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

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}