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

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