Issue |
ESAIM: COCV
Volume 5, 2000
|
|
---|---|---|
Page(s) | 71 - 85 | |
DOI | https://doi.org/10.1051/cocv:2000102 | |
Published online | 15 August 2002 |
Relaxation of singular functionals defined on Sobolev spaces
1
Département de Mathématiques,
Institut Préparatoire aux Études d'Ingénieurs de
Sfax, Route Menzel Chaker - Km 0,5, BP. 805, 3000 Sfax,
Tunisia; Fax: (00-216) 4. 246. 347.
2
Max-Planck Institute for Mathematics in the Sciences,
Inselstr. 22-26, 04103 Leipzig, Germany; Hafedh.Belgacem@mis.mpg.de.
Received:
1
December
1998
Revised:
17
November
1999
In this paper, we consider a Borel
measurable function on
the space of
matrices
taking the value
, such that its rank-one-convex
envelope
is finite and satisfies for some fixed
:
where
. Let
be a given
regular bounded
open domain of
. We define on
the functional
Then, under some technical restrictions on
, we show that the relaxed functional
for the weak topology
of
has the integral
representation:
where for a given function
,
denotes its
quasiconvex envelope.
Résumé
On considère une fonction Borel mesurable
qui prend la valeur
, dont l'enveloppe
rang-1-convexe Rf est finie et satisfait pour un certain p>1,
avec
fixés. Étant donné un ouvert borné Ω de
, on introduit
la
fonctionnelle
pour
. On montre alors sous quelques hypothèses
supplémentaires concernant f, que la relaxée
de I
pour la topolgie faible de
admet la représentation
suivante :
où pour une
fonction donnée g, Qg désigne son enveloppe quasi-convexe.
Mathematics Subject Classification: 49-xx
Key words: Rank-one convexity / quasiconvexity / weak lower semicontinuity.
© EDP Sciences, SMAI, 2000
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