Abstract

The paper establishes the existence of homeomorphisms between two planar domains that minimize the Dirichlet energy. Specifically, among all homeomorphisms f : R -> R* between bounded doubly connected domains such that Mod (R) < Mod (R*) there exists, unique up to conformal authomorphisms of R, an energy-minimal diffeomorphism. No boundary conditions are imposed on f. Although any energy-minimal diffeomorphism is harmonic, our results underline the major difference between the existence of harmonic diffeomorphisms and the existence of the energy-minimal diffeomorphisms. The existence of globally invertible energy-minimal mappings is of primary pursuit in the mathematical models of nonlinear elasticity and is also of interest in computer graphics.

Highlights

  • Throughout this text Ω and Ω∗ will be bounded domains in the complex plane C

  • The primary goal of this paper is to establish the existence of a diffeomorphism f : Ω −on→to Ω∗ of smallest Dirichlet energy

  • The existence of an energy-minimal diffeomorphism f : Ω −on→to Ω∗ may be interpreted as saying that the Cauchy-Riemann equation ∂ ̄f = 0 admits a diffeomorphic solution in the least squares sense, meaning that ∂ ̄f L2 assumes its minimum

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Summary

Introduction

Throughout this text Ω and Ω∗ will be bounded domains in the complex plane C. As we have already pointed out, energy-minimal diffeomorphisms for connected domains are obtained from the Riemann mapping theorem. For any bounded doubly connected domain Ω and any punctured domain Ω∗ there exists an energy-minimal diffeomorphism f : Ω −on→to Ω∗, which is unique up to a conformal change of variables in Ω. Among all homeomorphisms g : Ω∗ −on→to Ω there exists, unique up to a conformal automorphism of Ω, mapping of smallest L1-norm of the distortion. We conclude this introduction with a strategy of the proof of Theorem 1.1.

Statements
Basic properties of deformations
Harmonic replacement
Reich-Walczak-type inequalities
Hopf differentials
Monotonicity of the minimum energy function
Existence
Nonexistence
10. Convexity of the minimum energy function
11. Open questions and conjectures
12. Appendix
Full Text
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