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On the Lower Semicontinuity of Supremal Functionals

Published online by Cambridge University Press:  15 September 2003

Michele Gori  and
Francesco Maggi
Michele Gori
Affiliation:
Dipartimento di Matematica “L. Tonelli”, Università di Pisa, Via Buonarroti 2, 56127 Pisa, Italy; [email protected].
Francesco Maggi
Affiliation:
Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy; [email protected].
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Abstract

In this paper we study the lower semicontinuity problem for a supremal functional of the form $F(u,\Omega )= \underset{x\in\Omega}{\rm ess\,sup} f(x,u(x),Du(x))$ with respect to the strong convergence in L∞(Ω), furnishing a comparison with the analogous theory developed by Serrin for integrals. A sort of Mazur's lemma for gradients of uniformly converging sequences is proved.

Keywords

Supremal functionalslower semicontinuitylevel convexityCalculus of VariationsMazur's lemma.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 135 - 143
Copyright
© EDP Sciences, SMAI, 2003

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References

E. Acerbi, G. Buttazzo and F. Prinari, The class of functionals which can be represented by a supremum. J. Convex Anal. 9 (to appear).
L. Ambrosio, New lower semicontinuity results for integral functionals. Rend. Accad. Naz. Sci. XL 11 (1987) 1-42.
Aronsson, G., Minimization problems for the functional sup x F(x,f(x),f'(x)). Ark. Mat. 6 (1965) 33-53. CrossRef
Aronsson, G., Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6 (1967) 551-561. CrossRef
Aronsson, G., Minimization problems for the functional sup x F(x,f(x),f'(x)) II. Ark. Mat. 6 (1969) 409-431.
Aronsson, G., Minimization problems for the functional sup x F(x,f(x),f'(x)) III. Ark. Mat. 7 (1969) 509-512. CrossRef
Barron, E.N., Jensen, R.R. and Wang, C.Y., Lower semicontinuity of L ∞ functionals. Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001) 495-517. CrossRef
Barron, E.N. and Liu, W., Calculus of variations in L ∞. Appl. Math. Optim. 35 (1997) 237-263.
G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations. Longman, Harlow, Pitman Res. Notes Math. Ser. 207 (1989).
Carbone, L. and Sbordone, C., Some properties of Γ-limits of integral functionals. Ann. Mat. Pura Appl. 122 (1979) 1-60. CrossRef
Dal Maso, G., Integral representation on BV(Ω)of Γ-limits of variational integrals. Manuscripta Math. 30 (1980) 387-416.
E. De Giorgi, Teoremi di semicontinuità nel calcolo delle variazioni. Istituto Nazionale di Alta Matematica, Roma (1968).
E. De Giorgi, G. Buttazzo and G. Dal Maso, On the lower semicontinuity of certain integral functionals. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur., Rend. 74 (1983) 274-282.
Eisen, G., A counterexample for some lower semicontinuity results. Math. Z. 162 (1978) 241-243. CrossRef
Fonseca, I. and Leoni, G., Some remarks on lower semicontinuity. Indiana Univ. Math. J. 49 (2000) 617-635.
Fonseca, I. and Leoni, G., On lower semicontinuity and relaxation. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 519-565. CrossRef
M. Gori, F. Maggi and P. Marcellini, On some sharp lower semicontinuity condition in L 1. Differential Integral Equations (to appear).
Gori, M. and Marcellini, P., An extension of the Serrin's lower semicontinuity theorem. J. Convex Anal. 9 (2002) 1-28.
Ioffe, A.D., On lower semicontinuity of integral functionals. SIAM J. Control Optim. 15 (1977) 521-538. CrossRef
C.Y. Pauc, La méthode métrique en calcul des variations. Hermann, Paris (1941).
Serrin, J., On the definition and properties of certain variational integrals. Trans. Amer. Math. Soc. 101 (1961) 139-167. CrossRef