Mixed boundary condition for the Monge-Kantorovich equation

Abstract

In this work we give some equivalent formulations for the optimization problem \begin{multline*} \max\bigg\{ \int_{\Omega} \xi \,d\mu + \int_{\Gamma_{N}}\xi \,d\nu;\ \xi \in W^{1,\infty}(\Omega) \text{ such that } \\ \xi_{/\Gamma_{D}}= 0,\ |\nabla\xi(x)|\leq 1 \text{ a.e. } x\in \Omega\bigg\}, \end{multline*} where the boundary of $\Omega$ is $\Gamma=\Gamma_{N}\cup\Gamma_{D}$.