From Hopf--Lax Formula to Optimal Weak Transfer Plan

Abstract

We study the properties of the Hopf--Lax formula restricted to convex functions and provide a characterization of the optimal transfer plan for weak transport problems on the real line. On $n$ dimensional real space, we also provide a sufficient condition on the potential function such that the optimal plan of the classical Monge--Kantorovich problem does not depend on the cost function. As a byproduct, we establish a link between the combinatorial object (the permutation polytope) and the Hamilton--Jacobi equation.