# W2,2

Authors: Sara Monsurrò; Maria Salvato; Maria Transirico
Journal: International Journal of Differential Equations
 (2011)

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## Abstract

We obtain someW2,2 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.


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## Body

## 1. Introduction

LetΩ be an open subset of ℝn,n≥2. The uniformly elliptic second-order linear differential operator (1.1)L=-∑i,j=1naij∂2∂xi∂xj+∑i=1nai∂∂xi+a,
with leading coefficients aij=aji∈L∞(Ω),i,j=1,…,n, and the associated Dirichlet problem (1.2)u∈W2,2(Ω)∩W∘1,2(Ω),Lu=f,f∈L2(Ω),
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 W2,2 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 (aij)xk∈Ln(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 (aij)xk are in the weak-Ln space, while, in [4], they are supposed to be in an appropriate subspace of the classical Morrey space L2p,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 aij 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(aij)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 W2,2-bound, having the only term ∥Lu∥L2(Ω) in the right-hand side, (1.3)‖u‖W2,2(Ω)≤c‖Lu‖L2(Ω),∀u∈W2,2(Ω)∩W∘1,2(Ω),
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 aij 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.TheW2,2-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, (1.4)u∈Ws2,2(Ω)∩W∘s1,2(Ω),Lu=f,f∈Ls2(Ω),
where s∈ℝ, Ws2,2(Ω),W∘s1,2(Ω), and Ls2(Ω) are weighted Sobolev spaces where the weight ρs is power of a function ρ:Ω̅→ℝ+, of class C2(Ω̅), and such that (1.5)sup⁡x∈Ω|∂αρ(x)|ρ(x)<+∞,∀|α|≤2,lim⁡|x|→+∞(ρ(x)+1ρ(x))=+∞,lim⁡|x|→+∞ρx(x)+ρxx(x)ρ(x)=0,
see Sections 2 and 3 for more details. Also in this weighted case, we obtain the bound (1.6)‖u‖Ws2,2(Ω)≤c‖Lu‖Ls2(Ω),∀u∈Ws2,2(Ω)∩W∘s1,2(Ω),
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

LetΩ be an open subset of ℝn, not necessarily bounded, n≥2. We want to introduce a class of weight functions defined on Ω̅.To this aim, givenk∈ℕ0, we consider a function ρ:Ω̅→ℝ+ such that ρ∈Ck(Ω̅) and (2.1)sup⁡x∈Ω|∂αρ(x)|ρ(x)<+∞,∀|α|≤k.
As an example, we can think of the function (2.2)ρ(x)=(1+|x|2)t,t∈R.
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
(2.3)sup⁡x∈Ω|∂αρs(x)|ρs(x)<+∞∀s∈R,∀|α|≤k.Proof.
The proof is obtained by induction. From (2.1), we get
(2.4)|(ρs)xi|=|sρs-1ρxi|≤c1ρρs-1=c1ρs,i=1,…,n,
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
(2.5)|∂αρs|=|∂β(ρs)xi|=|∂β(sρs-1ρxi)|≤c2∑γ≤β|∂β-γρxi∂γρs-1|≤c3ρρs-1=c3ρs,fori=1,…,n,
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.Fork∈ℕ0, p∈[1,+∞[, and s∈ℝ, and given a weight function ρ satisfying (2.1), we define the space Wsk,p(Ω) of distributions u on Ω such that
(2.6)‖u‖Wsk,p(Ω)=∑|α|≤k‖ρs∂αu‖Lp(Ω)<+∞,
equipped with the norm given in (2.6). Moreover, we denote by W∘sk,p(Ω) the closure of C∘∞(Ω) in Wsk,p(Ω) and put Ws0,p(Ω)=Lsp(Ω).Lemma 2.2.
Letk∈ℕ0,  p∈[1,+∞[, and  s∈ℝ. If assumption (2.1) is satisfied, then there exist two constants  c1,c2∈ℝ+ such that
(2.7)c1‖u‖Wsk,p(Ω)≤‖ρtu‖Ws-tk,p(Ω)≤c2‖u‖Wsk,p(Ω),∀t∈R,∀u∈Wsk,p(Ω),
with c1=c1(t) and c2=c2(t).Proof.
