$L_{loc}^1$-convergence of Jacobians of Sobolev homeomorphisms via area formula

Abstract

We prove that given a sequence of homeomorphisms $f_k:\Omega \to {\mathbb{R}}^n$ convergent in $W^{1,p}(\Omega ,{\mathbb{R}}^n)$, $p\ge 1$ for $n=2$ and $p>n-1$ for $n\ge 3$, to a homeomorphism $f$ which maps sets of measure zero onto sets of measure zero, Jacobians $J{f}_k$ converge to $Jf$ in $L_{loc}^1(\Omega )$. We prove it via Federer's area formula and investigation of when $|f_k(E)|\to |f(E)|$ as $k\to \infty$ for Borel subsets $E⋐\Omega$.

arXiv comment17 pages