The co-area formula for Sobolev mappings

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...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2003-02, Vol.355 (2), p.477-492
Main Authors: Malý, Jan, Swanson, David, Ziemer, William P.
Format: Article
Language:English
Subjects:
Borel sets
Calculus of variations and optimal control
Exact sciences and technology
Functional analysis
Grants
Hausdorff measures
Integers
Jacobians
Mathematical analysis
Mathematical functions
Mathematical inequalities
Mathematical theorems
Mathematics
Orlicz space
Real functions
Sciences and techniques of general use
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-02-03091-X