Partial regularity of Brenier solutionsof the Monge-Ampère equation

Alessio Figalli; Young-Heon Kim

doi:10.3934/dcds.2010.28.559

Article Contents

2010, Volume 28, Issue 2: 559-565. Doi: 10.3934/dcds.2010.28.559 open access

This issue Previous Article Random dispersal vs. non-local dispersal Next Article Higher integrability for gradients of solutions to degenerate parabolic systems

Partial regularity of Brenier solutions of the Monge-Ampère equation

Alessio Figalli^1, and
Young-Heon Kim^2,

1.
Department of Mathematics, The University of Texas at Austin, Austin TX 78712
2.
Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2

Received: December 2009

Revised: April 2010

Published: July 2010

Abstract Related Papers Cited by

Abstract

Given $\Omega,\Lambda \subset \R^n$ two bounded open sets, and$f$ and $g$ two probability densities concentrated on $\Omega$ and $\Lambda$respectively, we investigate the regularity of the optimal map$\nabla \varphi$ (the optimality referring to the Euclidean quadratic cost) sending $f$ onto $g$. We show that if $f$ and$g$ are both bounded away from zero and infinity, we can findtwo open sets $\Omega'\subset \Omega$ and $\Lambda'\subset\Lambda$ such that $f$ and $g$ are concentrated on $\Omega'$ and$\Lambda'$ respectively, and $\nabla\varphi:\Omega' \to \Lambda'$is a (bi-Hölder) homeomorphism. This generalizes the $2$-dimensional partial regularityresult of [8].

Keywords:

Mathematics Subject Classification: Primary: 35J60; Secondary: 35D10, 49Q20.

Citation:

Article Metrics

HTML views() PDF downloads(278) Cited by(0)

Partial regularity of Brenier solutions of the Monge-Ampère equation

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

Partial regularity of Brenier solutions of the Monge-Ampère equation

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content