# W

Authors: Loredana Caso; Roberta D’Ambrosio; Maria Transirico
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

We give an overview on some results concerning the unique solvability of the Dirichlet problem inW2,p, p>1, for second-order linear elliptic partial differential equations in nondivergence form and with singular data in weighted Sobolev spaces. We also extend such results to the planar case.


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## Body

## 1. Introduction

LetΩ be an open subset of Rn, n≥2. Consider in Ω the Dirichlet problem
(1)u∈W2,pΩ∩W∘1,pΩ,iiiiLu=f,f∈LpΩ,
where p∈]1,+∞[ and L is the second-order linear elliptic differential operator in nondivergence form defined by
(2)L=-∑i,j=1n‍aij∂2∂xi∂xj+∑i=1n‍ai∂∂xi+a.Ifn≥3 and Ω is a bounded set with a suitable regularity property, the well-posedness of the Dirichlet problem (1) has been largely studied by several authors under various hypotheses of discontinuity on the coefficients. It must be mentioned the classical contribution by Miranda [1], where the author assumed that the aij belong to W1,n(Ω) and considered the case p=2. This result was later on generalized in [2, 3] by considering the coefficients aij belonging to the wider class of spaces VMO and p∈]1,+∞[.In the framework of open sets, nonnecessarily bounded, whose boundary has various singularities, for example, corners or edges, in accordance with the linear theory, it is natural to assume that the lower order coefficients and the right-hand side of the equation of problem (1) belong to some weighted Sobolev spaces, where the weight is usually a power of the distance function from the “singular set” of the boundary of domain. In these cases, near to the “singular set,” the solution u of the boundary value problem may have a singularity which can be often characterized by a weight of mentioned type. For instance, if ρ is a bounded weight function related to the distance function from nonempty subset Sρ of the boundary of an arbitrary domain Ω, not necessarily bounded and regular (see Section 2 for the definition of such weight function), a problem similar to (1) has been studied, in the weighted case, by several authors under suitable hypotheses on the weight function ρ and when the coefficients of lower order terms are singular near to Sρ. This kind of Dirichlet problem has been dealt, for example, in [4–6] under hypotheses as those in [2, 3] with p∈]1,+∞[. To be more precise, in [4–6], the authors consider the problem
(3)u∈Ws2,pΩ∩W∘s-11,pΩ,iiiiLu=f,f∈LspΩ,
where n≥3, Ws2,p(Ω),W∘s-11,p(Ω) and Lsp(Ω)(p∈1,+∞,s∈R) are some weighted Sobolev spaces whose weight functions are suitable powers of ρ. The existence and uniqueness of problem (3) have been firstly proved in [4, 5] when p≥2. Such results have been later on used in [6] to get the unique solvability also for 1<p<2. We point out that in this last case we needed to employ some variational results (see [6] and its bibliography).The aim of this work is to give an overview on the above-mentioned results concerning the solvability of (3) when p∈]1,+∞[ as well as to extend such results to the planar case. Here we note also that, for p=2, if n≥3 the required summability on (aij)xh can be equal to n while if n=2 we have to take a summability greater than n (see hypothesis h2).At last, we observe that the existence and uniqueness result for the problem (3) are based on the unique solvability of the problem (3) for p=2 and of a problem similar to (3) whose associated operator Ls-1 differs from that in (2) for a compact operator (see Section 4).For further results concerning elliptic boundary value problems similar to (3), involving different classes of weighted Sobolev spaces, we refer the reader also to [7, 8].


