Kellogg's theorem for diffeomophic minimisers of Dirichlet energy\n between doubly connected Riemann surfaces

2020/11/09 by David Kalaj, Kalaj, David
Mathematics · #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2011.04629

openalex publication_date 2020/11/09 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We extend the celebrated theorem of Kellogg for conformal diffeomorphisms to\nthe minimizers of Dirichlet energy. Namely we prove that a diffeomorphic\nminimiser of Dirichlet energy of Sobolev mappings between doubly connected\nRiemanian surfaces ( X,\σ) and ( Y,\ρ) having\n mathscrC^n,\α boundary, 0<\α<1, is mathscrC^n,\α up\nto the boundary, provided the metric \ρ is smooth enough. Here n is a\npositive integer. It is crucial that, every diffeomorphic minimizer of\nDirichlet energy is a harmonic mapping with a very special Hopf differential\nand this fact is used in the proof. This improves and extends a recent result\nby the author and Lamel in citekalam, where the authors proved a similar\nresult for double-connected domains in the complex plane but for \α'\nwhich is \≤ \α and \ρ\≡ 1. This is a complementary result of an\nexistence result proved by T. Iwaniec et al. in citeiwa and the author in\n citekal0\n

Related

• Minimisers and Kellogg's theorem
• Lipschitz property of minimisers between double connected surfaces
• Energy-minimal diffeomorphisms between doubly connected Riemann surfaces
• Lipschitz regularity of energy-minimal mappings between doubly connected Riemann surfaces
• Radially symmetry of minimizers to the weighted Dirichlet energy