$ L^{p, q} $ estimates on the transport density

Samer Dweik

doi:10.3934/cpaa.2019134

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2019, Volume 18, Issue 6: 3001-3009. Doi: 10.3934/cpaa.2019134

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$ L^{p, q} $ estimates on the transport density

Samer Dweik

Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France

Received: August 31, 2018

Revised: February 28, 2019

Published: May 2019

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Abstract

In this paper, we show a new regularity result on the transport density $ \sigma $ in the classical Monge-Kantorovich optimal mass transport problem between two measures, $ \mu $ and $ \nu $, having some summable densities, $ f^+ $ and $ f^- $. More precisely, we prove that the transport density $ \sigma $ belongs to $ L^{p,q}(\Omega) $ as soon as $ f^+,\,f^- \in L^{p,q}(\Omega) $.

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Mathematics Subject Classification: 35B65, 46N10, 49N60.

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