Summability estimates on transport densities with Dirichlet regions on the boundary via symmetrization techniques
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1167-1180.

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 bundary), 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.

DOI : 10.1051/cocv/2017018
Classification : 49J45, 35R06
Mots clés : optimal transport, Monge-Kantorovich system, transport density, symmetrization
Dweik, Samer 1 ; Santambrogio, Filippo 1

1
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Dweik, Samer; Santambrogio, Filippo. Summability estimates on transport densities with Dirichlet regions on the boundary via symmetrization techniques. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1167-1180. doi : 10.1051/cocv/2017018. http://www.numdam.org/articles/10.1051/cocv/2017018/

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