# Higher Integrability of Weak Solutions to a Class of Double Obstacle Systems

Authors: Zhenhua Hu; Shuqing Zhou
Journal: Journal of Mathematics
 (2013)

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## Abstract

We first introduce double obstacle systems associated with the second-order quasilinear elliptic differential equationdiv(A(x,∇u))=divf(x,u), where A(x,∇u), f(x,u) are two n×N matrices satisfying certain conditions presented in the context, then investigate the local and global higher integrability of weak solutions to the double obstacle systems, and finally generalize the results of the double obstacle problems to the double obstacle systems.


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## Body

## 1. Introduction

LetΩ⊂Rn be a bounded domain. We consider the following quasilinear elliptic systems:
(1)DiAαi(x,∇u)=Difαi(x,u),α=1,2,…,N,
where Aαi(x,h),fαi(x,u) satisfy the conditions given in the following context. If we denote A(x,h)=(Aαi(x,h)),f(x,u)=(fαi(x,u))(n×N matrices), then (1) turns into
(2)divA(x,∇u)=divf(x,u).
Our aim is to generalize the integrability results of double obstacle problems (N=1) to systems (N>1). In order to do that, first, we have to define the obstacle problems corresponding to systems (2), and then we investigate the integrability of the weak solutions to the double obstacle systems.In order to narrate our assumptions and our results, we give the following notations.Letf(x)=(f1(x),f2(x),…,fN(x)), g(x)=(g1(x),…,gN(x)) be two vector-valued functions defined on Ω, then we say that f(x)≤g(x) if and only if fα(x)≤gα(x)a.e.x∈Ω,∀1≤α≤N, and define max{f(x),g(x)}=(max{f1(x),g1(x)},…,max{fN(x),gN(x)}), min{f(x),g(x)}=-max{-f(x),-g(x)}, θ+(x)=(max{θ1(x),0},…,max{θN(x),0}), θ-(x)=(min{θ1(x),0},…,min{θN(x),0}).LetW1,p(Ω), and let W01,p(Ω),Wloc1,p(Ω) be usual Sobolev spaces, then define
(3)W1,p(Ω,RN)={W1,pf(x)∣f(x)=(f1(x),…,fN(x)),fα(x)∈W1,p(Ω),α=1,…,N},W01,p(Ω,RN)={W01,pf(x)∣f(x)=(f1(x),…,fN(x)),fα(x)∈W01,p(Ω),α=1,…,N},Wloc1,p(Ω,RN)={W01,pf(x)∣f(x)=(f1(x),…,fN(x)),fα(x)∈Wloc1,p(Ω),α=1,…,N}.
Let θ∈W1,p(Ω,RN) and φ,ψ:Ω→RN, then we denote
(4)Kφ,ψθ,p(Ω,RN)={u∈W1,p(Ω,RN):u-θ∈W01,p(Ω,RN),φ≤u≤ψa.e.inΩ{RN}},
and we call φ,ψ obstacles and θ boundary value.LetA=(aαi), B=(bαi) be two n×N matrixes, then define A•B=aαibαi; here and in the following we use the convention that repeated indices are summed: here α goes from 1 to N and i from 1 to n.We consider the higher integrability of weak solutions toKφ,ψθ,p(A)-double obstacle systems corresponding to (2).Definition 1.
We call a functionu∈Kφ,ψθ,p(Ω,RN) a weak solution to Kφ,ψθ,p(A)-double obstacle systems if
(5)∫Ω‍(A(x,∇u)-f(x,u))•∇(v-u)dx≥0,
holds for all v∈Kφ,ψθ,p(Ω,RN), here ∇u=(∇u1,∇u2,…,∇u1)T.Obstacle problems naturally appear in the nonlinear potential theory and variational inequalities (see [1, 2] and references therein). It can be applied to phase transitions in materials science, flame propagation, combustion theory, crystal growth, optimal control problems, elasto-plastic problems, or financial problems [3, 4]. Reference [5] obtained higher integrability and stability results of weak solutions to Kφ,ψθ,p(A)-obstacle problems under the conditions N=1, f=0, and A satisfies homogeneous conditions. In this paper, we investigate the local and global higher integrability of weak solutions associated with Kφ,ψθ,p(A)-double obstacle systems. This kind of higher integrability has been previously studied in [6] for single obstacle problems (N=1). Our notation is standard.


