Partial regularity for elliptic equations

In this paper we study partial and anisotropic Schauder estimatesfor linear and nonlinear elliptic equations. We prove that if theinhomogeneous term $f$ is Hölder continuous in the$x_n$-direction, then the mixed derivatives uxxn are Höldercontinuous; if $f$ satisfies an anisotropic Hölder continuitycondition, then the second derivatives $D^2 u$ satisfy relatedanisotropic Hölder continuity estimates.