Minimisers and Kellogg’s theorem

Abstract

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 ${\mathscr {C}}^{n,\alpha }$ boundary is ${\mathscr {C}}^{n,\alpha }$ up to the boundary, provided ${{\,\mathrm{Mod}\,}}(D)\geqslant {{\,\mathrm{Mod}\,}}(\Omega )$. If ${{\,\mathrm{Mod}\,}}(D)< {{\,\mathrm{Mod}\,}}(\Omega )$ and $n=1$ we obtain that the diffeomorphic minimizer has ${\mathscr {C}}^{1,\alpha '}$ 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).