# American Institute of Mathematical Sciences

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

## Partial regularity for elliptic equations

 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.
Citation: Guji Tian, Xu-Jia Wang. Partial regularity for elliptic equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 899-913. doi: 10.3934/dcds.2010.28.899
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