On Lipschitz approximations in second order Sobolev spaces and the change of variables formula

Abstract

In this paper we study approximations of functions of Sobolev spaces $W^2_{p,\loc}(Ω)$, $\Omega \subset {\mathbb{R}}^n$, by Lipschitz continuous functions. We prove that if $f\in W^2_{p,\loc}(Ω)$, $1\le p<\infty$, then there exists a sequence of closed sets $\{A_k{\}}_{k=1}^{\infty },A_k\subset A_{k+1}\subset \Omega$, such that the restrictions $f{|}_{A_k}$ are Lipschitz continuous functions and $\cp_p\left(S\right)=0$, $S=\Omega \setminus {\bigcup }_{k=1}^{\infty }{A}_k$. Using these approximations we prove the change of variables formula in the Lebesgue integral for mappings of Sobolev spaces $W^2_{p,\loc}(Ω;\mathbb R^n)$ with the Luzin capacity-measure $N$-property.

arXiv comment12 pages