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

Loading next page...
 
/lp/springer_journal/a-reciprocity-principle-for-constrained-isoperimetric-problems-and-UF1bUvZSLB
Publisher
Springer Berlin Heidelberg
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 EquationsSpringer Journals

Published: Mar 13, 2018

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off