## 2. Main Results

Let1<p<∞, and s>p. We assume that our mappings A:Ω×RnN→RnN, f:Ω×RN→RnN are Caratheodory functions and satisfy the following conditions for fixed 0<α<β<∞, 0<λ<∞:(A1)
for allh∈RnN and a.e.x∈Ω,
(6)A(x,h)•h≥α|h|p,|A(x,h)|≤β|h|p-1;(A2)
for allu∈RN and a.e.x∈Ω,
(7)|f(x,u)|≤ϕ(x)+λ|u|(p-1)γ.Fixx0∈Ω, let Qr be a cube with center x0 and side length r, and let Qλr(λ>0) be the cube parallel to Qr with the same center as Qr and side length λr. We denote
(8)fr≜⨍Qrfdx≜1|Qr|∫Qr‍fdx,
where |Qr| denotes the Lebesgue Measure of Qr.Theorem 2.
Suppose thatφ,ψ,θ∈W1,s(Ω,RN)(s>p), and let u∈Kφ,ψθ,p(Ω,RN) be a weak solution to Kφ,ψθ,p(A)-double obstacle systems under conditions (A1) and (A2) with 1≤γ<n/(n-p), ϕ(x)∈Ls/(p-1)(Ω,RN), then there exists a constant 0<ϵ0=ϵ0(n,N,p,s,α,β,γ,λ,diam(Ω))<s-p such that, for each ϵ∈[0,ϵ0), one has u∈Wloc1,p+ϵ(Ω,RN). Furthermore, for every x0∈Ω and every cube Qr⊂Ω (r<r0 small enough) centered at x0 such that Q2r⊂⊂Ω, one has
(9)[⨍Qr(|∇u|+|u|γ)p+ϵdx]1/(p+ϵ)≤C{[⨍Q2r(|∇u|+|u|γ)pdx]1/p+[⨍Q2rHsdx]1/s},
where H=|∇φ|+|∇ψ|+|ϕ|1/(p-1) and C=C(n,N,p,s,α,β,γ,λ,diam(Ω))<∞.In order to obtain the global higher integrability of weak solutions toKφ,ψθ,p(A)-double obstacle systems, it seems that we need to impose some regularity condition for ∂Ω, the boundary of Ω. We say that ∂Ω is p-Poincaré thick if there exists 0<a<∞ such that, for all open cubes Qr⊂Rn with side length r>0, there holds
(10)(∫Q2r|u|pdx)1/p≤ax(∫Q2r|∇u|pn/(p+n)dx)(p+n)/pn,
whenever u∈W1,p(Q2r), u=0,a.e. on (Rn∖Ω)∩Q2r and Q3r/2∩ΩC≠∅. Theorem 2.3 and Corollary 2.7 in [7] have given some simple conditions such that (10) holds for p≥n/(n-1).Theorem 3.
Suppose that the boundary∂Ω of Ω is p-Poincaré thick with p>n/(n-1), and φ,ψ,θ∈W1,s(Ω,RN)(s>p). If u∈Kφ,ψθ,p(Ω,RN) is a weak solution to Kφ,ψθ,p(A)-double obstacle systems under conditions (A1) and (A2) with 1≤γ<n/(n-p), ϕ(x)∈Ls/(p-1)(Ω,RN), then there exists a constant 0<ϵ0=ϵ0(n,N,p,s,a,α,β,γ,λ,diam(Ω))<s-p such that for each ϵ∈[0,ϵ0), one has u∈W1,p+ϵ(Ω,RN). Furthermore, we have
(11)⨍Ω(|∇u|+|u|γ)p+ϵdx]1/(p+ϵ)≤C{[⨍Ω(|∇u|+|u|γ)pdx]1/p+[⨍ΩHsdx]1/s},
where H=|∇φ|+|∇ψ|+|∇θ|+|ϕ|1/(p-1) and C=C(n,N,p,s,a,α,β,γ,λ,diam(Ω))<∞.


