Volume 24, Number 3, July–September 2018
|Page(s)||1167 - 1180|
|Published online||27 June 2018|
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,
Orsay Cedex, France
a Corresponding author: email@example.com
Revised: 22 February 2017
Accepted: 23 February 2017
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 Lp estimates (if f+ ∈ Lp, then σ ∈ Lp). 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.
Mathematics Subject Classification: 49J45 / 35R06
Key words: optimal transport / Monge-Kantorovich system / transport density / symmetrization
© EDP Sciences, SMAI 2018
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