  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}