## 3. Proofs of Main Results

The following lemma is due to Giaquinta and Modica [8].Lemma 4 (Reverse Hölder’s inequality).
LetQ be an n-cube and g,G be two nonnegative functions defined on Q. Suppose that for each x0∈Q and each r<min{(1/2)dist(x0,∂Q),r0}(12)⨍Qrgpdx≤b(⨍Q2rgdx)p+τ⨍Q2rgpdx+⨍Q2rGpdx,
where constants b>1, r0>0, 0≤τ<1. Then g∈Llocq(Q) for q∈[p,p+ϵ) and
(13)(⨍Qrgqdx)1/q≤C(⨍Q2rgpdx)1/p+(⨍Q2rGqdx)1/q,
for Q2r⊂⊂Q, r<r0, where C and ϵ are positive constants depending only on b,τ,p,n.The proofs of Theorems2 and 3 are stimulated by [5]. The general constant C denotes a constant whose value may change even on the same line.Proof of Theorem2.
For any fixedx0∈Ω and cube Qr centered at x0 with side length r such that Q2r⊂⊂Ω, let η∈C0∞(Q2r) be a cutoff function such that 0≤η≤1,|∇η|≤C/r and η≡1 on Qr.
Letv=(1-ηp)(u-u2r)+ηpw, where w=(ψ-u2r)-+min((φ-u2r)+,(ψ-u2r)+). Due to the boundedness of Ω, we have φ,ψ∈W1,p(Ω,RN). Moreover, v+u2r∈Kφ,ψθ,p(Ω,RN) because
(14)w={(φ-u2r)+,ψ≥u2r,ψ-u2r,ψ<u2r,
and this yields φ≤v+u2r≤ψa.e. in Ω. Hence we have, by (5),
(15)∫Ω‍(A(x,∇u)-f(x,u))•∇(v-u)dx≥0.
By the choice ofv, we get
(16)v-u=-u2r-ηp(u-u2r)+ηpw,
and hence
(17)∇(v-u)=-ηp∇u+ηp∇w+pηp-1∇η⊗[w-(u-u2r)].
This and (15), together with the structure assumptions (A1), (A2), yield
(18)α∫Q2r‍ηp|∇u|pdx≤∫Q2r‍ηpA(x,∇u)•∇udx≤β∫Q2r‍ηp|∇u|p-1|∇w|dx+∫Q2r‍ηp|ϕ||∇u|dx+λ∫Q2r‍ηp|u|(p-1)γ|∇u|dx+pβ∫Q2r‍ηp-1[|w|+|u-u2r|]|∇η||∇u|p-1dx+∫Q2r‍ηp|ϕ||∇w|dx+λ∫Q2r‍ηp|u|(p-1)γ|∇w|dx+p∫Q2r‍ηp-1|ϕ|[|w|+|u-u2r|]|∇η|dx+pλ∫Q2r‍ηp-1|u|(p-1)γ[|w|+|u-u2r|]|∇η|dx.
Using Hölder’s inequality, Young’s inequality, and Minikowski’s inequality, we can estimate each term in (18) as follows:
(19)β∫Q2r‍ηp|∇u|p-1|∇w|dx≤β(∫Q2r‍ηp|∇u|pdx)(p-1)/p(∫Q2r‍ηp|∇w|pdx)1/p≤α8∫Q2r‍ηp|∇u|pdx+C∫Q2r‍ηp|∇w|pdx,∫Q2r‍ηp|ϕ||∇u|dx≤α8∫Q2r‍ηp|∇u|pdx+C∫Q2r‍ηp|ϕ|p/(p-1)dx,λ∫Q2r‍ηp|u|(p-1)γ|∇u|dx≤α8∫Q2r‍ηp|∇u|pdx+C∫Q2r‍ηp|u|pγdx,pβ∫Q2r‍ηp-1[|w|+|u-u2r|]|∇η||∇u|p-1dx≤α8∫Q2r‍ηp|∇u|pdx+Cr-p∫Q2r‍[|w|p+|u-u2r|]pdx.∫Q2r‍ηp|ϕ||∇w|dx≤C∫Q2r‍ηp|∇w|pdx+C∫Q2r‍ηp|ϕ|p/(p-1)dx,λ∫Q2r‍ηp|u|(p-1)γ|∇w|dx≤C∫Q2r‍ηp|∇w|pdx+C∫Q2r‍ηp|u|pγdx,p∫Q2r‍ηp-1[|w|+|u-u2r|]|ϕ||∇η|dx≤C∫Q2r‍ηp|ϕ|p/(p-1)dx+Cr-p∫Q2r‍[|w|p+|u-u2r|p]dx,pλ∫Q2r‍ηp-1|u|(p-1)γ[|w|+|u-u2r|]|∇η|dx≤C∫Q2r‍ηp|u|pγdx+Cr-p∫Q2r‍[|w|p+|u-u2r|p]dx.
