Calculus of Variations and Geometric Measure Theory

A. Figalli

Regularity properties of optimal maps between nonconvex domains in the plane

created by figalli on 27 Aug 2009
modified on 19 Jan 2010


Accepted Paper

Inserted: 27 aug 2009
Last Updated: 19 jan 2010

Journal: Comm. Partial Differential Equations
Year: 2009


Given two bounded open subsets $\Omega,\Lambda \subset \mathbb{R}^2$, and two densities $f$ and $g$ concentrated on $\Omega$ and $\Lambda$ respectively, we investigate the regularity of the optimal map $\nabla \varphi$ sending $f$ onto $g$. We show that, if $f$ and $g$ are both bounded away from zero and infinity, then we can find two 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 homeomorphism. Moreover, if $f$ and $g$ are smooth, then $\nabla \varphi$ is a smooth diffeomorphism between $\Omega'$ and $\Lambda'$. Finally, we give a quite precise description of the singular set of $\varphi$, showing that it is a $1$-dimensional manifold of class $C^1$ out of a countable set.