Lipschitz Property of Minimisers Between Double Connected Surfaces
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Lipschitz Property of Minimisers Between Double Connected Surfaces David Kalaj1 Received: 20 March 2019 © Mathematica Josephina, Inc. 2019
Abstract We study the global Lipschitz character of minimisers of the Dirichlet energy of diffeomorphisms between doubly connected domains with smooth boundaries from Riemann surfaces. The key point of the proof is the fact that minimisers are certain Noether harmonic maps, with Hopf differential of special form, a fact invented by Iwaniec et al. (Invent Math 186:667–707, 2011) for Euclidean metric and by the author in Kalaj (Calc Var Partial Differ Equ 51:465–494, 2014) for the arbitrary metric, which depends deeply on a result of Jost (in: Yau (ed) Tsing Hua lectures on geometry and analysis, Taiwan, 1990–91. International Press, Cambridge, 1997). Keywords Minimizers · Lipschitz mappings · Annuli Mathematics Subject Classification Primary 31A05; Secondary 42B30
1 Introduction and Overview Let 0 < r < R, 0 < r∗ < R∗ and let X and Y be two domains in the complex plane C∼ = R2 . Let ρ be a continuous function on the closure of Y. The ρ-Dirichlet energy integral of a mapping h ∈ W 1,2 (X, Y) is defined by ρ
E [h] =
ρ(h(z))Dh(z)2 dz.
(1.1)
X
The central aim of this paper is to get some boundary regularity of the minimizer of the ρ-energy integral of homomorphisms from the Sobolev class W 1,2 (X, Y).
B 1
David Kalaj [email protected] Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b., 81000 Podgorica, Montenegro
123
D. Kalaj
The main result of this paper is Theorem 1.1 Suppose that X and Y are double connected domains in C with C 2 boundaries and let ρ ∈ C 2 (Y) be a real non-vanishing function in the closure of Y. Then every energy minimizing diffeomorphism of ρ-energy between X and Y is Lipschitz continuous up to the boundary of X. However it is not bi-Lipschitz in general. The paper consist of this section and three more sections. In the following subsections, we present three different types of harmonic mappings. Further in the Sect. 2 we make some background and reformulate main result in the therm of harmonic mappings. In Sect. 3 we define the class of (K , K )-quasiconformal mappings and prove that stationary points of the energy take part on this class. In the Sect. 4 we prove the main result. In the last subsection are performed some precise calculations of Lipschitz constants for minimisers of energy for radial metrics and circular annuli. 1.1 Harmonic Mappings Assume that X is domain in R2 (for example X is homeomorphic to an circular annulus {x ∈ R2 |1 < |x| < R}). The classical Dirichlet problem concerns the energy minimal mapping h : X → R2 of the Sobolev class h ∈ h ◦ + W◦1,2 (X, R2 ) whose boundary values are explicitly prescribed by means of a given mapping h ◦ ∈ W 1,2 (X, R2 ). Let us consider the variation h h + η, in which η ∈ C◦∞ (X, R2 ) and → 0, leads to the integral form of the familiar harmonic system of equations
∇ρ, η || Dh || 2 + ρ(h)Dh, Dη = 0, for every η ∈ C◦∞ (X, R2 ). (1.2) X
Equivale
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