We deduce from (14) that
(20)|w|≤{|φ-φ2r|,ψ≥u2r,|ψ-ψ2r|,ψ<u2r,|∇w|≤|∇φ|+|∇ψ|.
Hence
(21)⨍Q2rηp|∇u|pdx≤C⨍Q2rηp(|∇φ|+|∇ψ|+|ϕ|1/(p-1))pdx+C⨍Q2rηp|u|pγdx+Cr-p⨍Q2r(|φ-φ2r|p+|ψ-ψ2r|p+|u-u2r|p)dx.
Now choosingmax{1,np/(n+p)}≤t<p and using Poincaré’s inequality and Sobolev-Poincaré’s inequality, we get
(22)r-p⨍Q2r|φ-φ2r|pdx≤C⨍Q2r|∇φ|pdx,(23)r-p⨍Q2r|ψ-ψ2r|pdx≤C⨍Q2r|∇ψ|pdx,(24)r-p⨍Q2r|u-u2r|pdx≤C(⨍Q2r|∇u|tdx)p/t,(25)⨍Q2rηp|u|pγdx≤Cr-n∫Q2r‍|u-u2r|pγdx+C⨍Q2r|u2r|pγdx≤Cr[n((1/pγ)-(1/p))+1]pγ(∫Q2r‍|∇u|pdx)γ-1×⨍Q2r|∇u|pdx+C(⨍Q2r|u|γtdx)p/t≜ξ(r)⨍Q2r|∇u|pdx+C(⨍Q2r|u|γtdx)p/t,
where ξ(r)→0 as r→0 because of the absolute continuity of integrals and the fact that γ≥1.
The above five inequalities imply that(26)⨍Qr|∇u|pdx≤ξ(r)⨍Q2r|∇u|pdx+C(⨍Q2r|u|γtdx)p/t+C⨍Q2r(|∇φ|+|∇ψ|+|ϕ|1/(p-1))pdx+C(⨍Q2r|∇u|tdx)p/t.
To complete our proof, adding⨍Q2r|u|pγdx to each side of (26), using (25), and setting
(27)g=(|∇u|+|u|γ)t,k=pt,H=|∇φ|+|∇ψ|+|ϕ|1/(p-1),G=Ht.
Equation (26) can be rewritten as
(28)⨍Qrgkdx≤Cξ(r)⨍Qrgkdx+C[(⨍Q2rgdx)k+⨍Q2rGkdx].
Forr(r<r0) small enough, we have τ=Cξ(r)<1, and then Lemma 4 implies that there exists 0<ϵ0=ϵ0(n,N,p,s,α,β,γ,λ,diam(Ω))<s-p such that, for 0≤ϵ<ϵ0, we have u∈Wloc1,p+ϵ(Ω,RN), and for every cube Qr(r<r0) such that Q2r⊂Ω, we have
(29)[⨍Qr(|∇u|+|u|γ)p+ϵdx]1/(p+ϵ)≤C{[⨍Q2r(|∇u|+|u|γ)pdx]1/p+[⨍Q2rGsdx]1/s},
where C=C(n,N,p,s,α,β,γ,λ,diam(Ω))<∞.Proof of Theorem3.
Choose a cubeQ0=Q2r0 such that Ω⊂Q0. For an arbitrary cube Q2r⊂Q0, there are two possibilities to consider: (I) Q3r/2⊂Ω, or (II) Q3r/2∩ΩC≠∅.
In the case (I), following the proof of Theorem2, we have
(30)⨍Qrgkdx≤Cξ(r)⨍Q2rgkdx+C[(⨍Q2rgdx)k+⨍Q2rGkdx]
with g=(|∇u|+|u|γ)t, H=|∇φ|+|∇ψ|+|ϕ|1/(p-1), G=Ht in Q2r∩Ω, and g=G=0 in Q2r∖Ω, k=p/t, where max{1,np/(n+p)}≤t<p, and ξ(r)→0 as r→0.
In case (II), observing that replacingθ by θ1=min{ψ,max{φ,θ}}, we may as well assume that the boundary function θ satisfies φ≤θ≤ψ in Ω. Indeed, θ1=(φ-θ)+-(ψ-θ)-+θ, and since 0≤(φ-θ)+≤(u-θ)+∈W01,p(Ω,RN), 0≤-(ψ-θ)-≤(u-θ)-∈W01,p(Ω,RN), the functions (φ-θ)+,-(ψ-θ)-, and hence u-θ1 belongs to W01,p(Ω,RN). Next consider the function v=u-ηp(u-θ) in Ω, where η∈C0∞(Q2r) is a standard test function as in the proof of Theorem 2, then v∈Kφ,ψθ,p(Ω,RN). Indeed, because v-θ∈W01,p and φ≤u≤ψ,  φ≤θ≤ψ  a.e. in Ω, we have