## 2. Notation and Function Spaces

In this section we recall the definitions and the main properties of the class of weights we are interested in and of certain classes of function spaces where the coefficients of our operator belong. Thus, from now on, letΩ be an unbounded open subset of Rn,  n≥2. By ΣF we denote the σ-algebra of all Lebesgue measurable subsets F of Ω. For E∈ΣF, χE is its characteristic function, E is its Lebesgue measure and Ex,r=E∩Bx,rx∈Rn,r∈R+, where Bx,r is the open ball with center in x and radius r. The class of restrictions to E of functions ζ∈C0∞Rn (resp., ζ∈C00Rn) with E-∩supp⁡ζ⊆E is DE (resp., D0E). For p∈1,+∞, Lloc⁡pE is the class of all functions g, defined on E, such that ζg∈LpE for all ζ∈DE.We denote byA(Ω) the class of all measurable weight functions ρ:Ω→R+ such that
(4)γ-1ρ(y)≤ρ(x)≤γρ(y)∀y∈Ω,∀x∈Ω(y,ρ(y)),
where γ∈R+ is independent on x and y. Given ρ∈A(Ω), we put
(5)Sρ={z∈∂Ω∣lim⁡x→zρ(x)=0}.It is known that(6)ρ∈Lloc⁡∞(Ω¯),ρ-1∈Lloc⁡∞(Ω¯∖Sρ),
and if Sρ≠∅,
(7)ρ(x)≤dist⁡(x,Sρ)∀x∈Ω,
(see [9, 10] for further details).Fork∈N0,1≤p≤+∞,s∈R and ρ∈A(Ω), the related weighted Sobolev space Wsk,p(Ω) is made up of all the distributions u on Ω such that ρs+|α|-k∂αu∈Lp(Ω) for |α|≤k. We observe that Wsk,p(Ω) is a Banach space with the norm defined by
(8)uWsk,pΩ=∑α≤k‍ρs+α-k∂αuLp(Ω).Moreover, it is separable if1≤p<+∞, reflexive if 1<p<+∞, and, in particular, Wsk,2(Ω) is an Hilbert space. We also denote by W∘sk,p(Ω) the closure of C0∞(Ω) in Wsk,p(Ω) and we put Ws0,p(Ω)=Lsp(Ω).Clearly the following embeddings hold:(9)W∘sk,pΩ↪Wsk,p(Ω)↪Ls-kp(Ω).A more detailed account of the properties of the above-defined weighted Sobolev spaces can be found in [11–13].Forp∈[1,+∞[,s∈R and ρ∈A(Ω), let Ksp(Ω) be the set of all the functions g∈Lloc⁡p(Ω-∖Sρ) such that
(10)gKsp(Ω)=sup⁡Ωρs-n/pxgLpΩx,ρx<+∞,
equipped with the norm defined in (10). Obviously, the space Ksp(Ω) is a Banach space containing Ls∞(Ω) and C0∞(Ω) as well (see [10]). Therefore, we denote by K~sp(Ω) (resp., K∘sp(Ω)) the closure of Ls∞(Ω) (resp., C0∞(Ω)) in Ksp(Ω).Now, let us define the moduli of continuity of functions belonging toK~sp(Ω) or K∘sp(Ω).Given a functiong in Ksp(Ω), the following characterization holds:
