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 minimiser of Dirichlet energy of Sobolev mappings between double connected domains $D$ and $\Omega$ having ${\mathcal{C}}^{n,\alpha }$ boundary is ${\mathcal{C}}^{n,\alpha }$ up to the boundary, provided $\text{Mod}(D)\ge \text{Mod}(\Omega )$. If $\text{Mod}(D)<\text{Mod}(\Omega )$ and $n=1$ we obtain that the diffeomorphic minimiser has ${\mathcal{C}}^{1,{\alpha }^{\prime }}$ extension up to the boundary, for ${\alpha }^{\prime }=\alpha /(2+\alpha )$. 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.

26 pages. This version is published in Annalen