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 Wp,loc2(Ω), Ω⊂Rn, by Lipschitz continuous functions. We prove that if f∈Wp,loc2(Ω), 1≤p<∞, then there exists a sequence of closed sets {Ak}k=1∞,Ak⊂Ak+1⊂Ω, such that the restrictions f|Ak are Lipschitz continuous functions and capp(S)=0, S=Ω∖⋃k=1∞Ak. Using these approximations we prove the change of variables formula in the Lebesgue integral for mappings of Sobolev spaces Wp,loc2(Ω;Rn) with the Luzin capacity-measure N-property.