In this paper we consider the mass transportation problem in a bounded domain \n \Omega \n where a positive mass \n {f}^{+}\n in the interior is sent to the boundary \n \partial \Omega \n. This problems appears, for instance in some shape optimization issues. We prove summability estimates on the associated transport density \n \sigma \n, which is the transport density from a diffuse measure to a measure on the boundary \n {f}^{-}={P}_{\#}{f}^{+}(P\n being the projection on the bundary), hence singular. Via a symmetrization trick, as soon as \n \Omega \n is convex or satisfies a uniform exterior ball condition, we prove \n {L}^{p}\n estimates (if ${f}^{+}\in {L}^{p}$ then \n \sigma \in {L}^{p}\n). Finally, by a counter-example we prove that if \n {f}^{+}\in {L}^{\infty}\left(\Omega \right)\n and \n {f}^{-}\n has bounded density w.r.t. the surface measure on \n \partial \Omega \n, the transport density \n \sigma \n between \n {f}^{+}\n and \n {f}^{-}\n is not necessarily in \n {L}^{\infty}\left(\Omega \right)\n, which means that the fact that \n {f}^{-}={P}_{\#}{f}^{+}\n is crucial.
Keywords: optimal transport, Monge-Kantorovich system, transport density, symmetrization
@article{COCV_2018__24_3_1167_0, author = {Dweik, Samer and Santambrogio, Filippo}, title = {Summability estimates on transport densities with {Dirichlet} regions on the boundary via symmetrization techniques}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1167--1180}, publisher = {EDP-Sciences}, volume = {24}, number = {3}, year = {2018}, doi = {10.1051/cocv/2017018}, mrnumber = {3877197}, zbl = {1405.49036}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017018/} }
TY - JOUR AU - Dweik, Samer AU - Santambrogio, Filippo TI - Summability estimates on transport densities with Dirichlet regions on the boundary via symmetrization techniques JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1167 EP - 1180 VL - 24 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017018/ DO - 10.1051/cocv/2017018 LA - en ID - COCV_2018__24_3_1167_0 ER -
%0 Journal Article %A Dweik, Samer %A Santambrogio, Filippo %T Summability estimates on transport densities with Dirichlet regions on the boundary via symmetrization techniques %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1167-1180 %V 24 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017018/ %R 10.1051/cocv/2017018 %G en %F COCV_2018__24_3_1167_0
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, Volume 24 (2018) no. 3, pp. 1167-1180. doi : 10.1051/cocv/2017018. http://archive.numdam.org/articles/10.1051/cocv/2017018/
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