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

TL;DR

This paper investigates how functions in second order Sobolev spaces can be approximated by Lipschitz functions on large sets and uses these approximations to establish a change of variables formula for Sobolev mappings.

Contribution

It provides a method to approximate Sobolev functions by Lipschitz functions on sets with full capacity, enabling a new proof of the change of variables formula in Sobolev spaces.

Findings

Existence of Lipschitz approximations on sets with full capacity

Proof of change of variables formula for Sobolev mappings with capacity-measure property

Extension of classical results to second order Sobolev spaces

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.