Existence of energy-minimal diffeomorphisms between doubly connected domains

TL;DR

This paper proves the existence of unique energy-minimal diffeomorphisms between doubly connected planar domains, highlighting differences from harmonic maps and relevance to nonlinear elasticity and computer graphics.

Contribution

It establishes the existence and uniqueness of energy-minimal diffeomorphisms between doubly connected domains without boundary conditions, a novel result in geometric analysis.

Findings

Existence of energy-minimal diffeomorphisms under certain domain moduli conditions.

Energy-minimal diffeomorphisms are harmonic but not necessarily boundary-preserving.

Results have implications for nonlinear elasticity and computer graphics.

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.