Regularity of solutions to the Dirichlet problem for Monge-Amp\`ere equations

TL;DR

This paper investigates the regularity of solutions to the Dirichlet problem for Monge-Ampère equations, focusing on Hölder continuity under specific measure and boundary conditions in hyperconvex Lipschitz domains.

Contribution

It establishes Hölder continuity results for solutions with measures having densities in L^p and boundary data in Hölder spaces, extending regularity theory for Monge-Ampère equations.

Findings

Solutions are Hölder continuous under specified measure conditions.

Regularity depends on the measure's density and boundary data smoothness.

Results apply to strongly hyperconvex Lipschitz domains.

Abstract

We study H\"older continuity of solutions to the Dirichlet problem for measures having density in $L^p$, $p>1$, with respect to Hausdorff-Riesz measures of order $2n-2+\epsilon$ for $0<\epsilon \leq 2$, in a bounded strongly hyperconvex Lipschitz domain and the boundary data belongs to $ C^{0,\alpha}(\partial \Omega)$, $ 0<\alpha \leq 1$.