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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Regularity of the homogeneous Monge-Ampère equation

Pages: 6069 - 6084, Volume 35, Issue 12, December 2015      doi:10.3934/dcds.2015.35.6069

 
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Qi-Rui Li - Centre for Mathematics and Its Applications, the Australian National University, Canberra, ACT 0200, Australia (email)
Xu-Jia Wang - Centre for Mathematics and Its Applications, Australian National University, Canberra, ACT 0200, Australia (email)

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.

Received: September 2013;      Revised: February 2014;      Available Online: May 2015.

 References