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 $W^2_{p,\loc}(\Omega)$, $\Omega\subset\mathbb R^n$, by Lipschitz continuous functions. We prove that if $f\in W^2_{p,\loc}(\Omega)$, $1\leq p<\infty$, 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 $f \vert_{A_k}$ 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 $W^2_{p,\loc}(\Omega;\mathbb R^n)$ with the Luzin capacity-measure $N$-property.