Nonoccurrence of the Lavrentiev Phenomenon for Variational Problems

  • Alexander J. Zaslavski
Part of the Springer Optimization and Its Applications book series (SOIA, volume 82)


In this chapter we study nonoccurrence of the Lavrentiev phenomenon for a large class of nonconvex nonautonomous constrained variational problems. A state variable belongs to a convex subset H of a Banach space X with nonempty interior. Integrands belong to a complete metric space of functions \(\mathcal{M}_{B}\) which satisfy a growth condition common in the literature and are Lipschitzian on bounded sets. This space will be described below. In [97] we considered a class of nonconstrained variational problems with integrands belonging to a subset \(\mathcal{L}_{B} \subset \mathcal{M}_{B}\) and showed that for \(f \in \mathcal{L}_{B}\) the following property holds:


  1. 97.
    Zaslavski AJ (2005) Nonoccurrence of the Lavrentiev phenomenon for nonconvex variational problems. Ann Inst H Poincaré Anal Non linéaire 22:579–596MathSciNetzbMATHCrossRefGoogle Scholar
  2. 101.
    Zaslavski AJ (2007) Nonoccurrence of the Lavrentiev phenomenon for many nonconvex constrained variational problems. Calc Var 28:351–381MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Alexander J. Zaslavski
    • 1
  1. 1.Department of MathematicsTechnion - Israel Institute of TechnologyHaifaIsrael

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