(11)g∈K~spΩ⟺lim⁡h→+∞sup⁡E∈ΣΩsup⁡x∈Ω⁡Ωx,ρx∩E/ρn(x)≤1/hgχEKspΩ=0,
(see [10]).Thus, ifg is a function in K~sp(Ω), amodulus of continuity of g in K~sp(Ω) is a map ω~sp[g]:R+→R+ such that
(12)sup⁡E∈ΣΩsup⁡x∈Ω⁡Ωx,ρx∩E/ρnx≤1/h⁡gχEKsp(Ω)≤ω~sp[g](h),lim⁡h→+∞ω~sp[g](h)=0.Next, we introduce a class of mappings needed to define amodulus of continuity of g in K∘sp(Ω). Fix f in D(R-+) satisfying the conditions
(13)0≤f≤1,ft=1ift≤12,f(t)=0ift≥1,
and α∈C∞(Ω)∩C0,1(Ω-) equivalent to dist (·,∂Ω) (for more details on the existence of such an α see, for instance, Theorem  2, Chapter VI in [14] and Lemma  3.6.1 in [15]). Hence, for k∈N we define the functions
(14)ψk:x∈Ω-⟶1-fkαxfx2k.It is easy to prove that eachψk belongs to D(Ω¯∖Sρ) and
(15)0≤ψk≤1,ψk∣Ω-k=1,supp⁡ψk⊂Ω-2k,
where Ωk={x∈Ω∣|x|<k,α(x)>1/k}.Giveng in Ksp(Ω), it is known (see [10]) that
(16)g∈K∘sp(Ω)⟺lim⁡h→+∞1-ψhgKsp(Ω)=0.Thus, amodulus of continuity of g in K∘sp(Ω) is a map ω∘sp[g]:N→N such that
(17)1-ψhgKsp(Ω)+sup⁡E∈ΣΩsup⁡x∈Ω⁡Ωx,ρx∩E≤1/hψhgχEKspΩ≤ω∘spgh,lim⁡h→+∞ω∘sp[g](h)=0.Further properties of the spacesKsp(Ω), K~sp(Ω), and K∘sp(Ω) can be found in [10].In the end, we recall the definitions of two other classes of function spaces where the leading coefficients of our operator belong.IfΩ has the property
(18)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,t) (t∈R+) of functions g∈Lloc⁡1(Ω-) such that
(19)gBMO(Ω,t)=sup⁡x∈Ωr∈0,t⨍Ω(x,r)g-⨍Ω(x,r)g<+∞,
where
(20)⨍Ω(x,r)g=Ωx,r-1⨍Ω(x,r)g.Ifg∈BMO(Ω)=BMO(Ω,tA), where
(21)tA=sup⁡t∈R+sup⁡x∈Ωr∈0,trnΩx,r≤1A,
we say that g∈VMO(Ω) if gBMO(Ω,t)→0 for t→0+. A function η[g]:R+→R+ is called amodulus of continuity of g in VMO(Ω) if
(22)gBMO(Ω,t)≤η[g](t)∀t∈R+,lim⁡t→0+η[g](t)=0.A more detailed account of properties of the above-defined spacesBMO(Ω,t) and VMO(Ω) can be found in [16] or in some general reference books, for example, [17].


