Relaxation and gamma-convergence of supremal functionals

Francesca Prinari

Bollettino dell'Unione Matematica Italiana (2006)

  • Volume: 9-B, Issue: 1, page 101-132
  • ISSN: 0392-4041

Abstract

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We prove that the Γ -limit in L μ ∞ of a sequence of supremal functionals of the form F k ⁢ ( u ) = μ - ess ⁢ sup Ω ⁡ f k ⁢ ( x , u ) is itself a supremal functional. We show by a counterexample that, in general, the function which represents the Γ -lim F ⁢ ( ⋅ , B ) of a sequence of functionals F k ⁢ ( u , B ) = μ - 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 ) = μ - 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.

How to cite

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Prinari, Francesca. "Relaxation and gamma-convergence of supremal functionals." Bollettino dell'Unione Matematica Italiana 9-B.1 (2006): 101-132. <http://eudml.org/doc/289598>.

@article{Prinari2006,
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.},
author = {Prinari, Francesca},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {101-132},
publisher = {Unione Matematica Italiana},
title = {Relaxation and gamma-convergence of supremal functionals},
url = {http://eudml.org/doc/289598},
volume = {9-B},
year = {2006},
}

TY - JOUR
AU - Prinari, Francesca
TI - Relaxation and gamma-convergence of supremal functionals
JO - Bollettino dell'Unione Matematica Italiana
DA - 2006/2//
PB - Unione Matematica Italiana
VL - 9-B
IS - 1
SP - 101
EP - 132
AB - 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.
LA - eng
UR - http://eudml.org/doc/289598
ER -

References

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  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. Zbl1012.49010MR1917396
  2. ATTOUCH, H., Variational Convergence for Functions and Operators, Appl. Math.Ser.Pitman, Boston (1984). Zbl0561.49012MR773850
  3. BARRON, E. N. - CARDALIAGUET, P. - JENSEN, R. R., Radon-Nikodym Theorem in L ∞ , Appl. Math. Optim., 2 (2000), 103-126. MR1784171DOI10.1007/s002450010006
  4. BARRON, E. N. - JENSEN, R., Relaxed minimax control, SIAM J. Control Optimazion, 33 (1995), 1028-1039. MR1339052DOI10.1137/S0363012993250530
  5. BARRON, E. N. - LIU, W., Calculus of Variations in L ∞ , App. Math. Optim., 35 (1997), 237-263. Zbl0871.49017MR1431800DOI10.1007/s002459900047
  6. BERLIOCCHI, H. - LASRY, J. M., Integrand normales et mésures paramétres en calcul des variations, Bull. Soc. Math. France, 101 (1973), 129-184. MR344980
  7. BUTTAZZO, G. - DAL MASO, G., Γ -Limits of integral functionals, J. Analyse Math., 37 (1980), 145-185. MR583636DOI10.1007/BF02797684
  8. BUTTAZZO, G. - DAL MASO, G., On Nemyckii operators and integral representation of local functionals, Rendiconti di Matematica, 3 (1983), 491-510. Zbl0536.47027MR743394
  9. BUTTAZZO, G. - DAL MASO, G., Integral representation and relaxation of local functionals, Nonlinear Anal., 9 (1985), 515-532. Zbl0527.49008MR794824DOI10.1016/0362-546X(85)90038-0
  10. BUTTAZZO, G., Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Res. Notes Math. Ser., 207, Longman, Harlow (1989). MR1020296
  11. CASTAING, C. - VALADIER, M., Convex Analysis and measurable Multifunctions, Lecture Notes in Math., 590, Springer-Verlag, Berlin (1977). Zbl0346.46038MR467310
  12. DACOROGNA, B., Direct Methods in the Calculus of Variations, Appl. Math. Sciences78, Springer-Verlag, Berlin (1989). Zbl0703.49001MR990890DOI10.1007/978-3-642-51440-1
  13. DAL MASO, G., An introduction to Gamma-convergence, Birkhäuser, Boston (1993). Zbl0816.49001MR1201152DOI10.1007/978-1-4612-0327-8
  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. MR671757
  15. EKELAND, I. - TEMAM, R., Convex Analysis and variational problems, North Holland, Amsterdam (1978). Zbl0939.49002MR463994
  16. MARCELLINI, P. - SBORDONE, C., Dualità e perturbazioni di funzionali integrali, Ricerche Mat., 26 (1977), 383-421. MR467437
  17. MASCOLO, E. - MIGLIACCIO, L., Relaxation methods in control theory, Appl. Math. Optim.1 (1989), 97-103. Zbl0682.49007MR989434DOI10.1007/BF01447649
  18. MULLER, S., Variational models for microstructure and phase transitions, Lecture Notes in Math., 1713, Springer, Berlin (1999), 85-210. Zbl0968.74050MR1731640DOI10.1007/BFb0092670
  19. PRINARI, F., Calculus of Variations for Supremal Functionals, PhD Thesis, University of Pisa. Zbl1163.49012
  20. VALADIER, M., Young Measures, Lecture Notes in Math1446, Springer, Berlin(1990), 152-188. MR1079763DOI10.1007/BFb0084935
  21. VALADIER, M., A course on Young Measures, Rend. Istit. Mat. Univ. Trieste, 26 (1994), 349-394. MR1408956
  22. VOLLE, M., Duality for level sum of quasiconvex function and applications, ESAIM Control Optim. Calc. Var., 3 (1998), 329-343. Zbl0904.49023MR1641843DOI10.1051/cocv:1998114

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