On the Lower Semicontinuity of Supremal Functionals

Michele Gori; Francesco Maggi

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 9, page 135-143
  • ISSN: 1292-8119

Abstract

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In this paper we study the lower semicontinuity problem for a supremal functional of the form F ( u , Ω ) = ess sup x ∈ Ω f ( x , u ( x ) , D u ( 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.

How to cite

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Gori, Michele, and Maggi, Francesco. "On the Lower Semicontinuity of Supremal Functionals." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 135-143. <http://eudml.org/doc/90685>.

@article{Gori2010,
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. },
author = {Gori, Michele, Maggi, Francesco},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = { Supremal functionals; lower semicontinuity; level convexity; Calculus of Variations; Mazur's lemma.; supremal functionals; level convexity; Mazur lemma},
language = {eng},
month = {3},
pages = {135-143},
publisher = {EDP Sciences},
title = {On the Lower Semicontinuity of Supremal Functionals},
url = {http://eudml.org/doc/90685},
volume = {9},
year = {2010},
}

TY - JOUR
AU - Gori, Michele
AU - Maggi, Francesco
TI - On the Lower Semicontinuity of Supremal Functionals
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 9
SP - 135
EP - 143
AB - 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.
LA - eng
KW - Supremal functionals; lower semicontinuity; level convexity; Calculus of Variations; Mazur's lemma.; supremal functionals; level convexity; Mazur lemma
UR - http://eudml.org/doc/90685
ER -

References

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  11. G. Dal Maso, Integral representation on BV(Ω)of Γ-limits of variational integrals. Manuscripta Math.30 (1980) 387-416.  
  12. E. De Giorgi, Teoremi di semicontinuità nel calcolo delle variazioni. Istituto Nazionale di Alta Matematica, Roma (1968).  
  13. 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.  
  14. G. Eisen, A counterexample for some lower semicontinuity results. Math. Z.162 (1978) 241-243.  
  15. I. Fonseca and G. Leoni, Some remarks on lower semicontinuity. Indiana Univ. Math. J.49 (2000) 617-635.  
  16. I. Fonseca and G. Leoni, On lower semicontinuity and relaxation. Proc. Roy. Soc. Edinburgh Sect. A131 (2001) 519-565.  
  17. M. Gori, F. Maggi and P. Marcellini, On some sharp lower semicontinuity condition in L1. Differential Integral Equations (to appear).  
  18. M. Gori and P. Marcellini, An extension of the Serrin's lower semicontinuity theorem. J. Convex Anal.9 (2002) 1-28.  
  19. A.D. Ioffe, On lower semicontinuity of integral functionals. SIAM J. Control Optim.15 (1977) 521-538.  
  20. C.Y. Pauc, La méthode métrique en calcul des variations. Hermann, Paris (1941).  
  21. J. Serrin, On the definition and properties of certain variational integrals. Trans. Amer. Math. Soc.101 (1961) 139-167.  

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