## 3. Preliminary Results

In this section we recall some embedding and compactness estimates for a multiplication operator as well as a regularity result which will be also extended to the planar case.Let us assume thatΩ has the segment property (for all definitions of regularity properties of open subsets of Rn we will refer to [18]) and fix ρ∈A(Ω)∩L∞(Ω) such that Sρ≠∅.For our purposes, we suppose that the following condition onΩ holds.(h0) There exists an open subset Ωo of Rn with the uniform C1,1-regularity property such that
(23)Ω⊂Ωo,∂Ω∖Sρ⊂∂Ωo.We note that since the required segment property gives thatΩ lies on one side of the “singular” part Sρ of its boundary the hypothesis (ho), roughly speaking, means that it is possible to “widen” suitably Ω on the other side of Sρ.Remark 1.
We observe that, as a result of condition(h0), there exists θ∈]0,π/2[  such that
(24)∀x∈Ω∃Cθx:Cθ(x,ρ(x))¯⊂Ω,
where Cθ(x) is an open infinite cone with vertex in x and opening θ and Cθ(x,ρ(x))=Cθ(x)∩B(x,ρ(x)) (see Remark  5.1 in [10]). As a consequence, there exists a function σ∈A(Ω)∩C∞(Ω)∩C0,1(Ω-) which is equivalent to ρ and such that
(25)∂ασx≤cασ1-αx∀x∈Ω,∀α∈N0n,
where cα∈R+ is independent of x (see [9]).Let us start collecting two results of [10] which provide the boundedness and compactness of the multiplication operator
(26)u⟶gu,
where the function g belongs to suitable spaces Ksp(Ω).Theorem 2.
Letr,s,p,q be numbers such that
(27)r∈N,s∈R,1≤p≤q<+∞,q≥nr,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiq>nrifnr=p>1.
If condition(h0) holds, then for all u∈Wsr,p(Ω) and for any g∈K-s+rq(Ω) we have gu∈Lp(Ω) and
(28)guLp(Ω)≤cgK-s+rqΩuWsr,p(Ω),
where the constant c∈R+ is independent of g and u.
Furthermore, ifg∈K∘-s+rq(Ω) then for any ϵ∈R+ there exist a constant c(ϵ)∈R+ and a bounded open set Ωϵ⊂⊂Ω with the cone property such that
(29)guLpΩ≤ϵuWsr,pΩ+cϵuLpΩϵ∀u∈Wsr,p(Ω),
and we have that the multiplication operator
(30)u∈Wsr,p(Ω)⟶gu∈Lp(Ω)
is compact.We go on collecting a density result, an a priori estimate, and a regularity result, which will be some of the crucial analytic tools of our main results.Letp∈]1,+∞[, s∈R. At first we recall a density result (see Lemma  3.2 in [6]).Lemma 3.
IfΩ verifies condition (h0), then for every u∈Ws2,p(Ω)∩W∘s-11,p(Ω) there exists a sequence (uk)k∈N such that
(31)uk∈Ws2,pΩ∩W∘s-11,2Ω∩D0Ω¯∖Sρ,iiiiiiiiiiiiiiiiiiiiiiiiiuk⟶uinWs2,pΩ.Now consider inΩ the second-order linear differential operator in nondivergence form
(32)L=-∑i,j=1n‍aij∂2∂xi∂xj+∑i=1n‍ai∂∂xi+a.Assume that the leading coefficients satisfy the hypothesis.(h1) There exist extensions aijo of aij to Ωo such that
(33)aijo=ajio∈L∞(Ωo)∩VMO(Ωo),i,j=1,…n,∃νo∈R+:∑i,j=1n‍aijoξiξj≥νoξ2a.e.inΩo,∀ξ∈Rn.We note that the assumption(h1) holds if the coefficients aij are restrictions to Ω of functions in VMO (see also [2, 3]).For the lower order terms coefficients suppose that(i2)
(34)ai∈K~1t1Ω,i=1,…,n,wheret1≥p,t1≥n,kkkkkkkkkkkkkkkkkkkkkkkkkkit1>pifp=n,a∈K~2t2Ω,wheret2≥p,t2≥n2,kkkkkkkkkkkkkkit2>pifp=n2.We observe that, in view of Theorem2, under the assumptions (h0), (h1), and (i2), the operator L:Ws2,p(Ω)→Lsp(Ω) is bounded.The following a priori estimate holds.Lemma 4.
LetL be defined in (32). If hypotheses (h0), (h1), and (i2) are satisfied, then there exists c∈R+ such that
(35)uWs2,p(Ω)≤cLuLspΩ+uLs-2pΩii∀u∈Ws2,pΩ∩W∘s-11,pΩ,
where c depends on Ω, n, p, t1, t2, s, ρ, νo, aijoL∞(Ωo), aiK1t1(Ω), aK2t2(Ω), η[aijo], ω~1t1[ai], and ω~2t2[a].The proof of (35) can be found in [19] in the case n≥3 and it can be easily extended to the planar case, taking care to use Lemma  3.1 in [20] in place of Theorem  5.1 in [21].In the next Lemma we recall a regularity result of [5] (see Theorem 3.4) and we extend it to the case n=2.Lemma 5.
LetL be defined in (32). Suppose that conditions (h0), (h1), and (i2) hold with t1>n and t2>n/2. Then any solution u of the problem
(36)u∈Wloc⁡2,qΩ¯∖Sρ∩W∘loc⁡1,qΩ¯∖Sρ∩Lso-2poΩ,Lu∈LspΩ,
with q,po∈]1,p] and so=s-n((1/po)-(1/p)), belongs to the space Ws2,p(Ω) and verifies the bound
(37)uWs2,p(Ω)≤cLuLspΩ+uLso-2poΩ,
where c∈R+ depends only on Ω, n, p, po, t1, t2, s, ρ, νo, aijoL∞Ωo, aiK1t1Ω, aK2t2(Ω), η[aijo], ω~1t1[ai], and ω~2t2[a].Let us give an overview of the proof of Lemma5. The idea is to use a local regularity result which is based on an analogous result for solutions of boundary value problems in classical Sobolev spaces defined on regular domains. The mentioned regularity result has been proved in [19] when n≥3 (see, Theorem 5.1 of [19]) and it can be easily extended to the planar case, taking into account to apply, at right time, Lemma  4.2 in [20] in place of Lemma  4.2 in [22].


