# Elliptic Equations in Weighted Sobolev Spaces on Unbounded Domains

Authors: Serena Boccia; Sara Monsurrò; Maria Transirico
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

We study in this paper a class of second-order
linear elliptic equations in weighted Sobolev spaces on unbounded
domains ofℝn, n≥3. We obtain an a priori bound, and a regularity
result from which we deduce a uniqueness theorem.


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## Body

## 1. Introduction

LetΩ be an open subset of Rn, n≥3.
Assign in Ω the uniformly elliptic second-order linear
differential operator(1.1)L=−∑i,j=1naij∂2∂xi∂xj+∑i=1nai∂∂xi+a,with coefficients aij=aji∈L∞(Ω), i,j=1,…,n,
and consider the associate Dirichlet problem:(1.2)u∈W2,p(Ω)∩W∘1,p(Ω),Lu=f,f∈Lp(Ω),where p∈]1,+∞[.It is well known that ifΩ is a bounded and sufficiently regular set, the
above problem has been widely investigated by several authors under various
hypotheses of discontinuity on the leading coefficients, in the case p=2 or p sufficiently close to 2. In particular, some W2,p-bounds for the solutions of the problem (1.2)
and related existence and uniqueness results have been obtained. Among the
other results on this subject, we quote here those proved in [1], where the author assumed
that aij's belong to W1,n(Ω) (and considered the case p=2) and in [2–4] (where the coefficients belong to some classes wider
than W1,n(Ω)). More recently, a relevant contribution has
been given in [5–8], where the coefficients aij are assumed to be in the class VMO and p∈]1,+∞[;
observe here that VMO contains the space W1,n(Ω).If the setΩ is unbounded and regular enough, under
assumptions similar to those required in [1], problem (1.2) has for instance been studied in
[9–11]
with p=2,
and in [12] with p∈]1,+∞[.
Instead, in [13, 14] the leading
coefficients satisfy restrictions similar to those in [5, 6].In this paper, we extend some results of [13, 14] to a weighted case. More
precisely, we denote by ρ a weight function belonging to a suitable
class such that(1.3)inf⁢Ωρ>0,lim|x|→+∞ρ(x)=+∞,and consider the Dirichlet
problem:(1.4)u∈Ws2,p(Ω)∩W∘s1,p(Ω),Lu=f,f∈Lsp(Ω),where s∈R, Ws2,p(Ω), W∘s1,p(Ω),
and Lsp(Ω) are some weighted Sobolev spaces and the
weight functions are a suitable power of ρ.
We obtain an a priori bound for the solutions of (1.4). Moreover, we state a
regularity result that allows us to deduce a uniqueness theorem for the problem
(1.4). A similar weighted case was studied in [15] with the leading
coefficients satisfying hypotheses of Miranda's type and when p=2.


