# Calculus of variations in L ∞

Calculus of variations in L ∞ Given an arbitrary function we determine the greatest quasiconvex minorant of the function in a way analogous to the classical Legendre-Fenchel transform. The greatest quasiconvex minorant is shown to be the same as the lower semicontinuous regularization of the functional. This fact is used to produce the relaxation of functionals on L ∞ of the form F ( ξ, ξ′ )=ess sup 0≤ s ≤ T h(s, ξ(s), ξ′(s) ). The relaxed functional will be lower semicontinuous in the appropriate topology and yields the existence of a minimizer. Then the relaxation theorem is established, proving that the original problem and the relaxed problem have the same values under broad assumptions on h . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Calculus of variations in L ∞

Applied Mathematics and Optimization, Volume 35 (3) – May 1, 1997
27 pages

Loading next page...

/lp/springer_journal/calculus-of-variations-in-l-RfstT0kTD0
Publisher
Springer Journals
Copyright
Copyright © 1997 by Springer-Verlag New York Inc.
Subject
Mathematics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Methods
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/BF02683330
Publisher site
See Article on Publisher Site

### Abstract

Given an arbitrary function we determine the greatest quasiconvex minorant of the function in a way analogous to the classical Legendre-Fenchel transform. The greatest quasiconvex minorant is shown to be the same as the lower semicontinuous regularization of the functional. This fact is used to produce the relaxation of functionals on L ∞ of the form F ( ξ, ξ′ )=ess sup 0≤ s ≤ T h(s, ξ(s), ξ′(s) ). The relaxed functional will be lower semicontinuous in the appropriate topology and yields the existence of a minimizer. Then the relaxation theorem is established, proving that the original problem and the relaxed problem have the same values under broad assumptions on h .

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: May 1, 1997

## 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

### 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.

DeepDyve

DeepDyve

### Pro

Price

FREE

\$49/month
\$360/year

Save searches from
Google Scholar,
PubMed

Create folders to
organize your research

Export folders, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off