$C^{1,\alpha}$ theory for the prescribed mean curvature equation with Dirichlet data

TL;DR

This paper proves that solutions to the prescribed mean curvature equation with $C^{1,eta}$ boundary data are $C^{1,eta}$ manifolds-with-boundary, even when solutions do not necessarily attain the boundary data, under certain regularity conditions.

Contribution

It establishes $C^{1,eta}$ regularity of the support of associated currents for solutions with prescribed boundary data, extending regularity results to non-attaining solutions.

Findings

Support of the associated current is a $C^{1,eta}$ manifold-with-boundary.

Supports the prescribed boundary data exactly when the data is $C^{1,eta}$.

Solutions exhibit $C^{1,eta}$ regularity under the given conditions.

Abstract

In this work we study solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. To 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 $C^{1,\alpha}$ 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 $C^{1,\alpha}$.