Optimal $C^{1,\frac{1}{2}}$-regularity of $H$-surfaces with a free boundary

TL;DR

This paper establishes the optimal regularity of $H$-surfaces with free boundaries on $C^2$ manifolds, showing they are $C^{1,1/2}$ up to the boundary, even with non-perpendicular contact and boundary manifolds.

Contribution

It proves the optimal $C^{1,1/2}$ regularity of $H$-surfaces with free boundaries on $C^2$ manifolds, including non-perpendicular contact and boundary manifolds.

Findings

$H$-surfaces are $C^{1,1/2}$ up to the boundary.

Regularity result is optimal, matching known examples.

Includes cases with non-perpendicular boundary contact.

Abstract

We prove that a surface of prescribed mean curvature ($H$-surface) with free boundary on a two-dimensional $C^2$-manifold belongs to $C^{1,\frac{1}{2}}$ up to that the boundary, provided it is a-priori continuous. We allow the $H$-surface to meet the manifold non-perpendicularly and the manifold itself to have a boundary. Our result is optimal according to an example of Hildebrandt and Nitsche.