C 1,α Theory for the Prescribed Mean Curvature Equation with Dirichlet Data

Abstract

In this work we study solutions of the prescribed mean curvature equa- tion over a general domain that do not necessarily attain the given boundary data. With such a solution we can naturally associate a current with support in the closed cylinder above the domain and with boundary given by the prescribed boundary data and which inherits a natural minimizing property. Our main result is that its support is a C 1,α manifold-with-boundary, with boundary equal to the prescribed boundary data, provided that both the initial domain and the prescribed boundary data are of class C 1,α .