Monotone Hopf-Harmonics

We introduce the concept of monotone Hopf-harmonics in 2D as an alternative to harmonic homeomorphisms. 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 Radó–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 Wloc1,2, 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.