Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-28T12:03:12.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Γ -convergence and absolute minimizers for supremal functionals

Published online by Cambridge University Press:  15 February 2004

Thierry Champion ,
Luigi De Pascale  and
Francesca Prinari
Thierry Champion
Affiliation:
Laboratoire d'Analyse Non Linéaire Appliquée, U.F.R. des Sciences et Techniques, Université de Toulon et du Var, Avenue de l'Université, BP. 132, 83957 La Garde Cedex, France; [email protected].
Luigi De Pascale
Affiliation:
Dipartimento di Matematica Applicata, Universitá di Pisa, Via Bonanno Pisano 25/B, 56126 Pisa, Italy.
Francesca Prinari
Affiliation:
Dipartimento di Matematica, Universitá di Pisa, Via Buonarroti 2,56127 Pisa, Italy.
Get access

Abstract

In this paper, we prove that the Lp approximants naturally associated to a supremal functionalΓ-converge to it. This yields a lower semicontinuity result for supremalfunctionals whose supremand satisfy weak coercivity assumptions aswell as a generalized Jensen inequality. The existence of minimizersfor variational problems involving such functionals (together with aDirichlet condition) then easily follows. In the scalarcase we show the existence of at least one absolute minimizer (i.e. localsolution) among these minimizers. We provide two different proofs ofthis fact relying on different assumptions and techniques.

Keywords

Supremal functionalslower semicontinuitygeneralized Jensen inequalityabsolute minimizer (AMLlocal minimizer)Lp approximation.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 1 , January 2004 , pp. 14 - 27
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acerbi, E., Buttazzo, G. and Prinari, F., On the class of functionals which can be represented by a supremum. J. Convex Anal. 9 (2002) 225-236.
Aronsson, G., Minimization Problems for the Functional $\sup _{x}F(x,f(x),f'(x))$ . Ark. Mat. 6 (1965) 33-53. CrossRef
Aronsson, G., Minimization Problems for the Functional $\sup _{x}F(x,f(x),f'(x))$ . II. Ark. Mat. 6 (1966) 409-431.
Aronsson, G., Extension of Functions satisfying Lipschitz conditions. Ark. Mat. 6 (1967) 551-561. CrossRef
Aronsson, G., Minimization Problems for the Functional $\sup _{x}F(x,f(x),f'(x))$ . III. Ark. Mat. 7 (1969) 509-512. CrossRef
E.N. Barron, Viscosity solutions and analysis in L ∞. Nonlinear Anal. Differential Equations Control. Montreal, QC (1998) 1-60. Kluwer Acad. Publ., Dordrecht, NATO Sci. Ser. C Math. Phys. Sci. 528 (1999).
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., Jensen, R.R. and Wang, C.Y., The Euler equation and absolute minimizers of L ∞ functionals. Arch. Rational Mech. Anal. 157 (2001) 255-283. CrossRef
T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as p → ∞ of Δpup = ƒ and related extremal problems, Some topics in nonlinear PDEs. Turin (1989). Rend. Sem. Mat. Univ. Politec. Torino 1989, Special Issue (1991) 15-68.
Berliocchi, H. and Lasry, J.M., Intégrandes normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France 101 (1973) 129-184. CrossRef
M.G. Crandal and L.C. Evans, A remark on infinity harmonic functions, in Proc. of the USA-Chile Workshop on Nonlinear Analysis. Vina del Mar-Valparaiso (2000) 123-129. Electronic. Electron. J. Differential Equations Conf. 6 . Southwest Texas State Univ., San Marcos, TX (2001).
Crandal, M.G., Evans, L.C. and Gariepy, R.F., Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. Partial Differential Equations 13 (2001) 123-139.
B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, Berlin, Appl. Math. Sci. 78 (1989).
G. Dal Maso, An Introduction to Γ-Convergence. Birkhauser, Basel, Progr. in Nonlinear Differential Equations Appl. 8 (1993).
G. Dal Maso and L. Modica, A general theory of variational functionals. Topics in functional analysis (1980–81) 149-221. Quaderni, Scuola Norm. Sup. Pisa, Pisa (1981).
De Giorgi, E. and Franzoni, T., Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 (1975) 842-850.
Garroni, A., Nesi, V. and Ponsiglione, M., Dielectric Breakdown: Optimal bounds. Proc. Roy. Soc. London Sect. A 457 (2001) 2317-2335. CrossRef
Gori, M. and Maggi, F., On the lower semicontinuity of supremal functional. ESAIM: COCV 9 (2003) 135. CrossRef
Jensen, R.R., Uniqueness of Lipschitz Extensions: Minimizing the Sup Norm of the Gradient. Arch. Rational Mech. Anal. 123 (1993) 51-74. CrossRef
Juutinen, P., Absolutely Minimizing Lipschitz Extensions on a metric space. An. Ac. Sc. Fenn. Mathematica 27 (2002) 57-67.
Kinderlehrer, D. and Pedregal, P., Characterization of Young Measures Generated by Gradients. Arch. Rational Mech. Anal. 115 (1991) 329-365. CrossRef
Kinderlehrer, D. and Pedregal, P., Gradient Young Measures Generated by Sequences in Sobolev Spaces. J. Geom. Anal. 4(1994) 59-90. CrossRef
S. Muller, Variational models for microstructure and phase transitions. Calculus of variations and geometric evolution problems. Cetraro (1996) 85-210. Springer, Berlin, Lecture Notes in Math. 1713 (1999).
P. Pedregal, Parametrized measures and variational principles. Birkhäuser Verlag, Basel, Progr. in Nonlinear Differential Equations Appl. 30 (1997).