# Integrability for Solutions of Anisotropic Obstacle Problems

Authors: Hongya Gao; Yanjie Zhang; Shuangli Li
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

---

## Abstract

This paper deals with anisotropic obstacle problem for the𝒜-harmonic equation ∑i=1nDi(ai(x,Du(x)))=0. An integrability result is given under suitable assumptions, which show higher integrability of the boundary datum, and the obstacle force solutions u have higher integrability as well.


---

## Body

## 1. Introduction and Statement of Result

LetΩ be a bounded open subset of Rn. For pi>1, i=1,2,…,n, we denote pm=maxi=1,2,…,npi and p¯ is the harmonic mean of pi, that is,
(1.1)1p¯=1n∑i=1n‍1pi.
The anisotropic Sobolev space W1,(pi)(Ω) is defined by
(1.2)W1,(pi)(Ω)={v∈W1,1(Ω):Div∈Lpi(Ω)foreveryi=1,2,…,n}.Let us consider solutionsu∈W1,(pi)(Ω) of the following 𝒜-harmonic equation:
(1.3)∑i=1n‍Di(ai(x,Du(x)))=0,
where D=(D1,D2,…,Dn) is the gradient operator, and the Carathéodory functions ai(x,ξ):Ω×Rn→R, i=1,2,…,n, satisfy
(1.4)|ai(x,z)|≤c2(h(x)+|zi|)pi-1,
for almost every x∈Ω, for every z∈Rn, and for any i=1,2,…,n, and there exists ν~∈(0,+∞) such that
(1.5)ν~∑i=1n‍|zi-z~i|pi≤∑i=1n‍(ai(x,z)-ai(x,z~))(zi-z~i),
for almost every x∈Ω, for any z,z~∈Rn. The integrability condition for h(x)≥0 in (1.4) will be given later.Letψ be any function in Ω with values in R∪{±∞} and θ∈W1,(pi)(Ω), and we introduce
(1.6)Kψ,θ(pi)(Ω)={v∈W1,(pi)(Ω):v≥ψ,a.e.andv-θ∈W01,(pi)(Ω)}.
Note that
(1.7)W01,(pi)(Ω)={v∈W01,1(Ω):Div∈Lpi(Ω)foreveryi=1,2,…,n}.
The function ψ is an obstacle and θ determines the boundary values.Definition 1.1.
A solution to the𝒦ψ,θ(pi)-obstacle problem is a function u∈𝒦ψ,θ(pi)(Ω) such that
(1.8)∫Ω‍∑i=1n‍ai(x,Du(x))(Div(x)-Diu(x))dx≥0,
whenever v∈𝒦ψ,θ(pi)(Ω).Higher integrability property is important among the regularity theories of nonlinear elliptic PDEs and systems, see the monograph [1] by Bensoussan and Frehse. Meyers and Elcrat [2] first considered the higher integrability for weak solutions of (1.3) in 1975. Iwaniec and Sbordone [3] obtained a regularity result for very weak solutions of the 𝒜-harmonic equation (1.3) by using the celebrated Gehring's Lemma. Global integrability for anisotropic equation is contained in [4]. As far as higher integrability of ∇u is concerned, in problems with nonstandard growth a delicate interplay between the regularity with respect to x and the growth with respect to ξ appears: see [5]. For a global boundedness result of anisotropic variational problems, see [6]. For other related works, see [7]. We refer the readers to the classical books by Ladyženskaya and Ural’ceva [8], Morrey [9], Gilbarg and Trudinger [10] and Giaquinta [11] for some details of isotropic cases.In the present paper, we consider integrability for solutions of anisotropic obstacle problems of the𝒜-harmonic equation (1.3), which show higher integrability of the boundary datum, and the obstacle force solutions u, have higher integrability as well. The idea of this paper comes from [4], and the result can be considered as a generalization of [4, Theorem 2.1].Theorem 1.2.
Letu∈𝒦ψ,θ(pi)(Ω) be a solution to the 𝒦ψ,θ(pi) obstacle problem and θ∈W1,(qi)(Ω), qi∈(pi,+∞), i=1,2,…,n, 0≤h∈Lqm(Ω) with qm=maxi=1,…,nqi, ψ∈[-∞,+∞] is such that θ*=max{ψ,θ}∈θ+W01,(qi)(Ω). Moreover, p¯<n. Then
(1.9)u∈θ*+Lweakt(Ω),
where
(1.10)t=p¯*1-(bp¯*/p¯)(pm/pm-1)>p¯*,
and b is any number verifying
(1.11)0<b≤min⁡j=1,…,n(1-pjqj)(1-1pj),b<pm-1pmp¯p¯*.Remark 1.3.
Take the obstacle functionψ to be minus infinity in Theorem 1.2, and the condition (1.4) replaced by
(1.2)′|ai(x,z)|≤c2(1+|zi|)pi-1
for almost every x∈Ω, for every z∈Rn, and for any i=1,2,…,n, then we arrive at Theorem 2.1 in [4].


