Nonoccurrence of the Lavrentiev phenomenon for many nonconvex constrained variational problems

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 \(\mathcal {M}\) 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 \(\mathcal {L} \subset \mathcal {M}\) 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 \(\mathcal {L}\), then this property also holds for any integrand which is contained in a certain neighborhood of f in \(\mathcal {M}\). Using this result we establish nonoccurrence of the Lavrentiev phenomenon for most elements of \(\mathcal {M}\) in the sense of Baire category.