βˆ’π‘ (𝑠+1)𝑛𝑖=1𝜎2π‘₯π‘–πœŽ2+𝑠𝑛𝑖=1𝜎π‘₯𝑖π‘₯π‘–πœŽ|||||,(5.12) is uniquely solvable.

Proof. Observe that 𝑒 is a solution of problem (5.11) if and only if 𝑀=πœŽπ‘ π‘’ is a solution of the problem π‘€βˆˆπ‘Š2,2(Ξ©)βˆ©βˆ˜π‘Š1,2(Ξ©),βˆ’Ξ”(πœŽβˆ’π‘ π‘€)+π‘πœŽβˆ’π‘ π‘€=𝑓,π‘“βˆˆπΏ2𝑠(Ξ©).(5.13) Clearly, for any π‘–βˆˆ{1,…,𝑛}, πœ•2πœ•π‘₯2𝑖(πœŽβˆ’π‘ π‘€)=πœŽβˆ’π‘ π‘€π‘₯𝑖π‘₯π‘–βˆ’2π‘ πœŽβˆ’π‘ βˆ’1𝜎π‘₯𝑖𝑀π‘₯𝑖+𝑠(𝑠+1)πœŽβˆ’π‘ βˆ’2𝜎2π‘₯π‘–π‘€βˆ’π‘ πœŽβˆ’π‘ βˆ’1𝜎π‘₯𝑖π‘₯𝑖𝑀,(5.14) then (5.13) is equivalent to the problem π‘€βˆˆπ‘Š2,2(Ξ©)βˆ©βˆ˜π‘Š1,2(Ξ©),βˆ’Ξ”π‘€+π‘›βˆ‘π‘–=1𝛼𝑖𝑀π‘₯𝑖+𝛼𝑀=𝑔,π‘”βˆˆπΏ2(Ξ©),(5.15) where π›Όπ‘–πœŽ=2𝑠π‘₯π‘–πœŽ,𝑖=1,…,𝑛,𝛼=π‘βˆ’π‘ (𝑠+1)𝑛𝑖=1𝜎2π‘₯π‘–πœŽ2+𝑠𝑛𝑖=1𝜎π‘₯𝑖π‘₯π‘–πœŽ,𝑔=πœŽπ‘ π‘“.(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 β„Ž1β€“β„Ž3, the problem π‘’βˆˆπ‘Šπ‘ 2,2(Ξ©)βˆ©βˆ˜π‘Šπ‘ 1,2(Ξ©),𝐿𝑒=𝑓,π‘“βˆˆπΏ2𝑠(Ξ©),(5.17) is uniquely solvable.

Proof. For each 𝜏∈[0,1], we put 𝐿𝜏=𝜏(𝐿)+(1βˆ’πœ)(βˆ’Ξ”+𝑏).(5.18) In view of Theorem 5.1, β€–π‘’β€–π‘Šπ‘ 2,2(Ξ©)β€–β€–πΏβ‰€π‘πœπ‘’β€–β€–πΏπ‘π‘ (Ξ©),βˆ€π‘’βˆˆπ‘Šπ‘ 2,2(Ξ©)βˆ©βˆ˜π‘Šπ‘ 1,2[].(Ξ©),βˆ€πœβˆˆ0,1(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.


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