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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. [ABSTRACT FROM AUTHOR]

We consider a system of differential equations of Monge-Kantorovich type which describes the equilibrium configurations of granular material poured by a constant source on a network. Relying on the definition of viscosity solution for Hamilton-Jacobi equations on networks introduced in [P.-L. Lions and P. E. Souganidis, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27 (2016), pp. 535-545], we prove existence and uniqueness of the solution of the system and we discuss its numerical approximation. Some numerical experiments are carried out. [ABSTRACT FROM AUTHOR]