Calculus of Variations and Geometric Measure Theory

J. Louet - F. Santambrogio

A sharp inequality for transport maps in $W^{1,p}(\mathbb{R})$ via approximation

created by louet on 06 Oct 2011
modified on 05 Jul 2012


Published Paper

Inserted: 6 oct 2011
Last Updated: 5 jul 2012

Journal: Applied Math Letters
Volume: 25
Number: 3
Pages: 648-653
Year: 2012


For $f$ convex and increasing, we prove the inequality $ \int f(
) \geq \int f(nT')$, every time that $U$ is a Sobolev function of one variable and $T$ is the non-decreasing map defined on the same interval with the same image measure as $U$, and the function $n(x)$ takes into account the number of pre-images of $U$ at each point. This may be applied to some variational problems in a mass-transport framework or under volume constraints.

Keywords: calculus of variations, coarea formula, Semi-continuity, Monotone transport, Volume constraints