An alternative solvability criterion for the Dirichlet problem for the minimal surface equation and an application to the mean curvature flow

Abstract

We propose an alternative condition for the solvability of the Dirichlet problem for the minimal surface equation that applies to non-mean convex domains. We introduce a structural condition, obtained from a second-order ordinary differential equation, which allows the construction of explicit boundary barriers and it can also be applied to unbounded domains. In the setting of Hadamard manifolds, this condition relates the geometry of the domain to the admissible boundary data in a direct way. In Euclidean space, the condition leads to solvability under geometric hypotheses of a different nature from those in the classical Jenkins-Serrin theory, and in some configurations it applies where the Jenkins-Serrin method does not. A central point of the present approach is that the geometric restrictions and the boundary data enter independently. The same barrier construction can be used for graphical mean curvature flow. This yields short-time existence with prescribed boundary values even when the boundary of the domain is not mean convex. When mean convexity is present, one recovers the classical graphical setting.