Summability estimates on transport densities with Dirichlet regions on the boundary via symmetrization techniques

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

^a Corresponding author: amer.dweik@math.u-psud.fr

Received: 7 July 2016
Revised: 22 February 2017
Accepted: 23 February 2017

Abstract

In this paper we consider the mass transportation problem in a bounded domain Ω where a positive mass f⁺ in the interior is sent to the boundary ∂Ω. This problems appears, for instance in some shape optimization issues. We prove summability estimates on the associated transport density σ, which is the transport density from a diffuse measure to a measure on the boundary f⁻ = P_#f⁺ (P being the projection on the boundary), hence singular. Via a symmetrization trick, as soon as Ω is convex or satisfies a uniform exterior ball condition, we prove L^p estimates (if f⁺ ∈ L^p, then σ ∈ L^p). Finally, by a counter-example we prove that if f⁺ ∈ L^∞ (Ω) and f⁻ has bounded density w.r.t. the surface measure on ∂Ω, the transport density σ between f⁺ and f⁻ is not necessarily in L^∞ (Ω), which means that the fact that f⁻ = P_#f⁺ is crucial.