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 -convexity condition. But in this article we do not assume it on a portion of the lateral boundary. Under some assumptions on the exterior forces and the shape of we prove that the solution satisfies Dirichlet boundary condition on in a weak sense.
Kazuya HAYASIDA. Yoshiaki IKEDA. "Dirichlet problem for evolutionary surfaces of prescribed mean curvature in a non-convex domain." J. Math. Soc. Japan 56 (4) 1169 - 1201, October, 2004. https://doi.org/10.2969/jmsj/1190905454