Observe that from (2.3), we have
(2.8)|∂α(ρtu)|≤c1∑β≤α|∂α-βρt∂βu|≤c2|ρt∂βu|,∀|α|≤k,
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(2.9)|ρt∂αu|≤c3∑β≤α|∂β(ρtu)|,∀|α|≤k,
with c3∈ℝ+ depending only on t.
We will prove (2.9) by induction. From (2.3), one has
(2.10)|ρtuxi|=|(ρtu)xi-(ρt)xiu|≤c4((ρtu)x+ρt|u|),
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
(2.11)|ρt∂αu|≤|∂α(ρtu)|+c5∑β<α|∂α-βρt||∂βu|≤|∂α(ρtu)|+c6∑β<α|ρt∂βu|≤c7∑β≤α|∂β(ρtu)|,
with c7∈ℝ+ depending only on t.Let us specify a density result.Lemma 2.3.
Letk∈ℕ0,  p∈[1,+∞[, and  s∈ℝ. If  Ω has the segment property and assumption (2.1) is satisfied, then  𝒟(Ω̅) is dense in  Wsk,p(Ω).Proof.
The proof follows by Lemma 2.2 in [12], since clearly both ρ,ρ-1∈Lloc∞(Ω̅).This allows us to prove the following inclusion.Lemma 2.4.
Letk∈ℕ0,  p∈[1,+∞[, and  s∈ℝ. If  Ω has the segment property and assumption (2.1) is satisfied, then
(2.12)Wk,p(Ω)∩W∘k,p(Ω)⊂W∘sk,p(Ω).Proof.
The density result stated in Lemma2.3 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, ifΩ has the segment property, also Cok(Ω)⊂W∘sk,p(Ω).Lemma 2.5.
Letk∈ℕ0,  p∈[1,+∞[, and  s∈ℝ. If  Ω has the segment property and assumption (2.1) is satisfied, then the map
(2.13)u⟶ρsu
defines a topological isomorphism from Wsk,p(Ω) to Wk,p(Ω) and from W∘sk,p(Ω) to W∘k,p(Ω).Proof.
The first part of the proof easily follows from Lemma2.2 with t=s. Let us show that u∈W∘sk,p(Ω) if and only if ρsu∈W∘k,p(Ω).
Ifu∈W∘sk,p(Ω), there exists a sequence (ϕh)h∈ℕ⊂Co∞(Ω) converging to u in Wsk,p(Ω). Therefore, fixed ε∈ℝ+, there exists h0∈ℕ such that
(2.14)‖ρs(ϕh-u)‖Wk,p(Ω)<ε2,∀h>h0.
Fix h1>h0, clearly ρsϕh1∈W∘k,p(Ω), because of its compact support. Therefore, there exists a sequence (ψn)n∈ℕ⊂Co∞(Ω) converging to ρsϕh1 in Wk,p(Ω). Hence, there exists n0∈ℕ such that
(2.15)‖ψn-ρsϕh1‖Wk,p(Ω)<ε2,∀n>n0.
Putting together (2.14) and (2.15), we get
(2.16)‖ψn-ρsu‖Wk,p(Ω)≤‖ψn-ρsϕh1‖Wk,p(Ω)+‖ρsϕh1-ρsu‖Wk,p(Ω)<ε,
foralln>n0. Thus, ρsu∈W∘k,p(Ω).
Vice versa, if we assume thatρsu∈W∘k,p(Ω), we find a sequence (ϕh)h∈ℕ⊂Co∞(Ω) converging to ρsu in Wk,p(Ω). Hence, there exists h0∈ℕ such that
(2.17)‖ρ-sϕh-u‖Wsk,p(Ω)<ε2,∀h>h0.
Fix h1>h0, since ρ-sϕh1∈Cok(Ω), which is contained in W∘sk,p(Ω) by Lemma 2.4, there exists a sequence (ψn)n∈ℕ⊂Co∞(Ω) converging to ρ-sϕh1 in Wsk,p(Ω). Therefore, there exists n0∈ℕ such that
(2.18)‖ψn-ρ-sϕh1‖Wsk,p(Ω)<ε2,∀n>n0.
From (2.17) and (2.18), we get
(2.19)‖ψn-u‖Wsk,p(Ω)≤‖ψn-ρ-sϕh1‖Wsk,p(Ω)+‖ρ-sϕh1-u‖Wsk,p(Ω)<ε,
foralln>n0, so that u∈W∘sk,p(Ω).


