Characterization of gradient Young measures generated by homeomorphisms in the plane

TL;DR

This paper characterizes the Young measures generated by gradients of bi-Lipschitz, orientation-preserving maps in the plane, providing new insights into nonlinear elasticity problems and establishing existence results for minimizers of certain energy functionals.

Contribution

It offers a novel characterization of gradient Young measures for bi-Lipschitz homeomorphisms, enabling new lower semicontinuity results and existence proofs in nonlinear elasticity.

Findings

Characterization of Young measures for bi-Lipschitz maps in the plane

New weak* lower semicontinuity results for integral functionals

Existence of minimizers for nonpolyconvex energy densities

Abstract

We characterize Young measures generated by gradients of bi-Lipschitz orientation-preserving maps in the plane. This question is motivated by variational problems in nonlinear elasticity where the orientation preservation and injectivity of the admissible deformations are key requirements. These results enable us to derive new weak$^*$ lower semicontinuity results for integral functionals depending on gradients. As an application, we show the existence of a minimizer for an integral functional with nonpolyconvex energy density among bi-Lipschitz homeomorphisms.