The co-area formula for Sobolev mappings

Malý, Jan; Swanson, David; Ziemer, William

doi:10.1090/S0002-9947-02-03091-X

The co-area formula for Sobolev mappings
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by Jan Malý, David Swanson and William P. Ziemer PDF

Trans. Amer. Math. Soc. 355 (2003), 477-492 Request permission

Abstract:

We extend Federer’s co-area formula to mappings $f$ belonging to the Sobolev class $W^{1,p}(\mathbb {R}^n;\mathbb {R}^m)$, $1 \le m < n$, $p>m$, and more generally, to mappings with gradient in the Lorentz space $L^{m,1}(\mathbb {R}^n)$. This is accomplished by showing that the graph of $f$ in $\mathbb {R}^{n+m}$ is a Hausdorff $n$-rectifiable set.

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