## 3. Preliminary Results

From now on, we consider a weightρ:Ω̅→ℝ+, ρ∈C2(Ω̅), and such that (2.1) is satisfied (for k=2). Moreover, we assume that (3.1)lim⁡|x|→+∞(ρ(x)+1ρ(x))=+∞,lim⁡|x|→+∞ρx(x)+ρxx(x)ρ(x)=0.
An example of a function verifying our hypotheses is given by (3.2)ρ(x)=(1+|x|2)t,t∈R∖{0}.
We associate to ρ a function σ defined by (3.3)σ=ρifρ⟶+∞,for|x|⟶+∞,σ=1ρifρ⟶0,for|x|⟶+∞.
Clearly σ verifies (2.1) and (3.4)lim⁡|x|→+∞σ(x)=+∞,lim⁡|x|→+∞σx(x)+σxx(x)σ(x)=0.Now, let us fix a cutoff functionf∈C∘∞(ℝ̅+) such that (3.5)0≤f≤1,f(t)=1ift∈[0,1],f(t)=0ift∈[2,+∞[.
Then, set (3.6)ζk:x∈Ω̅⟶f(σ(x)k),k∈N,Ωk={x∈Ω:σ(x)<k},k∈N.
By our definition, it follows that ζk∈C∘∞(Ω¯) and (3.7)0≤ζk≤1,ζk=1on⁡Ωk¯,ζk=0on⁡Ω∖Ω2k¯,k∈N.Finally, we introduce the sequence(3.8)ηk:x∈Ω¯⟶2kζk(x)+(1-ζk(x))σ(x),k∈N.For anyk∈ℕ, one has (3.9)ηk=ζk(2k-σ)+σ≥σ,inΩ2k¯,(3.10)ηk≤2k+σ≤(2kinf⁡Ω2k¯⁡σ+1)σ=(ck+1)σ,inΩ2k¯,(3.11)ηk=σ,inΩ∖Ω2k¯,
where ck∈ℝ+ depends only on k. This entails that (3.12)σ∼ηk,∀k∈N.
Concerning the derivatives, easy calculations give that, for any k∈ℕ, (3.13)(ηk)x=(ηk)xx=0,inΩk¯,(3.14)(ηk)x=σx,(ηk)xx=σxx,inΩ∖Ω2k¯,(3.15)(ηk)x≤c1σx,(ηk)xx≤c2(σx2σ+σxx),inΩ∖Ω2k¯,
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, (3.16)(ηk)xηk≤c1′σxσ,inΩ¯,(ηk)xxηk≤c2′σx2+σσxxσ2,inΩ¯,
where c1′ and c2′ are positive constants independent of x and k.Combining (3.13) and (3.16), we have also, for any k∈ℕ, (3.17)(ηk)xηk≤c1′sup⁡Ω∖Ωk¯σxσ,inΩ¯,(3.18)(ηk)xxηk≤c2′sup⁡Ω∖Ωk¯σx2+σσxxσ2,inΩ¯.We conclude this section proving the following lemma.Lemma 3.1.
Letσ and  Ωk,k∈ℕ, be defined by (3.3) and (3.6), respectively. Then
(3.19)lim⁡k→+∞sup⁡Ω∖Ωk¯σx(x)+σxx(x)σ(x)=0.Proof.
Set(3.20)φ(x)=σx(x)+σxx(x)σ(x),x∈Ω̅,ψk=sup⁡Ω∖Ωk¯φ,k∈N.
By the second relation in (3.4), the supremum of φ over Ω∖Ωk¯ is actually a maximum; thus, for every k∈ℕ, there exists xk∈Ω∖Ωk¯ such that ψk=φ(xk).
To prove (3.19), we have to show that limk→+∞ψk=0.
We proceed by contradiction. Hence, let us assume that there existsε0>0 such that, for any k∈ℕ, there exists nk>k such that ψnk=φ(xnk)≥ε0.
If the sequence(xnk)k∈ℕ is bounded, there exists a subsequence (x′nk)k∈ℕ converging to a limit x∈Ω̅, and by the continuity of σ, (σ(x′nk))k∈ℕ converges to σ(x). On the other hand, x′nk∈Ω∖Ωk¯, thus σ(x′nk)≥nk, which is in contrast with the fact that (σ(x′nk))k∈ℕ is a convergent sequence.
