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Applied Mathematics and Optimization

, Volume 35, Issue 3, pp 237–263 | Cite as

Calculus of variations inL

  • E. N. Barron
  • W. Liu
Article

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 onL of the formF (ξ, ξ′)=ess sup0≤sT 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 onh.

Key Words

Quasiconvex Relaxed problem 

AMS Classification

49A10 49A50 49C20 

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Copyright information

© Springer-Verlag New York Inc 1997

Authors and Affiliations

  • E. N. Barron
    • 1
  • W. Liu
    • 1
  1. 1.Department of Mathematical SciencesLoyola UniversityChicagoUSA

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