Monotone Sobolev Mappings of planar domains and surfaces

TL;DR

This paper proves that monotone Sobolev mappings between surfaces can be approximated by homeomorphisms and establishes the existence of energy-minimal deformations within this class, bridging geometric function theory and nonlinear elasticity.

Contribution

It demonstrates that monotone Sobolev mappings are limits of homeomorphisms and diffeomorphisms, providing a new approximation result in the context of Sobolev spaces and energy minimization.

Findings

Monotone Sobolev mappings are limits of homeomorphisms and diffeomorphisms.

Existence of energy-minimal deformations within Sobolev monotone mappings.

Approximation theorem connecting monotone maps and energy-minimizing sequences.

Abstract

An approximation theorem of Youngs (1948) asserts that a continuous map between compact oriented topological 2-manifolds (surfaces) is monotone if and only if it is a uniform limit of homeomorphisms. Analogous approximation of Sobolev mappings is at the very heart of Geometric Function Theory (GFT) and Nonlinear Elasticity (NE). In both theories the mappings in question arise naturally as weak limits of energy-minimizing sequences of homeomorphisms. As a result of this, the energy-minimal mappings turn out to be monotone. In the present paper we show that, conversely, monotone mappings in the Sobolev space $W^{1,p}, 1<p < \infty$, are none other than $W^{1,p}$-weak (also strong) limits of homeomorphisms. In fact, these are limits of diffeomorphisms. By way of illustration, we establish the existence of energy-minimal deformations within the class of Sobolev monotone mappings for $\,p\,$-harmonic type energy integrals.