## 2. Proof of the Main Theorem

Proof of Theorem1.2.
Letu∈𝒦ψ,θ(pi)(Ω) be a solution to the 𝒦ψ,θ(pi)-obstacle problem. Take θ*=max{ψ,θ}∈θ+W01,(qi)(Ω). Let us consider L∈(0,+∞) and
(2.1)v={θ*-L,foru-θ*<-L,u,for-L≤u-θ*≤L,θ*+L,foru-θ*>L.
Then v∈𝒦ψ,θ(pi)(Ω). Indeed, for the second and the third cases of the above definition for v, we obviously have v≥ψ, and for the first case, u-θ*<-L, we have θ*>u+L≥ψ+L; this implies v=θ*-L≥ψ. Since u=θ*=θ on ∂Ω, then v=u on ∂Ω, this implies v=θ on ∂Ω. By Definition 1.1, one has
(2.2)0≤∫{|u-θ*|>L}‍∑i=1n‍ai(x,Du(x))(Div(x)-Diu(x))dx=∫{|u-θ*|>L}‍∑i=1n‍ai(x,Du(x))(Diθ*(x)-Diu(x))dx.
Monotonicity (1.5) allows us to write
(2.3)ν~∑i=1n‍∫{|u-θ*|>L}‍|Diu(x)-Diθ*(x)|pidx≤∫{|u-θ*|>L}‍∑i=1n‍(ai(x,Du(x))-ai(x,Dθ*(x)))(Diu(x)-Diθ*(x))dx,
which together with (2.2) implies
(2.4)ν~∑i=1n‍∫{|u-θ*|>L}‍|Diu(x)-Diθ*(x)|pidx≤-∫{|u-θ*|>L}‍∑i=1n‍ai(x,Dθ*)(Diu(x)-Diθ*(x))dx.
We now use anisotropic growth (1.4) and the Hölder inequality in (2.4), obtaining that
(2.5)ν~∑i=1n‍∫{|u-θ*|>L}‍|Diu-Diθ*|pidx≤-∑i=1n‍∫{|u-θ*|>L}‍ai(x,Dθ*)(Diu-Diθ*)dx≤c2∑i=1n‍∫{|u-θ*|>L}‍(h+|Diθ*|)pi-1|Diu-Diθ*|dx≤c2∑i=1n‍(∫{|u-θ*|>L}‍(h+|Diθ*|)pidx)(pi-1)/pi(∫{|u-θ*|>L}‍|Diu-Diθ*|pidx)1/pi.
Let ti be such that
(2.6)pi<ti≤qi,
for every i=1,…,n; ti will be chosen later. We use the Hölder inequality as follows:
(2.7)(∫{|u-θ*|>L}‍(h+|Diθ*|)pidx)(pi-1)/pi≤(∫{|u-θ*|>L}‍(h+|Diθ*|)tidx)(pi-1)/ti(|{|u-θ*|>L}|)(ti-pi)(pi-1)/tipi.
The following proof is similar to that of [4, Theorem 2.1]; we only list the necessary changes: instead of [4, (3.14)] by
(2.8)(∫{|u-θ*|>L}‍(h+|Diθ*|)pidx)(pi-1)/pi≤(∫{|u-θ*|>L}‍(h+|Diθ*|)tidx)(pi-1)/ti|{|u-θ*|>L}|b≤M|{|u-θ*|>L}|b,
where
(2.9)M=max⁡j=1,…,n(∫Ω‍(h+|Djθ*|)tjdx)(pj-1)/tj<∞,
and instead of [4, (3.19)] we use anisotropic Sobolev Embedding Theorem for v-u,
(2.10)(∫Ω‍|v-u|p¯*dx)1/p¯*≤c*[∏i=1n‍(∫Ω‍|Di(v-u)|pidx)1/pi]1/n≤c*[∏i=1n‍(∫{|u-θ*|>L}‍|Diu-Diθ*|pidx)1/pi]1/n.
By |v-u|=(|u-θ*|-L)1{|u-θ*|>L}, we obtain
(2.11)(∫{|u-θ*|>L}‍(|u-θ*|-L)p¯*dx)1/p¯*=(∫Ω‍|v-u|p¯*dx)1/p¯*.
Following the idea of the proof of Theorem 2.1 in [4], we complete the proof of Theorem 1.2.



---

*Source: 549285-2012-06-28.xml*