A reciprocity principle for constrained isoperimetric problems and existence of isoperimetric subregions in convex sets

A reciprocity principle for constrained isoperimetric problems and existence of isoperimetric... It is a well known fact that in $$\mathbb {R} ^n$$ R n a subset of minimal perimeter L among all sets of a given volume is also a set of maximal volume among all sets of the same perimeter L. This is called the reciprocity principle for isoperimetric problems. The aim of this note is to prove this relation in the case where the class of admissible sets is restricted to the subsets of some subregion $$G\subsetneq \mathbb {R} ^n$$ G ⊊ R n . Furthermore, we give a characterization of those (unbounded) convex subsets of $$\mathbb {R} ^2$$ R 2 in which the isoperimetric problem has a solution. The perimeter that we consider is the one relative to $$\mathbb {R} ^n$$ R n . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

A reciprocity principle for constrained isoperimetric problems and existence of isoperimetric subregions in convex sets

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-018-1325-y
Publisher site
See Article on Publisher Site

Abstract

It is a well known fact that in $$\mathbb {R} ^n$$ R n a subset of minimal perimeter L among all sets of a given volume is also a set of maximal volume among all sets of the same perimeter L. This is called the reciprocity principle for isoperimetric problems. The aim of this note is to prove this relation in the case where the class of admissible sets is restricted to the subsets of some subregion $$G\subsetneq \mathbb {R} ^n$$ G ⊊ R n . Furthermore, we give a characterization of those (unbounded) convex subsets of $$\mathbb {R} ^2$$ R 2 in which the isoperimetric problem has a solution. The perimeter that we consider is the one relative to $$\mathbb {R} ^n$$ R n .

Journal

Calculus of Variations and Partial Differential Equations – Springer Journals

Published: Mar 13, 2018

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