Weak limit of homeomorphisms in $W^{1,n-1}$ and (INV) condition

TL;DR

This paper investigates the weak limits of Sobolev homeomorphisms in three dimensions, establishing invertibility almost everywhere under certain conditions and exploring the differences between weak and strong limits.

Contribution

It proves that weak limits of Sobolev homeomorphisms in 3D satisfy the (INV) condition, extending previous planar results and providing a sharpness example.

Findings

Weak limits satisfy the (INV) condition and are invertible a.e.

The class of weak limits differs from strong limits in 3D Sobolev homeomorphisms.

An example demonstrates the sharpness of the integrability condition on the Jacobian.

Abstract

Let $\Omega,\Omega'\subset\mathbb{R}^3$ be Lipschitz domains, let $f_m:\Omega\to\Omega'$ be a sequence of homeomorphisms with prescribed Dirichlet boundary condition and $\sup_m \int_{\Omega}(|Df_m|^2+1/J^2_{f_m})<\infty$. Let $f$ be a weak limit of $f_m$ in $W^{1,2}$. We show that $f$ is invertible a.e., more precisely it satisfies the (INV) condition of Conti and De Lellis and thus it has all the nice properties of mappings in this class. Generalization to higher dimensions and an example showing sharpness of the condition $1/J^2_f\in L^1$ are also given. Using this example we also show that unlike the planar case the class of weak limits and the class of strong limits of $W^{1,2}$ Sobolev homeomorphisms in $\mathbb{R}^3$ are not the same.