Dirichlet problem for evolutionary surfaces of prescribed mean curvature in a non-convex domain

Abstract

We show the existence of solutions for Dirichlet problem of evolutionary surfaces of prescribed mean curvature. Usually the lateral boundary needs to satisfy a kind of convexity, more precisely $H$-convexity condition. But in this article we do not assume it on a portion $S$ of the lateral boundary. Under some assumptions on the exterior forces and the shape of $S$ we prove that the solution satisfies Dirichlet boundary condition on $S$ in a weak sense.