Partial regularity for elliptic equations

Guji Tian; Xu-Jia Wang

doi:10.3934/dcds.2010.28.899

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2010, Volume 28, Issue 3: 899-913. Doi: 10.3934/dcds.2010.28.899

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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
2.
Centre for Mathematics and Its Applications, Australian National University, Canberra, ACT 0200

Received: February 28, 2010

Revised: March 31, 2010

Published: August 2010

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Abstract

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 u_xxn are Hölder continuous; if $f$ satisfies an anisotropic Hölder continuity condition, then the second derivatives $D^2 u$ satisfy related anisotropic Hölder continuity estimates.

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Mathematics Subject Classification: 35J15, 35J05, 35J60.

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