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July  2010, 28(3): 899-913. doi: 10.3934/dcds.2010.28.899

Partial regularity for elliptic equations

Guji Tian 1, and  Xu-Jia Wang 2,

1. 

Wuhan Institute of Physics and Mathematics,Chinese Academy of Sciences, Wuhan 430071, China
2. 

Centre for Mathematics and Its Applications, Australian National University, Canberra, ACT 0200, Australia

Received  March 2010 Revised  April 2010 Published  April 2010

In this paper we study partial and anisotropic Schauder estimates for linear and nonlinear elliptic equations. We prove that if the inhomogeneous term $f$ is Hölder continuous in the $x_n$-direction, then the mixed derivatives uxxn are Hölder continuous; if $f$ satisfies an anisotropic Hölder continuity condition, then the second derivatives $D^2 u$ satisfy related anisotropic Hölder continuity estimates.
Keywords: Partial regularity, elliptic equation, perturbation argument..
Mathematics Subject Classification: 35J15, 35J05, 35J6.
Citation: Guji Tian, Xu-Jia Wang. Partial regularity for elliptic equations. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 899-913. doi: 10.3934/dcds.2010.28.899
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