## 2. Weight Functions and Weighted Spaces

LetG be any Lebesgue measurable subset of Rn and let Σ(G) be the collection of all Lebesgue measurable
subsets of G.
If F∈Σ(G),
denote by |F| the Lebesgue measure of F,
by χF the characteristic function of F,
by F(x,r) the intersection F∩B(x,r) (x∈Rn, r∈R+)—where B(x,r) is the open ball of radius r centered at x—and by𝒟(F) the class of restrictions to F of functions ζ∈C∘∞(Rn) with F¯∩suppζ⊆F.
Moreover, if X(F) is a space of functions defined on F,
we denote by Xloc(F) the class of all functions g:F→R,
such that ζg∈X(F) for any ζ∈𝒟(F).We introduce a class of weight functions defined on an
open subsetΩ of Rn.
Denote by 𝒜(Ω) the set of all measurable functions ρ:Ω→R+,
such that(2.1)γ−1ρ(y)≤ρ(x)≤γρ(y)∀y∈Ω,∀x∈Ω(y,ρ(y)),where γ∈R+ is independent of x and y.
Examples of functions in 𝒜(Ω) are the function(2.2)x∈Rn→1+a|x|,a∈]0,1[,and, if Ω≠Rn and S is a nonempty subset of ∂Ω,
the function(2.3)x∈Ω→adist(x,S),a∈]0,1[.For ρ∈𝒜(Ω),
we put(2.4)Sρ={z∈∂Ω:limx→zρ(x)=0}.It is known that(2.5)ρ∈Lloc∞(Ω¯),ρ−1∈Lloc∞(Ω¯∖Sρ)(see [16, 17]).We assign an unbounded open subsetΩ of Rn.Letρ1 be a function, such that ρ1∈𝒜(Rn) and(2.6)infΩρ1>0,lim|x|→+∞ρ1(x)=+∞.We put(2.7)ρ=ρ1∣Ω.For any a∈]0,1] and x∈Rn,
we set(2.8)Ia(x)=Ω(x,aρ1(x)).Ifk∈N0, 1≤p<+∞, s∈R,
and ρ∈𝒜(Ω),
consider the space Wsk,p(Ω) of distributions u on Ω,
such that ρs∂αu∈Lp(Ω) for |α|≤k,
equipped with the norm(2.9)∥u∥Wsk,p(Ω)=∑|α|≤k∥ρs∂αu∥Lp(Ω).Moreover, denote by W∘sk,p(Ω) the closure of C∘∞(Ω) in Wsk,p(Ω) and put Ws0,p(Ω)=Lsp(Ω).
A more detailed account of properties of the above defined spaces can be found,
for instance, in [18].From [15, Lemmas 1.1
and 2.1], we deduce the following two lemmas,
respectively.Lemma 2.1.
For anyp∈[1,+∞[, s∈R,
and a∈]0,1], g∈Lsp(Ω) if and only if g∈Llocp(Ω¯) and the function x∈Rn→ρ1s−n/p(x)||g||Lp(Ia(x)) belongs to Lp(Rn).
Moreover, there exist c1,c2∈R+,
such that(2.10)c1∥g∥Lsp(Ω)p≤∫Rnρ1sp−n(x)∥g∥Lp(Ia(x))pdx≤c2∥g∥Lsp(Ω)p∀g∈Lsp(Ω),
where c1 and c2 depend on n, p, s, a,
and ρ.Lemma 2.2.
IfΩ has the segment property, then for any k∈N0, p∈[1,+∞[,
and s∈R one has(2.11)Wsk,p(Ω)∩W∘lock,p(Ω¯)=W∘sk,p(Ω).


## 3. Some Embedding Lemmas

We now recall
the definitions of the function spaces in which the coefficients of the
operator will be chosen. IfΩ has the property(3.1)|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(3.2)[g]BMO(Ω,τ)=supx∈Ωr∈]0,τ]⨍Ω(x,r)|g−⨍Ω(x,r)g|<+∞,where(3.3)⨍Ω(x,r)⁢g=|Ω(x,r)|−1∫Ω(x,r)g.If g∈BMO(Ω)=BMO(Ω,τA),
where(3.4)τA=sup{τ∈R+:supx∈Ωr∈]0,τ]rn|Ω(x,r)|≤1A},we will say that g∈VMO(Ω) if [g]BMO(Ω,τ)→0 for τ→0+.
A function η[g]:]0,1]→R+ is called a modulus of continuity of g in VMO(Ω) if(3.5)[g]BMO(Ω,τ)≤η[g](τ)∀τ∈]0,1],limτ→0+η[g](τ)=0.For t∈[1,+∞[ and λ∈[0,n[,
we denote by Mt,λ(Ω) the set of all functions g in Lloct(Ω¯) such that(3.6)∥g∥Mt,λ(Ω)=supr∈]0,1]x∈Ωr−λ/t∥g∥Lt(Ω(x,r))<+∞,endowed with the norm defined by
(3.6). Then, we define M∘t,λ(Ω) as the closure of C∘∞(Ω) in Mt,λ(Ω).
In particular, we put Mt(Ω)=Mt,0(Ω),
and M∘t(Ω)=M∘t,0(Ω). In order to define the modulus of continuity
of a function g in M∘t,λ(Ω),
recall first that for a function g∈Mt,λ(Ω) the following characterization holds:(3.7)g∈M∘t,λ(Ω)⟺limτ→0+(pg(τ)+∥(1−ζ1/τ)g∥Mt,λ(Ω))=0,where(3.8)pg(τ)=supE∈Σ(Ω)supx∈Ω|E(x,1)|≤τ∥χEg∥Mt,λ(Ω),τ∈R+,and ζr, r∈R+,
is a function in C∘∞(Rn) such that(3.9)0≤ζr≤1,ζr∣Br=1,supp⁢ζr⊂B2r,with the position Br=B(0,r).
Thus, the modulus of continuity of g∈M∘t,λ(Ω) is a function(3.10)σ∘[g]:]0,1]→R+,such that(3.11)pg(τ)+∥(1−ζ1/τ)g∥Mt,λ(Ω)≤σ∘[g](τ)∀τ∈]0,1],limτ→0+σ∘[g](τ)=0.A more detailed account of
properties of the above defined function spaces can be found in [9, 19, 20].We consider the following condition:(h0)
Ω has the cone property, p∈]1,+∞[, s∈R, k, h, t are numbers such that(3.12)k∈N,h∈{0,1,…,k−1},t≥p,t>pifp=nk−h,g∈Mt(Ω).From [21, Theorem 3.1] we have the
following.Lemma 3.1.
If the assumption(h0) holds, then for any u∈Wsk,p(Ω) one has g∂hu∈Lsp(Ω) and
(3.13)∥g∂hu∥Lsp(Ω)≤c∥g∥Mt(Ω)∥u∥Wsk,p(Ω),with c dependent only on Ω, n, k, h, p,
and t.From [21, Theorem 3.2] it follows Lemma 3.2.Lemma 3.2.
If the assumption(h0) is satisfied and in addition g∈M∘t(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
(3.14)∥g∂hu∥Lsp(Ω)≤ε∥u∥Wsk,p(Ω)+c(ε)∥u∥Lp(Ωε)∀u∈Wsk,p(Ω),where c(ε), Ωε depend on ε, Ω, n, k, h, p, t, ρ, s,
and σ∘[g].


