# A relaxation result in BV for integral functionals with discontinuous integrands

Micol Amar; Virginia De Cicco; Nicola Fusco

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 13, Issue: 2, page 396-412
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topAmar, Micol, De Cicco, Virginia, and Fusco, Nicola. "A relaxation result in BV for integral functionals with discontinuous integrands." ESAIM: Control, Optimisation and Calculus of Variations 13.2 (2007): 396-412. <http://eudml.org/doc/249987>.

@article{Amar2007,

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.
},

author = {Amar, Micol, De Cicco, Virginia, Fusco, Nicola},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Lower semicontinuity; relaxation; BV-functions; blow-up; lower semicontinuity},

language = {eng},

month = {5},

number = {2},

pages = {396-412},

publisher = {EDP Sciences},

title = {A relaxation result in BV for integral functionals with discontinuous integrands},

url = {http://eudml.org/doc/249987},

volume = {13},

year = {2007},

}

TY - JOUR

AU - Amar, Micol

AU - De Cicco, Virginia

AU - Fusco, Nicola

TI - A relaxation result in BV for integral functionals with discontinuous integrands

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2007/5//

PB - EDP Sciences

VL - 13

IS - 2

SP - 396

EP - 412

AB -
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.

LA - eng

KW - Lower semicontinuity; relaxation; BV-functions; blow-up; lower semicontinuity

UR - http://eudml.org/doc/249987

ER -

## References

top- 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).
- G. Bouchitté, I. Fonseca and L. Mascarenhas, 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).
- G. Dal Maso, Integral representation on BV$\left(\Omega \right)$ of $\Gamma $-limits of variational integrals. Manuscripta Math.30 (1980) 387–416.
- G. Dal Maso, An Introduction to$\Gamma $-convergence. Birkhäuser, Boston (1993).
- V. De Cicco, N. Fusco and A. Verde, On L1-lower semicontinuity in BV$\left(\Omega \right)$. J. Convex Analysis12 (2005) 173–185.
- V. De Cicco, N. Fusco and A. Verde, A chain rule formula in BV$\left(\Omega \right)$ and its applications to lower semicontinuity. Calc. Var. Partial Differ. Equ.28 (2007) 427–447.
- V. De Cicco and G. Leoni, A chain rule in ${L}^{1}(\mathrm{div};\Omega )$ and its applications to lower semicontinuity. Calc. Var. Partial Differ. Equ.19 (2004) 23–51.
- E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.58 (1975) 842–850.
- E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Rend. Sem. Mat. Brescia3 (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).
- I. Fonseca and G. Leoni, Some remarks on lower semicontinuity. Indiana Univ. Math. J.49 (2000) 617–635.
- I. Fonseca and G. Leoni, On lower semicontinuity and relaxation. Proc. R. Soc. Edinb. Sect. A Math.131 (2001) 519–565.
- I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in L1. SIAM J. Math. Anal.23 (1992) 1081–1098.
- I. Fonseca and S. Müller, 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.
- N. Fusco, M. Gori and F. Maggi, A remark on Serrin's Theorem. NoDEA13 (2006) 425–433.
- E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston (1984).
- M. Gori and P. Marcellini, An extension of the Serrin's lower semicontinuity theorem. J. Convex Anal.9 (2002) 475–502.
- M. Gori, F. Maggi and P. Marcellini, On some sharp conditions for lower semicontinuity in L1. 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).

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.