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

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.