## 4. An a Priori Bound

Assume thatΩ is an unbounded open subset of Rn, n≥3,
with the uniform C1,1-regularity property, and let ρ be the function defined by (2.7). Moreover,
let p∈]1,+∞[ and s∈R.
Consider in Ω the differential operator:(4.1)L=−∑i,j=1naij∂2∂xi∂xj+∑i=1nai∂∂xi+a,with the following conditions on
the coefficients:(h1)
(4.2)aij=aji∈L∞(Ω)∩VMOloc(Ω¯),i,j=1,…,n,∃ν>0:∑i,j=1naijξiξj≥ν|ξ|2a.e.inΩ,∀ξ∈Rn,
there exist
functions eij, i,j=1,…,n, g and μ∈R+ such that(h2)
(4.3)eij=eji∈L∞(Ω),(eij)xh∈M0t,n−t(Ω),witht∈]2,n],i,j,h=1,…,n,∑i,j=1neijξiξj≥μ|ξ|2a.e.inΩ,∀ξ∈Rn,g∈L∞(Ω),g0=essinfΩg>0,limr→+∞∑i,j=1n∥eij−gaij∥L∞(Ω∖Br)=0,(h3)
(4.4)ai∈M∘t1(Ω),i=1,…,n,a=a′+a′′,a′∈M∘t2(Ω),a′′∈L∞(Ω),essinfΩa′′=a0′′>0,where(4.5)t1≥nifp<n,t1>nifp=n,t1=pifp>n,t2≥n/2ifp<n/2,t2>n/2ifp=n/2,t2=pifp>n/2.Observe that under the assumptions(h1)–(h3),
it follows that the operator L:Ws2,p(Ω)→Lsp(Ω) is bounded from Lemma 3.1.Theorem 4.1.
If the hypotheses(h1), (h2),
and (h3) are verified, then there exist a constant c∈R+ and a bounded open subset Ω0⊂⊂Ω,
with the cone property, such that(4.6)∥u∥Ws2,p(Ω)≤c(∥Lu∥Lsp(Ω)+∥u∥Lp(Ω0)),∀u∈Ws2,p(Ω)∩W∘s1,p(Ω),
with c and Ω0 depending on n, p, ρ, s, Ω, ν, μ, g0, a0′′, t, t1, t2, ∥aij∥L∞(Ω), ∥eij∥L∞(Ω), ∥g∥L∞(Ω), ∥a′′∥L∞(Ω), η[ζ2r0aij], σ∘[(eij)x], σ∘[ai], σ∘[a′],
where r0∈R+ depends on n, p, Ω, μ, g0, a0′′, t, ∥eij∥L∞(Ω), ∥g∥L∞(Ω), ∥a′′∥L∞(Ω), σ∘[(eij)x].Proof.
We consider a functionϕ∈C∘∞(Rn),
such that(4.7)ϕ|B1/2=1,suppϕ⊂B1,supRn|∂αϕ|≤cα∀α∈N0n,
where cα∈R+ depends only on α,
fix y∈Rn and put(4.8)ψ=ψy:x∈Rn→ϕ(x−yρ1(y)).Clearly we have(4.9)ψ|B(y,(1/2)ρ1(y))=1,suppψ⊂B(y,ρ1(y)),supRn|∂αψ|≤cαρ1−|α|(y)∀α∈N0n.
Now, we put(4.10)L0=−∑i,j=1naij∂2∂xi∂xjand fix u∈Ws2,p(Ω)∩W∘s1,p(Ω). Since ψu∈W2,p(Ω)∩W∘1,p(Ω),
from [14, Theorem 3.3], it follows that there exist c1∈R+ and a bounded open subset Ω1⊂⊂Ω,