## 4. Main Results

LetL be the operator defined in (32). Suppose that the coefficients of operator L satisfy the assumptions (h1),(h2)
(38)aijxh,ai∈K∘1t1Ω,i,j,h=1,…,n,(h3)
(39)a=a′+b,a′∈K∘2t2Ω,b∈K~2t2Ω,00000000000000essinf⁡Ωσ2b=bo>0,where t1,t2 are as in (i2) and σ is the function defined in Remark 1.Moreover, suppose that the following condition onρ holds:(h4)
(40)lim⁡x→xoσx(x)=lim⁡x→+∞σx(x)=0∀xo∈Sρ.We are interested in the study of the Dirichlet problem:(41)u∈Ws2,p(Ω)∩W∘s-11,pΩ,Lu=f,f∈LspΩ,
with p∈1,+∞ and s∈R.We point out that the unique solvability of (41) has been firstly proved in [4, 5] for p≥2 and n≥3. Later on, the results of [4, 5] have been used in [6] to get the existence and uniqueness of solution also for 1<p<2. Here, we collect the main results of the above-mentioned papers and we also extend them to the case n=2.For the case where the assumptions(h0)–(h4) are taken into account, with t1>n, t2>n/2 and a′=0, and for p=2, the unique solvability of (41) has been proved in Lemma 4.1 of [5], which can be easily extended to the planar case, applying our Lemmas 4 and 5 at the right time. If p≠2, the idea is to exploit the previous case and the unique solvability of a problem similar to (41), whose associated differential operator Ls-1 differs from that in (32) for a compact operator.Letp∈]1,+∞[, s∈R and consider in Ω the differential operator
(42)Ls-1=-∑i,j=1n‍aij∂2∂xi∂xj-∑i=1n∑j=1n‍aijxj+2s-10000000000000×aijσ-1σxjaijxj∑j=1n‍∂∂xi+b.In the following Lemma we put together some results of [4–6], showing an a priori estimate for the operator Ls-1 and the unique solvability of its associated Dirichlet problem.Lemma 6.
Suppose that conditions(h0)–(h4) hold. Then for any p∈]1,+∞[ and s∈R there exists c∈R+ such that
(43)uWs2,p(Ω)≤cLs-1uLsp(Ω)∀u∈Ws2,p(Ω)∩W∘s-11,p(Ω),
where c∈R+ depends on Ω, n, p, t1, t2, s, ρ, νo, aijoL∞(Ωo), (aij)xhK1t1(Ω), bK2t2(Ω), η[aijo], ω~1t1[(aij)xh], and ω~2t2[b].
Furthermore, ift1>n and t2>n/2 then the problem
(44)u∈Ws2,pΩ∩W∘s-11,pΩ,Ls-1u=f,f∈LspΩ,
is uniquely solvable.Here, we only give an overview of the proof, pointing out to the crucial aspects. Since the operatorLs-1 verifies the same assumptions of the operator L, we can write for it the estimate (35). Now a crucial point is to provide a bound for uLs-2p(Ω) in terms of Ls-1uLsp(Ω), when the function u is more regular; that is, u∈Ws2,p(Ω)∩W∘s-11,2(Ω)∩D0(Ω¯∖Sρ). To this aim, we build the following bilinear form as-1 related, in appropriate way, to the operator Ls-1:
(45)as-1u,w=∫Ω‍∑i,j=1naijuxiwxj+buwσ2(s-1)dx,
where u,w∈W∘s-11,2(Ω).By simple computations, we get(46)as-1u,w=∫ΩLs-1uwσ2(s-1)dx.Now we study separately the casesp≥2 and p<2. If p≥2, applying Lemma 4.1 of [4] in (46), with a suitable choice of function w, we easily deduce the claimed bound on uLs-2p(Ω). Thus, from Lemma 3 we get the estimate (43). Furthermore, in this case, the uniqueness of the solution of problem (44) easily follows from (43). For the existence of solution, we refer to Theorem  4.2 in [5], whose analytic technique exploits the bound (43), which also gives the closure of the range of operator Ls-1, the unique solvability of our problem when p=2, and the regularity result of Lemma 5, which allow us to go up in summability.Suppose now1<p<2. In this case, in order to get a bound on uLs-2p(Ω), when u is more regular, we use a variational result (see the proof of Lemma 3.3 in [6]) whose main analytic tools exploit the existence and uniqueness result of