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: firstname.lastname@example.org
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
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.