Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Partial regularity for elliptic equations

Pages: 899 - 913, Volume 28, Issue 3, November 2010      doi:10.3934/dcds.2010.28.899

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Guji Tian - Wuhan Institute of Physics and Mathematics,Chinese Academy of Sciences, Wuhan 430071, China (email)
Xu-Jia Wang - Centre for Mathematics and Its Applications, Australian National University, Canberra, ACT 0200, Australia (email)

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 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, 35J60.

Received: March 2010;      Revised: April 2010;      Available Online: April 2010.