Minimisers and Kellogg's theorem

TL;DR

This paper extends Kellogg's theorem to minimizers of Dirichlet energy in Sobolev mappings between doubly connected domains, establishing boundary regularity results based on conformal modulus conditions.

Contribution

It proves boundary regularity of Dirichlet energy minimizers in doubly connected domains, generalizing Kellogg's theorem to Sobolev mappings with specific boundary smoothness.

Findings

Minimizers are $ ext{C}^{n, ext{alpha}}$ up to boundary when $ ext{Mod}(D) extgreater= ext{Mod}( ext{Omega})$.

For $ ext{Mod}(D)< ext{Mod}( ext{Omega})$, minimizers have $ ext{C}^{1, ext{alpha}'}$ extension.

Every minimizer's Hopf differential relates to minimal surface theory.

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 $\mathscr{C}^{n,\alpha}$ boundary is $\mathscr{C}^{n,\alpha}$ 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 $\mathscr{C}^{1,\alpha'}$ 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.