Calculus of Variations and Geometric Measure Theory
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S. Dweik - F. Santambrogio

$L^p$ bounds for boundary-to-boundary transport densities, and $W^{1,p}$ bounds for the BV least gradient problem in 2D

created by santambro on 06 Mar 2018



Inserted: 6 mar 2018
Last Updated: 6 mar 2018

Year: 2018


The least gradient problem (minimizing the BV norm with given boundary data) is known to be equivalent, in the plane, to the Beckmann minimal-flow problem with source and target measures located on the boundary of the domain. Motivated by this fact, we prove $L^p$ summability results for the solution of the Beckmann problem in this setting, which improve upon previous results where the measures were themselves supposed to be $L^p$. This provides results about the $W^{1,p}$ regularity of the solution of the least gradient problem in uniformly convex domains.

Keywords: transport density, monge-kantorovich, regularity, BV


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