Gradient flow for a class of diffusion equations with Dirichlet boundary data

Abstract

In this paper we provide a variational characterisation for a class of non-linear evolution equations with constant non-negative Dirichlet boundary conditions on a bounded domain as gradient flows in the space of non-negative measures. The relevant geometry is given by the modified Wasserstein distance introduced by Figalli and Gigli that allows for a change of mass by letting the boundary act as a reservoir. We give a dynamic formulation of this distance as an action minimisation problem for curves of non-negative measures satisfying a continuity equation in the spirit of Benamou-Brenier. Then we characterise solutions to non-linear diffusion equations with Dirichlet boundary conditions as metric gradient flows of internal energy functionals in the sense of curves of maximal slope.