Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands

Fusco, Nicola; Cicco, Virginia De; Amar, Micol

doi:10.1051/cocv:2007061

Fusco, Nicola ¹ ; Cicco, Virginia De ; Amar, Micol

¹ Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Complesso di Monte Sant’Angelo, Via Cintia, 80126 Napoli, Italy;

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 3, pp. 456-477.

Résumé

New $L^{1}$ -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.

MR Zbl

DOI : 10.1051/cocv:2007061

Classification : 49J45, 49Q20, 49M20
Mots-clés : semicontinuity, relaxation, BV functions, capacity

Affiliations des auteurs :

Fusco, Nicola ¹ ; Cicco, Virginia De ; Amar, Micol

¹ Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Complesso di Monte Sant’Angelo, Via Cintia, 80126 Napoli, Italy;

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     author = {Fusco, Nicola and Cicco, Virginia De and Amar, Micol},
     title = {Lower semicontinuity and relaxation results in {BV} for integral functionals with {BV} integrands},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {456--477},
     publisher = {EDP-Sciences},
     volume = {14},
     number = {3},
     year = {2008},
     doi = {10.1051/cocv:2007061},
     mrnumber = {2434061},
     zbl = {1149.49016},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2007061/}
}

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Fusco, Nicola; Cicco, Virginia De; Amar, Micol. Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 3, pp. 456-477. doi : 10.1051/cocv:2007061. https://www.numdam.org/articles/10.1051/cocv:2007061/

Bibliographie
Cité par

[1] M. Amar, and G. Bellettini, A notion of total variation depending on a metric with discontinuous coefficients. Ann. Inst. Henri Poincaré 11 (1994) 91-133. | Numdam | MR | Zbl

[2] M. Amar and V. De Cicco, Relaxation in $B V$ for a class of functionals without continuity assumptions. NoDEA Nonlinear Differential Equations Appl. (to appear). | MR | Zbl

[3] M. Amar, V. De Cicco and N. Fusco, A relaxation result in BV for integral functionals with discontinuous integrands. ESAIM: COCV 13 (2007) 396-412. | Numdam | MR

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

[5] G. Anzellotti, G. Buttazzo and G. Dal Maso, Dirichlet problem for demi-coercive functionals. Nonlinear Anal. 10 (1986) 603-613. | MR | Zbl

[6] G. Bouchitté and M. Valadier, Integral representation of convex functionals on a space of measures. J. Funct. Anal. 80 (1988) 398-420. | MR | Zbl

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

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

[9] M. Carriero, G. Dal Maso, A. Leaci and E. Pascali, Relaxation of the non-parametric Plateau problem with an obstacle. J. Math. Pures Appl. 67 (1988) 359-396. | MR | Zbl

[10] G. Dal Maso, Integral representation on $B V (Ω)$ of $Γ$ -limits of variational integrals. Manuscripta Math. 30 (1980) 387-416. | MR | Zbl

[11] G. Dal Maso, On the integral representation of certain local functionals. Ricerche di Matematica 32 (1983) 85-113. | MR | Zbl

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

[13] V. De Cicco and G. Leoni, A chain rule in $L^{1} (div; Ω)$ and its applications to lower semicontinuity. Calc. Var. Partial Differential Equations 19 (2004) 23-51. | MR | Zbl

[14] V. De Cicco, N. Fusco and A. Verde, On $L^{1}$ -lower semicontinuity in $B V (Ω)$ . J. Convex Analysis 12 (2005) 173-185. | MR | Zbl

[15] V. De Cicco, N. Fusco and A. Verde, A chain rule formula in $B V (Ω)$ and its applications to lower semicontinuity. Calc. Var. Partial Differential Equations 28 (2007) 427-447. | MR | Zbl

[16] 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. | MR | Zbl

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

[18] H. Federer and W.P. Ziemer, The Lebesgue set of a function whose distribution derivatives are $p$ -th power summable. Indiana Un. Math. J. 22 (1972) 139-158. | MR | Zbl

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

[20] I. Fonseca and S. M $\ddot{u}$ ller, Quasi-convex integrands and lower semicontinuity in $L^{1}$ . SIAM J. Math. Anal. 23 (1992) 1081-1098. | MR | Zbl

[21] I. Fonseca and S. M $\ddot{u}$ ller, Relaxation of quasiconvex functionals in BV $(Ω, ℝ^{p})$ for integrands $f (x, u, \nabla u)$ . Arch. Rat. Mech. Anal. 123 (1993) 1-49. | MR | Zbl

[22] N. Fusco, F. Giannetti and A. Verde, A remark on the $L^{1}$ -lower semicontinuity for integral functionals in BV. Manuscripta Math. 112 (2003) 313-323. | MR | Zbl

[23] N. Fusco, M. Gori and F. Maggi, A remark on Serrin's Theorem. NoDEA Nonlinear Differential Equations Appl. 13 (2006) 425-433. | MR

[24] M. Gori and F. Maggi, The common root of the geometric conditions in Serrin's semicontinuity theorem. Ann. Mat. Pura Appl. 184 (2005) 95-114. | MR

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

[26] F. Maggi, On the relaxation on BV of certain non coercive integral functionals. J. Convex Anal. 10 (2003) 477-489. | MR | Zbl

[27] M. Miranda, Superfici cartesiane generalizzate ed insiemi di perimetro localmente finito sui prodotti cartesiani. Ann. Scuola Norm. Sup. Pisa 18 (1964) 515-542. | Numdam | MR | Zbl

[28] Y.G. Reshetnyak, Weak convergence of completely additive vector functions on a set. Siberian Math. J. 9 (1968) 1039-1045. | Zbl

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