Relaxation and gamma-convergence of supremal functionals

Prinari, Francesca

Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006), p. 101-132 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

We prove that the $\Gamma$-limit in $L^\infty_\mu$ of a sequence of supremal functionals of the form $F_k (u)=\operatorname{\mu-ess\,sup}_\Omega f_k(x, u)$ is itself a supremal functional. We show by a counterexample that, in general, the function which represents the $\Gamma$-lim $F(\cdot, B)$ of a sequence of functionals $F_k(u, B)= \operatorname{\mu-ess\,sup}_B f_k(x,u)$ can depend on the set $B$ and wegive a necessary and sufficient condition to represent $F$ in the supremal form$F(u, B)= \operatorname{\mu-ess\,sup}_B f(x,u)$. As a corollary, if $f$ represents a supremal functional, then the level convex envelope of $f$ represents its weak* lower semicontinuous envelope.

Résumé

Si prova che il $\Gamma$-limite in $L^\infty_\mu$ di una successione di funzionali supremali della forma $F_k (u)=\operatorname{\mu-ess\,sup}_\Omega f_k(x, u)$ è un funzionale supremale. In un controesempio si mostra che la funzione che rappresenta il $\Gamma$-limite $F(\cdot, B)$ di una successione di funzionali supremali della forma $F_k (u, B)= \operatorname{\mu-ess\,sup}_B f_k(x,u)$ può dipendere dall'insieme $B$ e si stabilisce una condizione necessaria e sufficiente al fine di rappresentare $F$ nella forma supremale $F(u, B) = \operatorname{\mu-ess\,sup}_B f(x,u)$. Come corollario, si dimostra che se $f$ rappresenta un funzionale supremale $F$, allora l'inviluppo level convex di $f$ rappresenta l'inviluppo semicontinuo inferiormente di $F$ rispetto alla topologia debole* di $L^\infty_\mu$.

Détail

Publié le : 2006-02-01

MR 2204903 | Zbl 1178.49018

Bibliographie

[1] Acerbi, E. - Buttazzo, G. - Prinari, F., The class of functional which can be represented by a supremum, J. Conv. Anal., 9 (2002), 225-236. | MR 1917396 | Zbl 1012.49010

[2] Attouch, H., Variational Convergence for Functions and Operators, Appl. Math.Ser.Pitman, Boston (1984). | MR 773850 | Zbl 0561.49012

[3] Barron, E. N. - Cardaliaguet, P. - Jensen, R. R., Radon-Nikodym Theorem in $L^\infty$, Appl. Math. Optim., 2 (2000), 103-126. | MR 1784171

[4] Barron, E. N. - Jensen, R., Relaxed minimax control, SIAM J. Control Optimazion, 33 (1995), 1028-1039. | MR 1339052

[5] Barron, E. N. - Liu, W., Calculus of Variations in $L^\infty$, App. Math. Optim., 35 (1997), 237-263. | MR 1431800 | Zbl 0871.49017

[6] Berliocchi, H. - Lasry, J. M., Integrand normales et mésures paramétres en calcul des variations, Bull. Soc. Math. France, 101 (1973), 129-184. | MR 344980

[7] Buttazzo, G. - Dal Maso, G., $\Gamma$-Limits of integral functionals, J. Analyse Math., 37 (1980), 145-185. | MR 583636

[8] Buttazzo, G. - Dal Maso, G., On Nemyckii operators and integral representation of local functionals, Rendiconti di Matematica, 3 (1983), 491-510. | MR 743394 | Zbl 0536.47027

[9] Buttazzo, G. - Dal Maso, G., Integral representation and relaxation of local functionals, Nonlinear Anal., 9 (1985), 515-532. | MR 794824 | Zbl 0527.49008

[10] Buttazzo, G., Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Res. Notes Math. Ser., 207, Longman, Harlow (1989). | MR 1020296

[11] Castaing, C. - Valadier, M., Convex Analysis and measurable Multifunctions, Lecture Notes in Math., 590, Springer-Verlag, Berlin (1977). | MR 467310 | Zbl 0346.46038

[12] Dacorogna, B., Direct Methods in the Calculus of Variations, Appl. Math. Sciences78, Springer-Verlag, Berlin (1989). | MR 990890 | Zbl 0703.49001

[13] Dal Maso, G., An introduction to Gamma-convergence, Birkhäuser, Boston (1993). | MR 1201152 | Zbl 0816.49001

[14] Dal Maso, G. - Modica, L., A general theory of variational functionals, In «Topics ifunctional analysis», Quaderni, Scuola Norm. Sup. Pisa, Pisa (1982), 149-221. | MR 671757

[15] Ekeland, I. - Temam, R., Convex Analysis and variational problems, North Holland, Amsterdam (1978). | MR 463994 | Zbl 0939.49002

[16] Marcellini, P. - Sbordone, C., Dualità e perturbazioni di funzionali integrali, Ricerche Mat., 26 (1977), 383-421. | MR 467437

[17] Mascolo, E. - Migliaccio, L., Relaxation methods in control theory, Appl. Math. Optim. 1 (1989), 97-103. | MR 989434 | Zbl 0682.49007

[18] Muller, S., Variational models for microstructure and phase transitions, Lecture Notes in Math., 1713, Springer, Berlin (1999), 85-210. | MR 1731640 | Zbl 0968.74050

[19] Prinari, F., Calculus of Variations for Supremal Functionals, PhD Thesis, University of Pisa. | Zbl 1163.49012

[20] Valadier, M., Young Measures, Lecture Notes in Math1446, Springer, Berlin(1990), 152-188. | MR 1079763

[21] Valadier, M., A course on Young Measures, Rend. Istit. Mat. Univ. Trieste, 26 (1994), 349-394. | MR 1408956

[22] Volle, M., Duality for level sum of quasiconvex function and applications, ESAIM Control Optim. Calc. Var., 3 (1998), 329-343. | MR 1641843 | Zbl 0904.49023