The refined area formula for Sobolev mappings $W^{k,p}$

TL;DR

This paper proves an area formula for Sobolev mappings in $W^{k,p}$, using Lipschitz approximations that match the original functions outside negligible sets, advancing the understanding of change-of-variable formulas in Sobolev spaces.

Contribution

It introduces a new method to establish the area formula for Sobolev mappings via Lipschitz approximations outside Riesz capacity zero sets.

Findings

Established the area formula for $W^{k,p}_{loc}$ Sobolev mappings.

Developed Lipschitz approximation techniques for Sobolev functions.

Extended change-of-variable formulas to broader Sobolev contexts.

Abstract

We establish the area formula for change-of-variable mappings in the Sobolev space $W^{k,p}_{\text{loc}}$. Our approach relies on constructing Lipschitz approximations of Sobolev functions that agree with the original functions outside a set of Riesz capacity zero.