2016/06/02 by Samer Dweik, Filippo Santambrogio, Dweik, Samer +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Point processes and geometric inequalities #math.AP #math.FA #math.OC
paper · pdf · doi:10.48550/arxiv.1606.00705
arxiv created 2016/06/02 · openalex publication_date 2016/06/02 · arxiv updated 2016/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, appearing for instance in some shape optimization problems, and 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 counterexample 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.