Optimal regularity for $A$-harmonic type equationsunder the natural growth

Shenzhou Zheng; Xueliang Zheng; Zhaosheng Feng

doi:10.3934/dcdsb.2011.16.669

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2011, Volume 16, Issue 2: 669-685. Doi: 10.3934/dcdsb.2011.16.669

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Optimal regularity for $A$-harmonic type equations under the natural growth

1.
Department of Mathematics, Beijing Jiaotong University, Beijing 100044
2.
Department of Mathematics, Taizhou University, Linhai, Zhejiang 317000
3.
Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539

Received: December 31, 2009

Revised: September 30, 2010

Published: August 2011

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Abstract

In this paper we are concerned with a class of nonlinear degenerate elliptic equations under the natural growth. We show that each bounded weak solution of $A$-harmonic type equations under the natural growth belongs to local Hölder continuity based on a density lemma and the Moser-Nash's argument. Then we show that its weak solution is of optimal regularity with the Hölder exponent for any $\gamma$: $0\le \gamma<\kappa$, where $\kappa$ is the same as the Hölder's index for homogeneous $A$-harmonic equations.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 35B45, 35D30.

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