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

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