Weak limit of homeomorphisms in $W^{1,n-1}$: invertibility and lower semicontinuity of energy

TL;DR

This work studies weak limits of Sobolev homeomorphisms in the borderline space $W^{1,n-1}$ and shows that, under growth conditions on convex energy terms $A$ and $\varphi$, the limit $f$ satisfies the (INV) condition and the Lusin (N) property, with polyconvex energies lower semicontinuous. It introduces a framework combining degree theory, equiintegrability of inverses, and boundary-boundary comparisons to control topological images and cavitation, connecting to prior results by DHM and Henao–Mora-Corral while providing a geometrically flavored approach and new (N)–dependent semicontinuity conclusions. The paper analyzes two main energies, $\mathcal{G}(f)=\int_{\Omega}(|Df|^{n-1}+\varphi(J_f)+(\frac{|Df|^n}{J_f})^{\frac{1}{n-1}})\,dx$ and $\mathcal{F}(f)=\int_{\Omega}(|Df|^{n-1}+A(|\operatorname{cof} Df|)+\varphi(J_f))\,dx$, proving lower semicontinuity and (INV) under the stated hypotheses, with additional commentary on cavitation and the necessity of the Lusin (N) condition. The results furnish a variationally viable class of deformations in nonlinear elasticity at the critical regularity $W^{1,n-1}$.

Abstract

Let $Ω$, $Ω'\subset\mathbb{R}^n$ be bounded domains and let $f_m\colonΩ\toΩ'$ be a sequence of homeomorphisms with positive Jacobians $J_{f_m} >0$ a.e. and prescribed Dirichlet boundary data. Let all $f_m$ satisfy the Lusin (N) condition and $\sup_m \int_Ω(|Df_m|^{n-1}+A(|\text{cof} Df_m|)+φ(J_f))<\infty$, where $A$ and $\varphi$ are positive convex functions. Let $f$ be a weak limit of $f_m$ in $W^{1,n-1}$. Provided certain growth behaviour of $A$ and $\varphi$, we show that $f$ satisfies the (INV) condition of Conti and De Lellis, the Lusin (N) condition, and polyconvex energies are lower semicontinuous.