Global Integrability of the Reciprocal of Jacobians for Homeomorphisms of Finite Distortion

TL;DR

This paper determines the optimal integrability conditions for the reciprocal of Jacobians in homeomorphisms with finite distortion, with implications for nonlinear elasticity models.

Contribution

It establishes the optimal global integrability degree for the reciprocal of Jacobians in finite distortion homeomorphisms, extending previous results on weak limits.

Findings

Optimal integrability degree obtained

Strengthened previous results on weak limits

Implications for nonlinear elasticity models

Abstract

For a homeomorphism with $p$-integrable distortion, we obtain the optimal global degree of integrability for the reciprocal of its Jacobian determinant. As an application, we strengthen the result of Dole\v{z}alov\'a, Hencl and Mal\'y concerning weak limits of Sobolev homeomorphisms with finite distortion. Such limits represent physically admissible deformations, as they remain injective almost everywhere and thus adhere as closely as possible to the principle of non-interpenetration of matter in mathematical models of nonlinear elasticity.