Nonoccurrence of the Lavrentiev phenomenon for many nonconvex constrained variational problems |
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Authors: | Alexander J Zaslavski |
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Institution: | (1) Department of Mathematics, The Technion-Israel Institute of Technology, 32000 Haifa, Israel |
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Abstract: | In this paper 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 of a Banach space with nonempty interior. Integrands belong
to a complete metric space of functions
which satisfy a growth condition common in the literature and are Lipschitzian on bounded sets. In our previous work Zaslavski
(Ann. Inst. H. Poincare, Anal. non lineare, 2006) we considered a class of nonconstrained variational problems with integrands
belonging to a subset
and showed that for any such integrand the infimum on the full admissible class is equal to the infimum on a subclass of
Lipschitzian functions with the same Lipschitzian constant. In the present paper we show that if an integrand f belongs to
, then this property also holds for any integrand which is contained in a certain neighborhood of f in
. Using this result we establish nonoccurrence of the Lavrentiev phenomenon for most elements of
in the sense of Baire category.
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Keywords: | Banach space Integrand Lavrentiev phenomenon Variational problem |
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