Minimisers and Kellogg's theorem

2019/08/27 by Kalaj, David, Lamel, Bernhard
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1908.10106

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 Ω having \mathscrC^n,α boundary is \mathscrC^n,α up to the boundary, provided Mod(D)≥ Mod(Ω). If Mod(D)< Mod(Ω) and n=1 we obtain that the diffeomorphic minimiser has \mathscrC^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.

Related

• Kellogg's theorem for diffeomophic minimisers of Dirichlet energy\n between doubly connected Riemann surfaces
• Lipschitz property of minimisers between double connected surfaces
• Lipschitz regularity of energy-minimal mappings between doubly connected Riemann surfaces
• Energy-minimal diffeomorphisms between doubly connected Riemann surfaces
• Stability of Minimising Harmonic Maps under W^1,p Perturbations of Boundary Data: p≥ 2