Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains

Chao Zhang; Lihe Wang; Shulin Zhou; Yun-Ho Kim

doi:10.3934/cpaa.2014.13.2559

Article Contents

2014, Volume 13, Issue 6: 2559-2587. Doi: 10.3934/cpaa.2014.13.2559

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Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains

1.
Department of Mathematics, Harbin Institute of Technology, Harbin 150001
2.
Department of Mathematics, Shanghai Jiaotong University, Shang hai 200240
3.
LMAM, School of Mathematical Sciences, Peking University, Bejing 100871
4.
Department of Mathematics Education, Sangmyung University, Seoul 110--743

Received: March 31, 2014

Revised: April 30, 2014

Published: October 2014

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Abstract

In this paper we consider the global gradient estimates for weak solutions of $p(x)$-Laplacian type equation with small BMO coefficients in a $\delta$-Reifenberg flat domain. The modified Vitali covering lemma, good $\lambda$-inequalities, the maximal function technique and the appropriate localization method are the main analytical tools. The global Caldéron--Zygmund theory for such equations is obtained. Moreover, we generalize the regularity estimates in the Lebesgue spaces to the Orlicz spaces.

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Mathematics Subject Classification: Primary: 35R05, 35J92; Secondary: 35J15, 35J25.

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