Regularity of the homogeneous Monge-Ampère equation
Qi-Rui Li Xu-Jia Wang
Discrete & Continuous Dynamical Systems - A 2015, 35(12): 6069-6084 doi: 10.3934/dcds.2015.35.6069
In this paper, we study the regularity of convex solutions to the Dirichlet problem of the homogeneous Monge-Ampère equation $\det D^2 u=0$. We prove that if the domain is a strip region and the boundary functions are locally uniformly convex and $C^{k+2,\alpha}$ smooth, then the solution is $C^{k+2,\alpha}$ smooth up to boundary. By an example, we show the solution may fail to be $C^{2}$ smooth if boundary functions are not locally uniformly convex. Similar results have also been obtained for the Dirichlet problem on bounded convex domains.
keywords: regularity. Degenerate Monge-Ampère equation
Partial regularity for elliptic equations
Guji Tian Xu-Jia Wang
Discrete & Continuous Dynamical Systems - A 2010, 28(3): 899-913 doi: 10.3934/dcds.2010.28.899
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.
Quasilinear elliptic equations with signed measure
Neil S. Trudinger Xu-Jia Wang
Discrete & Continuous Dynamical Systems - A 2009, 23(1&2): 477-494 doi: 10.3934/dcds.2009.23.477
This paper treats quasilinear elliptic equations in divergence form whose inhomogeneous term is a signed measure. We first prove the existence and continuity of generalized solutions to the Dirichlet problem. The main result of this paper is a weak convergence result, extending previous work of the authors for subharmonic functions and non-negative measures. We also prove a uniqueness result for uniformly elliptic operators and for operators of $p$-Laplacian type.
keywords: Elliptic equation existence weak convergence

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