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Weighted Cheeger sets are domains of isoperimetry

Weighted Cheeger sets are domains of isoperimetry We consider a generalization of the Cheeger problem in a bounded, open set $$\Omega $$ Ω by replacing the perimeter functional with a Finsler-type surface energy and the volume with suitable powers of a weighted volume. We show that any connected minimizer A of this weighted Cheeger problem such that $$\mathcal {H}^{n-1}(A^{(1)} \cap \partial A)=0$$ H n - 1 ( A ( 1 ) ∩ ∂ A ) = 0 satisfies a relative isoperimetric inequality. If $$\Omega $$ Ω itself is a connected minimizer such that $$\mathcal {H}^{n-1}(\Omega ^{(1)} \cap \partial \Omega )=0$$ H n - 1 ( Ω ( 1 ) ∩ ∂ Ω ) = 0 , then it allows the classical Sobolev and BV embeddings and the classical BV trace theorem. The same result holds for any connected minimizer whenever the weights grant the regularity of perimeter-minimizer sets and $$\Omega $$ Ω is such that $$|\partial \Omega |=0$$ | ∂ Ω | = 0 and $$\mathcal {H}^{n-1}(\Omega ^{(1)} \cap \partial \Omega )=0$$ H n - 1 ( Ω ( 1 ) ∩ ∂ Ω ) = 0 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Manuscripta Mathematica Springer Journals

Weighted Cheeger sets are domains of isoperimetry

Manuscripta Mathematica , Volume 156 (4) – Sep 11, 2017

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany
Subject
Mathematics; Mathematics, general; Algebraic Geometry; Topological Groups, Lie Groups; Geometry; Number Theory; Calculus of Variations and Optimal Control; Optimization
ISSN
0025-2611
eISSN
1432-1785
DOI
10.1007/s00229-017-0974-z
Publisher site
See Article on Publisher Site

Abstract

We consider a generalization of the Cheeger problem in a bounded, open set $$\Omega $$ Ω by replacing the perimeter functional with a Finsler-type surface energy and the volume with suitable powers of a weighted volume. We show that any connected minimizer A of this weighted Cheeger problem such that $$\mathcal {H}^{n-1}(A^{(1)} \cap \partial A)=0$$ H n - 1 ( A ( 1 ) ∩ ∂ A ) = 0 satisfies a relative isoperimetric inequality. If $$\Omega $$ Ω itself is a connected minimizer such that $$\mathcal {H}^{n-1}(\Omega ^{(1)} \cap \partial \Omega )=0$$ H n - 1 ( Ω ( 1 ) ∩ ∂ Ω ) = 0 , then it allows the classical Sobolev and BV embeddings and the classical BV trace theorem. The same result holds for any connected minimizer whenever the weights grant the regularity of perimeter-minimizer sets and $$\Omega $$ Ω is such that $$|\partial \Omega |=0$$ | ∂ Ω | = 0 and $$\mathcal {H}^{n-1}(\Omega ^{(1)} \cap \partial \Omega )=0$$ H n - 1 ( Ω ( 1 ) ∩ ∂ Ω ) = 0 .

Journal

Manuscripta MathematicaSpringer Journals

Published: Sep 11, 2017

References