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Journal of Convex Analysis 08 (2001), No. 2, 511--532
Copyright Heldermann Verlag 2001

Lavrentieff Phenomenon and Non Standard Growth Conditions

Giuseppe Cardone
Dip. di Ingegneria Civile, Seconda UniversitÓ di Napoli, Real Casa dell'Annunziata, Via Roma 29, 81031 Aversa, Italy

C. D'Apice
Dip. di Ingegneria dell'Informazione e Matematica Applicata, UniversitÓ di Salerno, Via Ponte don Melillo, 84084 Fisciano, Italy

U. De Maio
Dip. di Matematica ed Applicazioni, UniversitÓ di Napoli, Complesso Monte S. Angelo, 80126 Napoli, Italy


The functional $F(u) = \int_B f(x,Du)\,dx$ is considered, where $B$ is the unit ball in $\mathbb{R}^n$, $u$ varies in the set of the locally Lipschitz functions on $\mathbb{R}^n$, and $f$ belongs to a family of integrands containing, as model case, the following one \[ f:(x,z)\in \mathbb{R}^{n}\times \mathbb{R}^{n}\mapsto \frac{|\lt z,x \lt|}{|x|^{n}}% + |z|^{p},\text{ \ \ \ }1 \lt p \lt n. \] The computation of the relaxed functional of $F$ is provided. The formula obtained shows the persistence of the Lavrentieff Phenomenon. Examples of integrands not exhibiting the Lavrentieff Phenomenon are also presented, showing that this phenomenon is not linked only to the non standard growth behaviour of integrands.

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