A relaxation result in BV for integral functionals with discontinuous integrands
ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 2, pp. 396-412.

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.

DOI : 10.1051/cocv:2007015
Classification : 49J45, 26B30
Mots clés : lower semicontinuity, relaxation, BV-functions, blow-up
Amar, Micol  ; Cicco, Virginia De  ; Fusco, Nicola 1

1 Dipartimento di Matematica e Applicazioni Monte Sant’Angelo, via Cintia, 80126 Napoli, Italy;
@article{COCV_2007__13_2_396_0,
     author = {Amar, Micol and Cicco, Virginia De and Fusco, Nicola},
     title = {A relaxation result in {BV} for integral functionals with discontinuous integrands},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {396--412},
     publisher = {EDP-Sciences},
     volume = {13},
     number = {2},
     year = {2007},
     doi = {10.1051/cocv:2007015},
     mrnumber = {2306643},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2007015/}
}
TY  - JOUR
AU  - Amar, Micol
AU  - Cicco, Virginia De
AU  - Fusco, Nicola
TI  - A relaxation result in BV for integral functionals with discontinuous integrands
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2007
SP  - 396
EP  - 412
VL  - 13
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2007015/
DO  - 10.1051/cocv:2007015
LA  - en
ID  - COCV_2007__13_2_396_0
ER  - 
%0 Journal Article
%A Amar, Micol
%A Cicco, Virginia De
%A Fusco, Nicola
%T A relaxation result in BV for integral functionals with discontinuous integrands
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2007
%P 396-412
%V 13
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2007015/
%R 10.1051/cocv:2007015
%G en
%F COCV_2007__13_2_396_0
Amar, Micol; Cicco, Virginia De; Fusco, Nicola. A relaxation result in BV for integral functionals with discontinuous integrands. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 2, pp. 396-412. doi : 10.1051/cocv:2007015. http://archive.numdam.org/articles/10.1051/cocv:2007015/

[1] M. Amar and V. De Cicco, Relaxation in BV for a class of functionals without continuity assumptions. NoDEA (to appear).

[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press, New York (2000). | MR | Zbl

[3] G. Bouchitté, I. Fonseca and L. Mascarenhas, A global method for relaxation. Arch. Rat. Mech. Anal. 145 (1998) 51-98. | Zbl

[4] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation Problems in the Calculus of Variations. Pitman Res. Notes Math., Longman, Harlow (1989). | Zbl

[5] G. Dal Maso, Integral representation on BV(Ω) of Γ-limits of variational integrals. Manuscripta Math. 30 (1980) 387-416. | Zbl

[6] G. Dal Maso, An Introduction to Γ-convergence. Birkhäuser, Boston (1993). | MR | Zbl

[7] V. De Cicco, N. Fusco and A. Verde, On L 1 -lower semicontinuity in BV(Ω). J. Convex Analysis 12 (2005) 173-185. | Zbl

[8] V. De Cicco, N. Fusco and A. Verde, A chain rule formula in BV(Ω) and its applications to lower semicontinuity. Calc. Var. Partial Differ. Equ. 28 (2007) 427-447. | Zbl

[9] V. De Cicco and G. Leoni, A chain rule in L 1 ( div ;Ω) and its applications to lower semicontinuity. Calc. Var. Partial Differ. Equ. 19 (2004) 23-51. | Zbl

[10] 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. | Zbl

[11] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Rend. Sem. Mat. Brescia 3 (1979) 63-101.

[12] L.C. Evans and R.F. Gariepy, Lecture Notes on Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992). | Zbl

[13] H. Federer, Geometric measure theory. Springer-Verlag, Berlin (1969). | MR | Zbl

[14] I. Fonseca and G. Leoni, Some remarks on lower semicontinuity. Indiana Univ. Math. J. 49 (2000) 617-635. | Zbl

[15] I. Fonseca and G. Leoni, On lower semicontinuity and relaxation. Proc. R. Soc. Edinb. Sect. A Math. 131 (2001) 519-565. | Zbl

[16] I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in L 1 . SIAM J. Math. Anal. 23 (1992) 1081-1098. | Zbl

[17] I. Fonseca and S. Müller, Relaxation of quasiconvex functionals in BV(Ω, p ) for integrands f(x,u,u). Arch. Rat. Mech. Anal. 123 (1993) 1-49. | Zbl

[18] N. Fusco, M. Gori and F. Maggi, A remark on Serrin's Theorem. NoDEA 13 (2006) 425-433.

[19] E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston (1984). | MR | Zbl

[20] M. Gori and P. Marcellini, An extension of the Serrin's lower semicontinuity theorem. J. Convex Anal. 9 (2002) 475-502. | Zbl

[21] M. Gori, F. Maggi and P. Marcellini, On some sharp conditions for lower semicontinuity in L 1 . Diff. Int. Eq. 16 (2003) 51-76. | Zbl

[22] A.I. Vol'Pert and S.I. Hudjaev, Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics. Martinus & Nijhoff Publishers, Dordrecht (1985). | Zbl

Cité par Sources :