Gradient Young measures generated by quasiconformal maps in the plane

TL;DR

This paper explicitly characterizes Young measures generated by gradients of quasiconformal maps in the plane, extending previous results to a broader class of maps with applications in nonlinear elasticity.

Contribution

It generalizes existing characterizations of Young measures to quasiconformal maps, emphasizing injectivity for non-linear elasticity applications.

Findings

Complete explicit characterization of Young measures for quasiconformal maps

Extension of previous results from quasiregular and bi-Lipschitz maps

Relevance to ensuring injectivity in nonlinear elasticity models

Abstract

In this contribution, we completely and explicitly characterize Young measures generated by gradients of quasiconformal maps in the plane. By doing so, we generalize the results of Astala and Faraco \cite{AstalaFaraco} who provided a similar result for quasiregular maps and Bene\v{s}ov\'a and Kru\v{z}\'ik \cite{bbmk2013} who characterized Young measures generated by gradients of bi-Lipschitz maps. Our results are motivated by non-linear elasticity where injectivity of the functions in the generating sequence is essential in order to assure non-interpenetration of matter.