with the cone property, such that(4.11)∥ψu∥W2,p(Ω)≤c1(∥(L0+a′′)(ψu)∥Lp(Ω)+∥ψu∥Lp(Ω1)),with c1 and Ω1 depending on n, p, Ω, ν, μ, g0, a0′′, t, ∥aij∥L∞(Ω), ∥eij∥L∞(Ω), ∥g∥L∞(Ω), ∥a′′∥L∞(Ω), η[ζ2r0aij], σ∘[(eij)x],
where r0∈R+ depends on n, p, Ω, μ, g0, a0′′, t, ∥eij∥L∞(Ω), ∥g∥L∞(Ω), ∥a′′∥L∞(Ω), σ∘[(eij)x]. Since(4.12)L0(ψu)=ψL0u−2∑i,j=1naijψxiuxj−∑i,j=1naijψxixju,from (4.11) and (4.12), we have(4.13)∥ψu∥W2,p(Ω)≤c2(∥ψ(L0+a′′)u∥Lp(Ω)+∑i,j=1n∥ψxiuxj∥Lp(Ω)+∑i,j=1n∥ψxixju∥Lp(Ω)+∥ψu∥Lp(Ω1)),
with c2 dependent on the same parameters of c1.
On the other hand, sinceρ∈Lloc∞(Ω¯),
we have that(4.14)∥ψu∥Lp(Ω1)≤c3ρ1−2(y)∥u∥Lp(I1(y)),where c3∈R+ depends only on ρ.
Therefore, by (4.13) and (4.14), we deduce the bound:(4.15)∥u∥W2,p(I1/2(y))≤∥ψu∥W2,p(Ω)≤c4(∥L0u+a′′u∥Lp(I1(y))+ρ1−1(y)∥ux∥Lp(I1(y))+ρ1−2(y)∥u∥Lp(I1(y)),
where c4∈R+ depends on the same parameters of c2 and on ρ.
From (4.15) it follows(4.16)∫Rnρ1ps−n(y)∥u∥W2,p(I1/2(y))pdy≤c5(∫Rnρ1ps−n(y)∥L0u+a′′u∥Lp(I1(y))pdy+∫Rnρ1ps−n−p(y)∥ux∥Lp(I1(y))pdy+∫Rnρ1ps−n−2p(y)∥u∥Lp(I1(y))pdy),
where c5∈R+ depends on the same parameters of c4.
Since(4.17)Lsp(Ω)↪Ls−1p(Ω),Lsp(Ω)↪Ls−2p(Ω),from (4.16) and from Lemma 2.1 we
have that(4.18)∥u∥Ws2,p(Ω)≤c6(∥L0u+a′′u∥Lsp(Ω)+∥ux∥Ls−1p(Ω)+∥u∥Ls−2p(Ω)),with c6∈R+ dependent on the same parameters of c5 and also on s.
Moreover, from Lemma3.2 it follows that for any ε∈R+,
there exist c′(ε), c′′(ε)∈R+, and two bounded open sets Ωε′, Ωε′′⊂⊂Ω,
both with the cone property, such that(4.19)∥ux∥Ls−1p(Ω)+∥u∥Ls−2p(Ω)≤ε∥u∥Ws2,p(Ω)+c′(ε)∥u∥Lp(Ωε′),∥∑i=1naiuxi+a′u∥Lsp(Ω)≤ε∥u∥Ws2,p(Ω)+c′′(ε)∥u∥Lp(Ωε′′),
where c′(ε), Ωε′ depend on ε, Ω, n, p, ρ, s,
and c′′(ε), Ωε′′ depend on ε, Ω, n, p, t1, t2, ρ, s, σ∘[ai],
and σ∘[a′].
From (4.18) and  (4.19) it follows (4.6) and
then we have the result.