Therefore,(xnk)k∈ℕ is unbounded, so that there exists a subsequence (xnk′′)k∈ℕ such that limk→+∞|xnk′′|=+∞. Thus, by the second relation in (3.4), one has limk→+∞φ(xnk′′)=0. This gives the contradiction since φ(xnk′′)≥ε0.


## 4. A No Weighted A Priori Bound

We want to prove aW2,2 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 subsetG 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 χE 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.Forn≥2, λ∈[0,n[, p∈[1,+∞[, and fixed t in ℝ+, the space of Morrey type Mp,λ(Ω,t) is the set of all functions g in Llocp(Ω¯) such that (4.1)‖g‖Mp,λ(Ω,t)=sup⁡x∈Ωτ∈]0,t]τ-λ/p‖g‖Lp(Ω(x,τ))<+∞,
endowed with the norm defined in (4.1). It is easily seen that, for any t1,t2∈ℝ+, a function g belongs to Mp,λ(Ω,t1) if and only if it belongs to Mp,λ(Ω,t2); moreover, the norms of g in these two spaces are equivalent. This allows us to restrict our attention to the space Mp,λ(Ω)=Mp,λ(Ω,1).We now introduce three subspaces ofMp,λ(Ω) needed in the sequel. The set VMp,λ(Ω) is made up of the functions g∈Mp,λ(Ω) such that (4.2)lim⁡t→0‖g‖Mp,λ(Ω,t)=0,
while M̃p,λ(Ω) and M∘p,λ(Ω) denote the closures of L∞(Ω) and C∘∞(Ω) in Mp,λ(Ω), respectively. We point out that (4.3)M∘p,λ(Ω)⊂M̃p,λ(Ω)⊂VMp,λ(Ω).
We put Mp(Ω)=Mp,0(Ω), VMp(Ω)=VMp,0(Ω), M̃p(Ω)=M̃p,0(Ω), and M∘p(Ω)=M∘p,0(Ω).We want to define the moduli of continuity of functions belonging toM̃p,λ(Ω) or M∘p,λ(Ω). To this aim, let us put, for h∈ℝ+ and g∈Mp,λ(Ω), (4.4)F[g](h)=sup⁡sup⁡x∈Ω|E(x,1)|≤1/hE∈Σ(Ω)‖gχE‖Mp,λ(Ω).
Recall first that for a function g∈Mp,λ(Ω) the following characterization holds: (4.5)g∈M̃p,λ(Ω)⟺lim⁡h→+∞F[g](h)=0,
while (4.6)g∈M∘p,λ(Ω)⟺lim⁡h→+∞(F[g](h)+‖(1-ζh)g‖Mp,λ(Ω))=0,
where ζh denotes a function of class Co∞(Rn) such that (4.7)0≤ζh≤1,ζh|B(0,h)¯=1,supp⁡ζh⊂B(0,2h).
Thus, if g is a function in M̃p,λ(Ω), a modulus of continuity of g in M̃p,λ(Ω) is a map σ̃p,λ[g]:ℝ+→ℝ+ such that (4.8)F[g](h)≤σ̃p,λ[g](h),lim⁡h→+∞σ̃p,λ[g](h)=0.
While, if g belongs to Mop,λ(Ω), a modulus of continuity of g in Mop,λ(Ω) is an application σop,λ[g]:ℝ+→ℝ+ such that (4.9)F[g](h)+‖(1-ζh)g‖Mp,λ(Ω)≤σop,λ[g](h),lim⁡h→+∞σop,λ[g](h)=0.
If Ω has the property (4.10)|Ω(x,r)|≥Arn∀x∈Ω,∀r∈]0,1],
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∈Lloc1(Ω¯) such that (4.11)[g]BMO(Ω,τ)=sup⁡r∈]0,τ]x∈Ω⨍Ω(x,r)|g-⨍Ω(x,r)gdy|dy<+∞,
where (4.12)⨍Ω(x,r)gdy=|Ω(x,r)|-1∫Ω(x,r)gdy.
