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

Abstract

Let $\Omega$, ${\Omega }^{\prime }\subset {\mathbb{R}}^n$ be bounded domains and let $f_m:\Omega \to {\Omega }^{\prime }$ 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 ${\sup }_m{\int }_{\Omega }(|D{f}_m{|}^{n-1}+A(|\text{cof}D{f}_m|)+\varphi (J_f))<\infty$, where $A$ and $\phi$ 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 $\phi$, we show that $f$ satisfies the (INV) condition of Conti and De Lellis, the Lusin (N) condition, and polyconvex energies are lower semicontinuous.

Cited by in corpus (1)

• $(INV)$ condition and regularity of the inverse