Integrability for Solutions of Anisotropic Obstacle Problems

Correspondence should be addressed to Hongya Gao, ghy@hbu.edu.cn Received 30 March 2012; Revised 26 April 2012; Accepted 23 May 2012 Academic Editor: Martino Bardi Copyright q 2012 Hongya Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper deals with anisotropic obstacle problem for the A-harmonic equation ∑n i 1Di 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.


Introduction and Statement of Result
Let Ω be a bounded open subset of R n .For p i > 1, i 1, 2, . . ., n, we denote p m max i 1,2,...,n p i and p is the harmonic mean of p i , that is, The anisotropic Sobolev space W 1, p i Ω is defined by Let us consider solutions u ∈ W 1, p i Ω of the following A-harmonic equation: where D D 1 , D 2 , . . ., D n is the gradient operator, and the Carathéodory functions a i x, ξ : for almost every x ∈ Ω, for every z ∈ R n , and for any i 1, 2, . . ., n, and there exists ν ∈ 0, ∞ such that for almost every x ∈ Ω, for any z, z ∈ R n .The integrability condition for h x ≥ 0 in 1.4 will be given later.
Let ψ be any function in Ω with values in R ∪ {±∞} and θ ∈ W 1, p i Ω , and we introduce The function ψ is an obstacle and θ determines the boundary values.
Definition 1.1.A solution to the K 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 the A-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 . where and b is any number verifying

1.11
Remark 1.3.Take the obstacle function ψ to be minus infinity in Theorem 1.2, and the condition 1.4 replaced by for almost every x ∈ Ω, for every z ∈ R n , and for any i 1, 2, . . ., n, then we arrive at Theorem 2.1 in 4 .

Monotonicity 1.5 allows us to write
which together with 2.2 implies

2.4
We now use anisotropic growth 1.4 and the H ölder inequality in 2.4 , obtaining that

2.5
Let t i be such that for every i 1, . . ., n; t i will be chosen later.We use the H ölder inequality as follows:

2.7
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 . 1
Then v ∈ Kp i ψ,θ 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 International Journal of Mathematics and Mathematical Sciences 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 Following the idea of the proof of Theorem 2.1 in 4 , we complete the proof of Theorem 1.2.