If g∈BMO(Ω)=BMO(Ω,τA), where (4.13)τA=sup⁡{τ∈R+:sup⁡r∈]0,τ]x∈Ωrn|Ω(x,r)|≤1A},
we say that g∈VMO(Ω) if [g]BMO(Ω,τ)→0 for τ→0+.Ifg belongs to VMO(Ω), a modulus of continuity of g in VMO(Ω) is function η[g]:]0,1]→ℝ+ such that (4.14)[g]BMO(Ω,τ)≤η[g](τ)∀τ∈]0,1],lim⁡τ→0+η[g](τ)=0.
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.
IfΩ has the uniform C1,1-regularity property and
(4.15)g,gx∈{VMr(Ω),r>2,forn=2,VMr,n-r(Ω),r∈]2,n],forn>2,
then g∈VMO(Ω).Proof.
Forn>2, the result can be found in [16], combining Lemma 4.1 and the argument in the proof of Lemma 4.2.
Concerningn=2, we firstly apply a known extension result, see [9, Corollary 2.2], stating that any function g such that g,gx∈VMr(Ω) admits an extension p(g) such that p(g),(p(g))x∈VMr(ℝ2).
Then, we prove that for allx0∈ℝ2 and t∈ℝ+, there exists a constant c∈ℝ+ such that
(4.16)⨍B(x0,t)|p(g)-⨍B(x0,t)p(g)dx|dx≤c(t(r-2/r)‖(p(g))x‖Lr(B(x0,t))).
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(4.17)g*:z∈R2⟶p(g)(x0+tz)∈R.
By Poincaré-Wirtinger inequality and Hölder inequality, one gets
(4.18)⨍B(x0,t)|p(g)(x)-⨍B(x0,t)p(g)(x)dx|dx=π-1∫B(0,1)|g*(z)-⨍B(0,1)g*(z)dz|dz≤c1∫B(0,1)|(g*)z(z)|dz=c1t-1∫B(x0,t)|(p(g))x(x)|dx≤c1t-1|B(x0,t)|(r-1/r)‖(p(g))x‖Lr(B(x0,t)),
this gives (4.16).For reader’s convenience, we recall here some results proved in [17], adapted to our needs.Lemma 4.2.
IfΩ is an open subset of  ℝn having the cone property and  g∈Mr,λ(Ω), with  r>2 and  λ=0 if  n=2, and  r∈]2,n] and  λ=n-r if  n>2, then
(4.19)u⟶gu
is a bounded operator from W1,2(Ω) to L2(Ω). Moreover, there exists a constant c∈ℝ+, such that
(4.20)‖gu‖L2(Ω)≤c‖g‖Mr,λ(Ω)‖u‖W1,2(Ω),
with c=c(Ω,n,r).
Furthermore, ifg∈M̃r,λ(Ω), then for any ε>0 there exists a constant cε∈ℝ+, such that
(4.21)‖gu‖L2(Ω)≤ε‖u‖W1,2(Ω)+cε‖u‖L2(Ω),
with cε=cε(ε,Ω,n,r,σ̃r,λ[g]).
Ifg∈Mt,μ(Ω), with t≥2 and μ>n-2t, then the operator in (4.19) is bounded from W2,2(Ω) to L2(Ω). Moreover, there exists a constant c′∈ℝ+, such that
(4.22)‖gu‖L2(Ω)≤c′‖g‖Mt,μ(Ω)‖u‖W2,2(Ω),
with c′=c′(Ω,n,t,μ).
Furthermore, ifg∈M̃t,μ(Ω), then for any ε>0 there exists a constant cε′∈ℝ+, such that
(4.23)‖gu‖L2(Ω)≤ε‖u‖W2,2(Ω)+cε′‖u‖L2(Ω),
with cε′=cε′(ε,Ω,n,t,μ,σ̃t,μ[g]).Proof.
The proof easily follows from Theorem 3.2 and Corollary 3.3 of [17].From now on, we assume thatΩ is an unbounded open subset of ℝn,n≥2, with the uniform C1,1-regularity property.We consider the differential operator(4.24)L=-∑i,j=1naij∂2∂xi∂xj+∑i=1nai∂∂xi+a,
with the following conditions on the coefficients: (h1)aij=aji∈L∞(Ω),i,j=1,…,n,∃ν>0:∑i,j=1naijξiξj≥ν|ξ|2,a.e.inΩ,∀ξ∈Rn,(h2)(aij)xj,ai∈Mor,λ(Ω),i,j=1,…,n,withr>2,λ=0ifn=2,withr∈]2,n],λ=n-rifn>2,(h3)a∈M̃t,μ(Ω),witht≥2,μ>n-2t,essinf⁡Ωa=a0>0.
