Regularity of the homogeneous Monge-Ampère equation

Qi-Rui  Li; Xu-Jia Wang

doi:10.3934/dcds.2015.35.6069

Article Contents

2015, Volume 35, Issue 12: 6069-6084. Doi: 10.3934/dcds.2015.35.6069

This issue Previous Article Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations Next Article Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials

Regularity of the homogeneous Monge-Ampère equation

Qi-Rui Li^1, and
Xu-Jia Wang^2,

1.
Centre for Mathematics and Its Applications, the Australian National University, Canberra, ACT 0200
2.
Centre for Mathematics and Its Applications, Australian National University, Canberra, ACT 0200

Received: September 2013

Revised: February 2014

Published: December 2015

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Abstract

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:

Degenerate Monge-Ampère equation,
regularity.

Mathematics Subject Classification: Primary: 35J96; Secondary: 35J70.

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