Beckmann-type problem for degenerate Hamilton-Jacobi equations

Abstract

The aim of this note is to give a Beckmann-type problem as well as the corresponding optimal mass transportation problem associated with a degenerate Hamilton-Jacobi equation H ( x , ∇ u ) = 0 , H(x,\nabla u)=0, coupled with non-zero Dirichlet condition u = g u=g on ∂ Ω \partial \Omega . In this article, the Hamiltonian H H is continuous in both arguments, coercive and convex in the second, but not enjoying any property of existence of a smooth strict sub-solution. We also provide numerical examples to validate the correctness of theoretical formulations.

L1-Theory for Hele-Shaw flow with linear drift

“…And, Dirichlet one for p is connected to the possibility of crossing other part of the boundary (exits) without any charge. One can see [33] for other possibilities of boundary conditions and their interpretations.…”