Minimisers and Kellogg's theorem

We extend the celebrated theorem of Kellogg for conformal mappings to the minimizers of Dirichlet energy. Namely we prove that a diffeomorphic minimizer of Dirichlet energy of Sobolev mappings between doubly connected domains D and Omega having Cn,alpha boundary is Cn,alpha up to the boundary, provided Mod(D)> Mod(Omega). If Mod(D)<Mod(Omega) and n=1 we obtain that the diffeomorphic minimizer has C1,alpha ' extension up to the boundary, for alpha '=alpha/(2+alpha). It is crucial that, every diffeomorphic minimizer of Dirichlet energy has a very special Hopf differential and this fact is used to prove that every diffeomorphic minimizer of Dirichlet energy can be locally lifted to a certain minimal surface near an arbitrary point inside and at the boundary. This is a complementary result of an existence results proved by Iwaniec et al. (Invent Math 186(3):667-707, 2011).