We explicitly observe that under the assumptions h1–h3 the operator L:W2,2(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.
LetL be defined in (4.24). Under hypotheses h1–h3, there exists a constant c∈ℝ+ such that
(4.25)‖u‖W2,2(Ω)≤c‖Lu‖L2(Ω),∀u∈W2,2(Ω)∩W∘1,2(Ω),
with c=c(Ω,n,ν,r,t,μ,||aij||L∞(Ω),σor,λ[(aij)xj],σor,λ[ai],σ̃t,μ[a],a0).Proof.
Let us put(4.26)L0=-∑i,j=1naij∂2∂xi∂xj
and fix u∈W2,2(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
(4.27)‖u‖W2,2(Ω)≤c1(‖L0u‖L2(Ω)+‖u‖L2(Ω)),
with c1=c1(Ω,n,ν,||aij||L∞(Ω),σor,λ[(aij)xj]). Therefore,
(4.28)‖u‖W2,2(Ω)≤c1(‖Lu‖L2(Ω)+‖u‖L2(Ω)+∑i=1n‖aiuxi‖L2(Ω)+‖au‖L2(Ω)).
On the other hand, from Lemma 4.2, one has
(4.29)‖aiuxi‖L2(Ω)≤ε‖u‖W2,2(Ω)+cε‖uxi‖L2(Ω),‖au‖L2(Ω)≤ε‖u‖W2,2(Ω)+cε′‖u‖L2(Ω),
with cε=cε(ε,Ω,n,r,σor,λ[ai]) and cε′=cε′(ε,Ω,n,t,μ,σ̃t,μ[a]).
Furthermore, classical interpolation results give that there exists a constantK∈ℝ+ such that
(4.30)‖ux‖L2(Ω)≤Kε‖u‖W2,2(Ω)+Kε‖u‖L2(Ω),
with K=K(Ω). Combining (4.28), (4.29) and (4.30) we conclude that there exists c2∈ℝ+ such that
(4.31)‖u‖W2,2(Ω)≤c2(‖Lu‖L2(Ω)+‖u‖L2(Ω)),
with c2=c2(Ω,n,ν,r,t,μ,||aij||L∞(Ω),σor,λ[(aij)xj],σor,λ[ai],σ̃t,μ[a]).
To show (4.25), it remains to estimate ∥u∥L2(Ω). To this aim let us rewrite our operator in divergence form
(4.32)Lu=-∑i,j=1n(aijuxi)xj+∑i=1n(∑j=1n(aij)xj+ai)uxi+au,
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
(4.33)‖u‖L2(Ω)≤c3‖Lu‖L2(Ω),
with c3=c3(n,ν,r,σor,λ[(aij)xj],σor,λ[ai],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 spaceWs2,2(Ω). The W2,2-bound obtained in Theorem 4.3 allows us to show the following a priori estimate in the weighted case.Theorem 5.1.
LetL be defined in  (4.24). Under hypotheses  h1–h3, there exists a constant  c∈ℝ+ such that
(5.1)‖u‖Ws2,2(Ω)≤c‖Lu‖Ls2(Ω),∀u∈Ws2,2(Ω)∩W∘s1,2(Ω),
with c=c(Ω,n,s,ν,r,t,μ,||aij||L∞(Ω),||ai||Mr,λ(Ω),σor,λ[(aij)xj],σor,λ[ai],σ̃t,μ[a],a0).Proof.
Fixu∈Ws2,2(Ω)∩W∘s1,2(Ω). In the sequel, for sake of simplicity, we will write ηk=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 ηsu∈W2,2(Ω)∩W∘1,2(Ω). Therefore, in view of Theorem 4.3, there exists c0∈ℝ+, such that
(5.2)‖ηsu‖W2,2(Ω)≤c0‖L(ηsu)‖L2(Ω),
with c0=c0(Ω,n,ν,r,t,μ,||aij||L∞(Ω),σor,λ[(aij)xj],σor,λ[ai],σ̃t,μ[a],a0). Easy computations give
(5.3)L(ηsu)=ηsLu-s∑i,j=1naij((s-1)ηs-2ηxiηxju+ηs-1ηxixju+2ηs-1ηxiuxj)+s∑i=1naiηs-1ηxiu.
