## Journals

- Advances in Mathematics of Communications
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### Open Access Journals

DCDS

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.
DCDS

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.

DCDS

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.

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