Kellogg's theorem for diffeomophic minimisers of Dirichlet energy between doubly connected Riemann surfaces

Abstract: We extend the celebrated theorem of Kellogg for conformal diffeomorphisms to the minimizers of Dirichlet energy. Namely we prove that a diffeomorphic minimiser of Dirichlet energy of Sobolev mappings between doubly connected Riemanian surfaces $(\X,\sigma)$ and $(\Y,\rho)$ having $\mathscr{C}^{n,\alpha}$ boundary, $0<\alpha<1$, is $\mathscr{C}^{n,\alpha}$ up to the boundary, provided the metric $\rho$ is smooth enough. Here $n$ is a positive integer. It is crucial that, every diffeomorphic minimizer of Dirichlet energy is a harmonic mapping with a very special Hopf differential and this fact is used in the proof. This improves and extends a recent result by the author and Lamel in \cite{kalam}, where the authors proved a similar result for double-connected domains in the complex plane but for $\alpha'$ which is $\le \alpha$ and $\rho\equiv 1$. This is a complementary result of an existence result proved by T. Iwaniec et al. in \cite{iwa} and the author in \cite{kal0}