(31)v=(1-ηp)u+ηpθ≥φ,v=(1-ηp)u+ηpθ≤ψa.e.inΩ.
Since
(32)∇v-∇u=-ηp∇u-pηp-1(u-θ)∇η+ηp∇θ,
we have, by (5) and assumptions (A1) and (A2)
(33)α∫Ω‍ηp|∇u|pdx≤∫Ω‍ηpA(x,∇u)•∇udx≤∫Ω‍[(A(x,∇u)-f(x,u))•(ηp∇θ-pηp-1∇η⊗(u-θ))+ηpf(x,u)•∇u]dx≤β∫Ω‍ηp|∇u|p-1|∇θ|dx+∫Ω‍ηp|ϕ||∇θ|dx+∫Ω‍ηp|ϕ||∇u|dx+λ∫Ω‍ηp|u|(p-1)γ|∇θ|dx+λ∫Ω‍ηp|u|(p-1)γ|∇u|dx+pβ∫Ω‍ηp-1|u-θ||∇η||∇u|p-1dx+pβ∫Ω‍ηp-1|ϕ||u-θ||∇η|dx+pβ∫Ω‍ηp-1|u|(p-1)γ|u-θ||∇η|dx≤α2∫Ω‍ηp|∇u|pdx+C∫Ω‍ηp|∇θ|pdx+C∫Ω‍ηp|ϕ|p/(p-1)dx+C∫Ω‍ηp|u|pγdx+C∫Ω‍|u-θ|p|∇η|pdx,
where we have used Hölder’s inequality and Young’s inequality several times. Hence
(34)∫Ω‍ηp|∇u|pdx≤C[∫Ω‍ηp(|∇θ|p+|ϕ|p/(p-1))dx+∫Ω‍ηp|u|pγdx+∫Ω‍|u-θ|p|∇η|pdx],
where the generic constant C is depending only on n,N,p,α,β,λ.
To estimate the last term in (34), we employ the p-Poincaré thickness of ∂Ω. Indeed, the function u-θ can be extended continuously to be 0 to CΩ, and therefore
(35)∫Ω‍|u-θ|p|∇η|pdx≤Cr-px[∫Q2r∩Ω‍|∇(u-θ)|np/(n+p)d](n+p)/n.
Using Minikowski’s inequality and Hölder’s inequality, we obtain the following:(36)r-p[∫Q2r∩Ω‍|∇(u-θ)|np/(n+p)dx](n+p)/n≤Cr-p[(∫Q2r∩Ω‍|∇θ|np/(n+p)dx)(n+p)/np+(∫Q2r∩Ω‍|∇u|np/(n+p)dx)(n+p)/np]p≤C∫Q2r∩Ω‍|∇θ|pdx+Crn-(np/t)(∫Q2r∩Ω‍|∇u|tdx)p/t.
Hence we derive from (25), (34), (35), and (36) that
(37)⨍Qr∩Ω|∇u|pdx≤Cξ(r)⨍Q2r∩Ω|∇u|pdx+C⨍Q2r∩Ω(|∇θ|+|ϕ|1/(p-1))pdx+C[⨍Q2r∩Ω(|u|γt+|∇u|t)dx]p/t.
Adding⨍Qr|u|pγdx to each side of (37) and using (25), setting g=(|∇u|+|u|γ)t, H=|∇φ|+|∇ψ|+|∇θ|+|ϕ|1/(p-1), G=Ht in Q2r∩Ω, g=G=0 in Q2r∖Ω, k=p/t, where max{1,np/(n+p)}≤t<p, we obtain
(38)⨍Qr∩Ωgkdx≤Cξ(r)⨍Q2r∩Ωgkdx+C[(⨍Q2r∩Ωgdx)k+⨍Q2r∩ΩGkdx],
where ξ(r)→0 as r→0 and C=C(n,N,p,s,a,α,β,γ,λ,diam(Ω))>0.
Forr(r<r0) small enough, we have τ=Cξ(r)<1, and then Lemma 4 implies that there exists 0<ϵ0=ϵ0(n,N,p,s,a,α,β,γ,λ,diam(Ω))<s-p such that, for 0≤ϵ<ϵ0, we have u∈W1,p+ϵ(Ω,RN), and (11) holds. Hence the theorem follows.



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