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The co-area formula for Sobolev mappings

Author(s): Jan Maly; David Swanson; William P. Ziemer
Journal: Trans. Amer. Math. Soc. 355 (2003), 477-492.
MSC (2000): Primary 46E35, 46E30; Secondary 26B10, 26B35, 49Q15
Posted: August 27, 2002
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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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Additional Information:

Jan Maly
Affiliation: Faculty of Mathematics and Physics, Charles University -- KMA, Sokolovská 83, 18675 Praha 8, Czech Republic
Email: maly@karlin.mff.cuni.cz

David Swanson
Affiliation: Department of Mathematics, Texas A{&}M University, College Station, Texas 77843
Email: dswanson@math.tamu.edu

William P. Ziemer
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: ziemer@indiana.edu

DOI: 10.1090/S0002-9947-02-03091-X
PII: S 0002-9947(02)03091-X
Keywords: Sobolev mapping, Orlicz space, co-area formula, area formula, rectifiability
Received by editor(s): December 3, 2001
Posted: August 27, 2002
Additional Notes: The research of the first author is supported in part by the Research Project MSM 113200007 from the Czech Ministry of Education, Grant No. 201/00/0767 from the Grant Agency of the Czech Republic (GACR) and Grant No. 165/99 from the Grant Agency of Charles University (GAUK)
Copyright of article: Copyright 2002, American Mathematical Society

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