Optimal regularity of minimal graphs in the hyperbolic space

Abstract

We discuss the global regularity of solutions $f$ to the Dirichlet problem for minimal graphs in the hyperbolic space when the boundary of the domain $\Omega \subset {\mathbb{R}}^n$ has a nonnegative mean curvature and prove an optimal regularity $f\in C^{\frac{1}{n+1}}(\bar{\Omega })$. We can improve the Hölder exponent for $f$ if certain combinations of principal curvatures of the boundary do not vanish, a phenomenon observed by F.-H. Lin.

Accepted by Calc. Var. Partial Differential Equations