previous case and Lemma 5. The mentioned variational result gives us a bound for wLs-2p′(Ω), where w is the unique solution of the Dirichlet problem associated with as-1(u,w) and p′ is the conjugate exponent of p. From such estimate and (46), with a suitable choice of function w, we obtain our claim. Using again Lemma 3, we get (43) also in this case. Finally, for the existence and uniqueness of problem (44) we can refer to Lemma 3.4 of [6], whose proof can be easily extended also to the planar case, in view of previous considerations.In the following Theorem we collect two results of [4, 6], giving an a priori bound for the operator L when n≥2.Theorem 7.
Suppose that conditions(h0)–(h4) hold. Then for any p∈]1,+∞[ and s∈R there exist c∈R+ and a bounded open set Ω1⊂⊂Ω, with the cone property, such that
(47)uWs2,pΩ≤cLuLspΩ+uLpΩ1ii∀u∈Ws2,pΩ∩W∘s-11,pΩ,
where c∈R+ and Ω1 depend on Ω, n, p, t1, t2, s, ρ, νo, ∥aijo∥L∞Ωo, ∥(aij)xh∥K1t1Ω, ∥ai∥K1t1Ω, ∥a′∥K2t2Ω, ∥b∥K2t2Ω, η[aijo], ω∘1t1aijxh, ω∘1t1[ai], ω∘2t2[a′], and ω~2t2[b].Proof.
Fixu∈Ws2,p(Ω)∩W∘s-11,p(Ω). We observe that
(48)L=Ls-1+∑i=1n‍bi∂∂xi+a′,
where
(49)bi=∑j=1n‍aijxj+2s-1aijσ-1σxj+ai,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=1,…,n.
Thus, from estimate (43) we deduce that there exists c∈R+, depending on Ω, n, p, t1, t2, s, ρ, νo, ∥aijo∥L∞(Ωo), ∥(aij)xh∥K1t1(Ω), ∥b∥K2t2(Ω), η[aijo], ω~1t1aijxh, and ω~2t2[b], such that
(50)uWs2,pΩ≤cLuLspΩ+∑i=1n‍biuxi+a′uLspΩ.
On the other hand, by Theorem2 it follows that for any ϵ∈R+ there exist c(ϵ)∈R+ and a bounded open set Ωϵ⊂⊂Ω, with the cone property, such that
(51)∑i=1n‍biuxi+a′uLsp(Ω)≤ϵuWs2,p(Ω)+cϵuxLpΩϵ+uLpΩϵ,
where c(ϵ) and Ωϵ depend on ϵ, Ω, n, p, t1, t2, s, ρ, ∥(aij)xh∥K1t1(Ω), ∥ai∥K1t1(Ω), ∥a′∥K2t2(Ω), ω∘1t1[(aij)xh], ω∘1t1[ai], and ω∘2t2[a′]. The result easily follows from relations (50) and (51).Finally, in the next theorem we put together Theorem 4.2 of [5] with Theorem 4.2 of [6], showing the existence and uniqueness of problem (41) for p∈]1,+∞[ and n≥2.Theorem 8.
Under the same hypotheses of Theorem7 with t1>n and t2>n/2, for any p∈]1,+∞[ and s∈R, the problem (41) is an index problem with index equal to zero. Moreover, if a′=0, then the problem is uniquely solvable.Proof.
First we prove that the problem (41) is an index problem with index equal to zero. In fact, from Lemma 6 we deduce that the operator Ls-1 is a Fredholm operator with index zero. On the other hand, by Theorem 2 it follows that the operator
(52)u∈Ws2,pΩ⟶∑i=1n‍∑j=1n‍aijxj+2s-1aijσ-1σxj+ai×uxi+a′u∈Lsp(Ω)
is a compact operator. Thus, from (48), (49), and well known results of the classical Fredholm index theory, we deduce that the problem (41) is an index problem with index equal to zero.
Assume nowa′=0. We prove only the uniqueness of solution of problem (41); the existence will easily follow from uniqueness and from what we have proved in the case a′≠0. At first, suppose p≥2. Then, by the unique solvability of problem (41) for p=2 there exists a unique u∈Wt2,2(Ω)∩W∘t-11,2(Ω) such that Lu=0. Moreover, by our Lemma 5 and by Lemma 3.2 of [5] it follows that u belongs to Ws2,p(Ω)∩W∘s-11,p(Ω) and thus we deduce that u=0. Now let 1<p<2 and u∈Ws2,p(Ω)∩W∘s-11,p(Ω) such that Lu=0. Using again our Lemma 5 and Lemma 3.2 of [5] we get u∈Wt2,2(Ω)∩W∘t-11,2(Ω) with t=s+n(1/p-1/2) and Lu=0. Exploiting again the existence and uniqueness of (41) for p=2, we obtain that u=0. This concludes the proof.



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