## 5. A Uniqueness Result

In this
section, we will prove our uniqueness theorem. We begin to prove a regularity
result.Lemma 5.1.
Suppose that the assumptions(h1), (h2),
and (h3) (with t1>n and t2>n/2) hold, and let u be a solution of the problem
(5.1)u∈Wloc2,q(Ω¯)∩W∘loc1,q(Ω¯)∩Lmp(Ω),Lu∈Lsp(Ω),where q∈]1,p] and m∈R.
Then, u belongs to Ws2,p(Ω).Proof.
By [13, Lemma 4.1] we have(5.2)u∈Wloc2,p(Ω¯)∩W∘loc1,p(Ω¯).We choose r,r′∈R+,
with r<r′<1, and a function ϕ∈C∘∞(Rn),
such that(5.3)ϕ|Br=1,supp⁢ϕ⊂Br′,supRn|∂αϕ|≤cα(r′−r)−|α|∀α∈N0n,
where cα∈R+ depends only on α.
We fixy∈Rn and put(5.4)ψ=ψy:x∈Rn→ϕ(x−yρ1(y)).Clearly we have(5.5)ψ|B(y,rρ1(y))=1,suppψ⊂B(y,r′ρ1(y)),supRn|∂αψ|≤cαρ1−|α|(y)(r′−r)−|α|∀α∈N0n.
Sinceψu∈W2,p(Ω)∩W∘1,p(Ω),
from [14, Theorem 3.3] it follows that there exist c1∈R+ and a bounded open subset Ω1⊂⊂Ω,
with the cone property, such that(5.6)∥ψu∥W2,p(Ω)≤c1(∥L(ψu)∥Lp(Ω)+∥ψu∥Lp(Ω1)),with c1 and Ω1 depending on n, p, Ω, ν, μ, g0, a0′′, t, t1, t2, ∥aij∥L∞(Ω), ∥eij∥L∞(Ω), ∥g∥L∞(Ω), ∥a′′∥L∞(Ω), η[ζ2r0aij], σ∘[(eij)x], σ∘[ai], σ∘[a′],
where r0∈R+ depends on n, p, Ω, μ, g0, a0′′, t, ∥eij∥L∞(Ω), ∥g∥L∞(Ω), ∥a′′∥L∞(Ω), σ∘[(eij)x].
Since(5.7)L(ψu)=−∑i,j=1naij(ψu)xixj+∑i=1nai(ψu)xi+aψu=ψLu−2∑i,j=1naij(ψxiu)xj+∑i,j=1naijψxixju+∑i=1naiψxiu,
from (5.6) and (5.7), we have(5.8)∥ψu∥W2,p(Ω)≤c2(∥ψLu∥Lp(Ω)+∑i,j=1n∥(ψxiu)xj∥Lp(Ω)+∑i,j=1n∥ψxixju∥Lp(Ω)+∑i=1n∥aiψxiu∥Lp(Ω)+∥ψu∥Lp(Ω1)),
with c2 dependent on the same parameters of c1.
From Lemma3.1 with s=0,
we have that(5.9)∥aiψxiu∥Lp(Ω)≤c3∥ai∥Mt1(Ω)(∥ψxiu∥Lp(Ω)+∥(ψxiu)x∥Lp(Ω)),with c3 dependent on Ω, n, p,
and t1.
Using [22, Corollary 4.5], we can obtain the following interpolation estimate:(5.10)∥ψxiu∥Lp(Ω)+∥(ψxiu)xj∥Lp(Ω)≤c4(∥(ψxiu)xx∥Lp(Ω)1/2∥ψxiu∥Lp(Ω)1/2+∥ψxiu∥Lp(Ω)),
where the constant c4 depends on Ω, n, p.
Thus, by (5.8)–(5.10), with easy computations, we
