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

TL;DR

This paper investigates the weak limit behavior of homeomorphisms in the Sobolev space $W^{1,n-1}$, establishing conditions under which invertibility and energy lower semicontinuity are preserved in the limit.

Contribution

It proves that under certain growth conditions, the weak limit of homeomorphisms retains invertibility, the Lusin (N) condition, and ensures lower semicontinuity of polyconvex energies.

Findings

Weak limits satisfy the (INV) condition of Conti and De Lellis.

Weak limits preserve the Lusin (N) condition.

Polyconvex energies are lower semicontinuous under the given conditions.

Abstract

Let $\Omega$, $\Omega'\subset\mathbb{R}^n$ be bounded domains and let $f_m\colon\Omega\to\Omega'$ 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 $\sup_m \int_{\Omega}(|Df_m|^{n-1}+A(|\text{cof} Df_m|)+\phi(J_f))<\infty$, 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.