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Complete representation of some functionals showing the Lavrentieff phenomenon

Published online by Cambridge University Press:  12 July 2007

A. Corbo Esposito  and
T. Durante
A. Corbo Esposito
Affiliation:
Università di Cassino, Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell'Informazione e Matematica Industriale, via G. Di Biasio no. 43, 03043 Cassino (FR), Italy (corbo@unicas.it)
T. Durante
Affiliation:
Università di Cassino, Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell'Informazione e Matematica Industriale, via G. Di Biasio no. 43, 03043 Cassino (FR), Italy (durante@unicas.it)
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Abstract

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The functional F(u) = ∫Bf(x, Du) is considered, where B is the unit ball in R2, u varies in the set of the locally Lipschitz functions on R2 and f belongs to a family containing, as model case, the following integrand:

The computation of the relaxed functional F̄ is provided yielding an explicit representation formula.

This formula nevertheless is not integral, because F̄ is not a measure and does not coincide with the obvious extension of F over all W1,p(B).

This phenomenon is essentially due to the non-standard growth behaviour of f(x, z) in the variable z.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 131 , Issue 4 , August 2001 , pp. 811 - 832
Copyright
Copyright © Royal Society of Edinburgh 2001