deduce the bound:(5.11)∥u∥W2,p(Ir(y))≤∥ψu∥W2,p(Ω)≤c5(r′−r)−2×(∥Lu∥Lp(Ir′(y))+∥u∥W2,p(Ir′(y))1/2(ρ1−1(y)∥u∥Lp(Ir′(y)))1/2+ρ1−1(y)∥u∥Lp(Ir′(y))),
where c5∈R+ depends on n, p, ρ, Ω, ν, μ, g0, a0′′, t, t1, t2, ∥aij∥L∞(Ω), ∥eij∥L∞(Ω), ∥g∥L∞(Ω), ∥a′′∥L∞(Ω), η[ζ2r0aij], σ∘[(eij)x], ∥ai∥Mt1(Ω), σ∘[ai], σ∘[a′].
By a well-known lemma of monotonicity of Miranda (see
[23, Lemma 3.1]), it
follows from (5.11) that(5.12)∥u∥W2,p(I1/2(y))≤c6(∥Lu∥Lp(I1(y))+ρ1−1(y)∥u∥Lp(I1(y))+(ρ1−1(y)∥u∥Lp(I1(y)))1/2∥u∥W2,p(I1/2(y))1/2),
and then, using Young's
inequality, we deduce from (5.12) that(5.13)∥u∥W2,p(I1/2(y))≤c7(∥Lu∥Lp(I1(y))+ρ1−1(y)∥u∥Lp(I1(y))),with c6,c7∈R+ dependent on the same parameters of c5.
From (5.13) it follows(5.14)∫Rnρ1ps−n(y)∥u∥W2,p(I1/2(y))pdy≤c8(∫Rnρ1ps−n(y)∥Lu∥Lp(I1(y))pdy+∫Rnρ1ps−n−p(y)∥u∥Lp(I1(y))pdy),
where c8∈R+ depends on the same parameters of c7.
Ifm≥s−1,
since(5.15)Lmp(Ω)↪Ls−1p(Ω),from (5.14) and from Lemma 2.1
we have that(5.16)∥u∥Ws2,p(Ω)≤c9(∥Lu∥Lsp(Ω)+∥u∥Ls−1p(Ω)),with c9∈R+ dependent on the same parameters of c8 and on s.
Therefore, u belongs to Ws2,p(Ω).
Ifm<s−1,
we denote by k the positive integer, such that(5.17)s−m−1≤k<s−m.Then, for i=1,…,k,
we have that(5.18)Lsp(Ω)↪Lm+ip(Ω).Therefore, using (5.14) and (5.16)
with m+i, i=1,…,k,
instead of s,
we deduce that u∈Wm+12,p(Ω),…,u∈Wm+k2,p(Ω). On the other hand, we have that(5.19)Wm+k2,p(Ω)↪Ls−1p(Ω)and then, since u∈Ls−1p(Ω),
(5.14) holds. Thus, u satisfies (5.16) and then u∈Ws2,p(Ω).Theorem 5.2.
If
conditions(h1), (h2),
and (h3) (with t1>n and t2>n/2) hold, and a≥a0>0 a.e. in Ω,
then the problem(5.20)u∈Ws2,p(Ω)∩W∘s1,p(Ω),Lu=0,admits only the zero solution.Proof.
Fixu∈Ws2,p(Ω)∩W∘s1,p(Ω),
such that Lu=0.
From Lemma 5.1 it follows that u∈W2,p(Ω).
On the other hand, since u∈W1,p(Ω)∩W∘loc1,p(Ω¯),
from Lemma 2.2 we have that u∈W∘1,p(Ω).
Thus, from [13, Theorem 5.2] we deduce that u=0.



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*Source: 582435-2008-10-16.xml*