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Lower semicontinuity and relaxation results in BV for integral functionalswith BV integrands

Published online by Cambridge University Press:  21 November 2007

Micol Amar ,
Virginia De Cicco  and
Nicola Fusco
Micol Amar
Affiliation:
Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, Via A. Scarpa 16, 00161 Roma, Italy; amar@dmmm.uniroma1.it; decicco@dmmm.uniroma1.it
Virginia De Cicco
Affiliation:
Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, Via A. Scarpa 16, 00161 Roma, Italy; amar@dmmm.uniroma1.it; decicco@dmmm.uniroma1.it
Nicola Fusco
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Complesso di Monte Sant'Angelo, Via Cintia, 80126 Napoli, Italy; nicola.fusco@unina.it
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Abstract

New L1-lower semicontinuity and relaxation results for integral functionals defined in BV(Ω) are proved, under a very weak dependence of the integrand with respect to the spatial variable x. More precisely, only the lower semicontinuity in the sense of the 1-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to x. Under this further BV dependence, a representation formula for the relaxed functional is also obtained.

Keywords

SemicontinuityrelaxationBV functionscapacity
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 3 , July 2008 , pp. 456 - 477
Copyright
© EDP Sciences, SMAI, 2007

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