# Monotone Hopf-Harmonics

## TL;DR

This paper introduces monotone Hopf-harmonics as a new class of mappings in 2D that generalize harmonic homeomorphisms, motivated by nonlinear elasticity, and explores their properties and applications.

## Contribution

It establishes monotone Hopf-harmonics as a novel concept in Geometric Function Theory, extending harmonic homeomorphisms to more general target domains and linking them to nonlinear elasticity.

## Key findings

- Monotone Hopf-harmonics are valid generalizations of harmonic homeomorphisms.
- These mappings are continuous, monotone, and Sobolev class $ ext{ extbackslash mathscr W}^{1,2}_{{loc}}$.
- They serve as legitimate deformations in nonlinear elasticity theory.

## Abstract

The present paper introduces the concept of monotone Hopf-harmonics in $2D$ as an alternative to harmonic homeomorphisms. It opens a new area of study in Geometric Function Theory (GFT). Much of the foregoing is motivated by the principle of non-interpenetration of matter in the mathematical theory of Nonlinear Elasticity (NE). The question we are concerned with is whether or not a Dirichlet energy-minimal mapping between Jordan domains with a prescribed boundary homeomorphism remains injective in the domain. The classical theorem of Rad\'{o}-Kneser-Choquet asserts that this is the case when the target domain is convex. An alternative way to deal with arbitrary target domains is to minimize the Dirichlet energy subject to only homeomorphisms and their limits. This leads to the so called Hopf-Laplace equation. Among its solutions (some rather surreal) are continuous monotone mappings of Sobolev class $\mathscr W^{1,2}_{{loc}}$, called monotone Hopf-harmonics. It is at the heart of the present paper to show that such solutions are correct generalizations of harmonic homeomorphisms and, in particular, are legitimate deformations of hyperelastic materials in the modern theory of NE. We make this clear by means of several examples.