Weighted Beckmann problem with boundary costs

Abstract

We show that a solution to a variant of the Beckmann problem can be obtained by studying the limit of some weighted p − p- Laplacian problems. More precisely, we find a solution to the following minimization problem: min { ∫ Ω k d | w | + ∫ ∂ Ω g − d ν − − ∫ ∂ Ω g + d ν + : w ∈ M d ( Ω ) , ν ∈ M ( ∂ Ω ) , − ∇ ⋅ w = f + ν } , \begin{equation*} \min \bigg \{\int _\Omega k \,\mathrm {d}|w| + \int _{\partial \Omega } g^-\,\mathrm {d}\nu ^- - \int _{\partial \Omega } g^+\,\mathrm {d}\nu ^+\,:\,w \in \mathcal {M}^d(\Omega ),\,\nu \in \mathcal {M}(\partial \Omega ),\,-\nabla \cdot w =f + \nu \bigg \}, \end{equation*} where f , k f,\,k , and g ± g^\pm are given. In addition, we connect this problem to a formulation with Kantorovich potentials with Dirichlet boundary conditions.

Quasi-convex Hamilton--Jacobi equations via limits of Finsler $p$-Laplace problems as $p\to \infty$

“…Variants of Monge-Kantorovich problems with boundary costs were addressed in [23] where the boundary costs can be seen as some import/export taxes. In the same spirit, similar results were obtained in [10] with some weighted Euclidean distance as a cost. The use of PDE techniques à la Evans-Gangbo in the Finsler framework was addressed recently in [18].…”

“…and the systems (2.3)-(3.11) reduce the ones studied in [10]. Moreover, if the Finsler metric is defined via the so called Minkowski functional (or gauge function)…”

“…Let us recall that we can derive a dual problem to (KR) H using perturbation techniques (as in [10,13]), to get the following Kantorovich problem (K) H : min…”

Optimal transport approach to Sobolev regularity of solutions to the weighted least gradient problem

“…where f := ∂ τ g. In particular, we have that u is a solution for Problem (wLGP) if and only if the corresponding v := R π 2 Du is an optimal flow for Problem (wBP) with |v|(∂Ω) = 0. Yet, one can also show that Problem (wBP) is equivalent to the following optimal transport problem with Riemannian cost (see also [12,33]):…”

Optimal free export/import regions

“…But, as the total mass of f + can be different than the one for f − , we are allowed to export or import masses from the boundary ∂Ω paying two additional costs on the boundary (called export/import taxes) g + (x) and g − (y), for each unit of mass that comes out/enters at some point of ∂Ω. We note that this problem has already been considered in many papers [8,3,4,5,2]. In other words, we consider the following problem (1.1) min…”