Putting together (5.2) and (5.3), we deduce that
(5.4)‖ηsu‖W2,2(Ω)≤c1(‖ηsLu‖L2(Ω)+∑i,j=1n(‖ηs-2ηxiηxju‖L2(Ω)+‖ηs-1ηxixju‖L2(Ω)+‖ηs-1ηxiuxj‖L2(Ω))+∑i=1n‖aiηs-1ηxiu‖L2(Ω)∑i,j=1n(‖ηs-2ηxiηxju‖L2(Ω)+‖ηs-1ηxixju‖L2(Ω)),
where c1∈ℝ+ depends on the same parameters as c0 and on s.
On the other hand, from Lemma4.2 and (3.17), we get
(5.5)‖aiηs-1ηxiu‖L2(Ω)≤c2sup⁡Ω∖Ωk¯σxσ‖ai‖Mr,λ(Ω)‖ηsu‖W1,2(Ω),
with c2=c2(Ω,n,r).
Combining (3.17), (3.18), (5.4), and (5.5), with simple calculations we obtain the bound
(5.6)‖ηsu‖W2,2(Ω)≤c3[‖ηsLu‖L2(Ω)+(sup⁡Ω∖Ωk¯σx2+σσxxσ2+sup⁡Ω∖Ωk¯σxσ)‖ηsu‖W2,2(Ω)],
where c3 depends on the same parameters as c1 and on ∥ai∥Mr,λ(Ω).
By Lemma3.1, it follows that there exists ko∈ℕ such that
(5.7)(sup⁡Ω∖Ωko¯σx2+σσxxσ2+sup⁡Ω∖Ωko¯σxσ)≤12c3.
Now, if we still denote byη the function ηko, from (5.6) and (5.7), we deduce that
(5.8)‖ηsu‖W2,2(Ω)≤2c3‖ηsLu‖L2(Ω).
Then, by Lemma 2.2 and by (3.12), written for k=ko, we have
(5.9)∑|α|≤2‖σs∂αu‖L2(Ω)≤c4‖σsLu‖L2(Ω),
with c4 depending on the same parameters as c3 and on ko.
This last estimate being true for everys∈ℝ, we also have
(5.10)∑|α|≤2‖σ-s∂αu‖L2(Ω)≤c5‖σ-sLu‖L2(Ω).
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(5.11)u∈Ws2,2(Ω)∩W∘s1,2(Ω),-Δu+bu=f,f∈Ls2(Ω),
where
(5.12)b=1+|-s(s+1)∑i=1nσxi2σ2+s∑i=1nσxixiσ|,
is uniquely solvable.Proof.
Observe thatu is a solution of problem (5.11) if and only if w=σsu is a solution of the problem
(5.13)w∈W2,2(Ω)∩W∘1,2(Ω),-Δ(σ-sw)+bσ-sw=f,f∈Ls2(Ω).
Clearly, for any i∈{1,…,n},
(5.14)∂2∂xi2(σ-sw)=σ-swxixi-2sσ-s-1σxiwxi+s(s+1)σ-s-2σxi2w-sσ-s-1σxixiw,
then (5.13) is equivalent to the problem
(5.15)w∈W2,2(Ω)∩W∘1,2(Ω),-Δw+∑i=1nαiwxi+αw=g,g∈L2(Ω),
where
(5.16)αi=2sσxiσ,i=1,…,n,α=b-s(s+1)∑i=1nσxi2σ2+s∑i=1nσxixiσ,g=σsf.
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.
LetL be defined in  (4.24). Under hypotheses  h1–h3, the problem
(5.17)u∈Ws2,2(Ω)∩W∘s1,2(Ω),Lu=f,f∈Ls2(Ω),
is uniquely solvable.Proof.
For eachτ∈[0,1], we put
(5.18)Lτ=τ(L)+(1-τ)(-Δ+b).
In view of Theorem 5.1,
(5.19)‖u‖Ws2,2(Ω)≤c‖Lτu‖Lsp(Ω),∀u∈Ws2,2(Ω)∩W∘s1,2(Ω),∀τ∈[0,1].
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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*Source: 572824-2011-10-11.xml*