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

/lp/springer-journals/calculus-of-variations-in-l-RfstT0kTD0
Publisher
Springer Journals
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
DOI
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

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