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A relaxation result in BV for integral functionalswith discontinuous integrands

Published online by Cambridge University Press:  12 May 2007

Micol Amar,
Virginia De Cicco and
Nicola Fusco
Micol Amar
Affiliation:
Dipartimento di Metodi e Modelli Matematici, 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, via A. Scarpa 16, 00161 Roma, Italy; amar@dmmm.uniroma1.it; decicco@dmmm.uniroma1.it
Nicola Fusco
Affiliation:
Dipartimento di Matematica e Applicazioni Monte Sant'Angelo, via Cintia, 80126 Napoli, Italy; n.fusco@unina.it
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Abstract

We prove a relaxation theorem in BV for a non coercive functional with linear growth. No continuity of the integrand with respect to the spatial variable is assumed.

Keywords

Lower semicontinuityrelaxationBV-functionsblow-up

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 2 , April 2007, pp. 396 - 412
Copyright
© EDP Sciences, SMAI, 2007

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References

M. Amar and V. De Cicco, Relaxation in BV for a class of functionals without continuity assumptions. NoDEA (to appear).
L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press, New York (2000).
Bouchitté, G., Fonseca, I. and Mascarenhas, L., A global method for relaxation. Arch. Rat. Mech. Anal. 145 (1998) 51–98.
G. Buttazzo, Semicontinuity, Relaxation and Integral Representation Problems in the Calculus of Variations. Pitman Res. Notes Math., Longman, Harlow (1989).
Dal Maso, G., Integral representation on BV $(\Omega)$ of $\Gamma$ -limits of variational integrals. Manuscripta Math. 30 (1980) 387–416. CrossRef
G. Dal Maso, An Introduction to $\Gamma$ -convergence. Birkhäuser, Boston (1993).
De Cicco, V., Fusco, N. and Verde, A., On L 1-lower semicontinuity in BV $(\Omega)$ . J. Convex Analysis 12 (2005) 173–185.
De Cicco, V., Fusco, N. and Verde, A., A chain rule formula in BV $(\Omega)$ and its applications to lower semicontinuity. Calc. Var. Partial Differ. Equ. 28 (2007) 427–447. CrossRef
V. De Cicco and G. Leoni, A chain rule in $L^1({\rm div};\Omega)$ and its applications to lower semicontinuity. Calc. Var. Partial Differ. Equ. 19 (2004) 23–51.
De Giorgi, E. and Franzoni, T., Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (1975) 842–850.
De Giorgi, E. and Franzoni, T., Su un tipo di convergenza variazionale. Rend. Sem. Mat. Brescia 3 (1979) 63–101.
L.C. Evans and R.F. Gariepy, Lecture Notes on Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992).
H. Federer, Geometric measure theory. Springer-Verlag, Berlin (1969).
Fonseca, I. and Leoni, G., Some remarks on lower semicontinuity. Indiana Univ. Math. J. 49 (2000) 617–635. CrossRef
Fonseca, I. and Leoni, G., On lower semicontinuity and relaxation. Proc. R. Soc. Edinb. Sect. A Math. 131 (2001) 519–565. CrossRef
Fonseca, I. and Müller, S., Quasi-convex integrands and lower semicontinuity in L 1. SIAM J. Math. Anal. 23 (1992) 1081–1098. CrossRef
Fonseca, I. and Müller, S., Relaxation of quasiconvex functionals in BV $(\Omega,{\mathbb R}^p)$ for integrands $f(x,u,\nabla u)$ . Arch. Rat. Mech. Anal. 123 (1993) 1–49. CrossRef
Fusco, N., Gori, M. and Maggi, F., A remark on Serrin's Theorem. NoDEA 13 (2006) 425–433. CrossRef
E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston (1984).
Gori, M. and Marcellini, P., An extension of the Serrin's lower semicontinuity theorem. J. Convex Anal. 9 (2002) 475–502.
Gori, M., Maggi, F. and Marcellini, P., On some sharp conditions for lower semicontinuity in L 1. Diff. Int. Eq. 16 (2003) 51–76.
A.I. Vol'pert and S.I. Hudjaev, Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics. Martinus & Nijhoff Publishers, Dordrecht (1985).