Γ-convergence and absolute minimizers for supremal functionals

Thierry Champion; Luigi De Pascale; Francesca Prinari

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

  • Volume: 10, Issue: 1, page 14-27
  • ISSN: 1292-8119

Abstract

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

How to cite

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Champion, Thierry, De Pascale, Luigi, and Prinari, Francesca. "Γ-convergence and absolute minimizers for supremal functionals." ESAIM: Control, Optimisation and Calculus of Variations 10.1 (2010): 14-27. <http://eudml.org/doc/90718>.

@article{Champion2010,
abstract = { In this paper, we prove that the Lp approximants naturally associated to a supremal functional Γ-converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer (i.e. local solution) among these minimizers. We provide two different proofs of this fact relying on different assumptions and techniques. },
author = {Champion, Thierry, De Pascale, Luigi, Prinari, Francesca},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Supremal functionals; lower semicontinuity; generalized Jensen inequality; absolute minimizer (AML; local minimizer); Lp approximation. ; supremal functionals; absolute minimizer; local minimizer; approximation},
language = {eng},
month = {3},
number = {1},
pages = {14-27},
publisher = {EDP Sciences},
title = {Γ-convergence and absolute minimizers for supremal functionals},
url = {http://eudml.org/doc/90718},
volume = {10},
year = {2010},
}

TY - JOUR
AU - Champion, Thierry
AU - De Pascale, Luigi
AU - Prinari, Francesca
TI - Γ-convergence and absolute minimizers for supremal functionals
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 1
SP - 14
EP - 27
AB - In this paper, we prove that the Lp approximants naturally associated to a supremal functional Γ-converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer (i.e. local solution) among these minimizers. We provide two different proofs of this fact relying on different assumptions and techniques.
LA - eng
KW - Supremal functionals; lower semicontinuity; generalized Jensen inequality; absolute minimizer (AML; local minimizer); Lp approximation. ; supremal functionals; absolute minimizer; local minimizer; approximation
UR - http://eudml.org/doc/90718
ER -

References

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  1. E. Acerbi, G. Buttazzo and F. Prinari, On the class of functionals which can be represented by a supremum. J. Convex Anal.9 (2002) 225-236.  Zbl1012.49010
  2. G. Aronsson, Minimization Problems for the Functional sup x F ( x , f ( x ) , f ' ( x ) ) . Ark. Mat.6 (1965) 33-53.  Zbl0156.12502
  3. G. Aronsson, Minimization Problems for the Functional sup x F ( x , f ( x ) , f ' ( x ) ) . II. Ark. Mat.6 (1966) 409-431.  Zbl0156.12502
  4. G. Aronsson, Extension of Functions satisfying Lipschitz conditions. Ark. Mat.6 (1967) 551-561.  Zbl0158.05001
  5. G. Aronsson, Minimization Problems for the Functional sup x F ( x , f ( x ) , f ' ( x ) ) . III. Ark. Mat.7 (1969) 509-512.  Zbl0181.11902
  6. 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).  
  7. E.N. Barron, R.R. Jensen and C.Y. Wang, Lower Semicontinuity of L∞ functionals. Ann. Inst. H. Poincaré Anal. Non Linéaire18(2001) 495-517.  Zbl1034.49008
  8. E.N. Barron, R.R. Jensen and C.Y. Wang, The Euler equation and absolute minimizers of L∞ functionals. Arch. Rational Mech. Anal.157 (2001) 255-283.  Zbl0979.49003
  9. 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.  
  10. H. Berliocchi and J.M. Lasry, Intégrandes normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France101 (1973) 129-184.  Zbl0282.49041
  11. 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).  
  12. M.G. Crandal, L.C. Evans and R.F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. Partial Differential Equations13 (2001) 123-139.  Zbl0996.49019
  13. B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, Berlin, Appl. Math. Sci. 78 (1989).  Zbl0703.49001
  14. G. Dal Maso, An Introduction to Γ-Convergence. Birkhauser, Basel, Progr. in Nonlinear Differential Equations Appl. 8 (1993).  
  15. 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).  
  16. E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)58 (1975) 842-850.  
  17. A. Garroni, V. Nesi and M. Ponsiglione, Dielectric Breakdown: Optimal bounds. Proc. Roy. Soc. London Sect. A457 (2001) 2317-2335.  Zbl0993.78015
  18. M. Gori and F. Maggi, On the lower semicontinuity of supremal functional. ESAIM: COCV9 (2003) 135.  Zbl1066.49010
  19. R.R. Jensen, Uniqueness of Lipschitz Extensions: Minimizing the Sup Norm of the Gradient. Arch. Rational Mech. Anal.123 (1993) 51-74.  Zbl0789.35008
  20. P. Juutinen, Absolutely Minimizing Lipschitz Extensions on a metric space. An. Ac. Sc. Fenn. Mathematica27 (2002) 57-67.  Zbl1064.54027
  21. D. Kinderlehrer and P. Pedregal, Characterization of Young Measures Generated by Gradients. Arch. Rational Mech. Anal.115 (1991) 329-365.  Zbl0754.49020
  22. D. Kinderlehrer and P. Pedregal, Gradient Young Measures Generated by Sequences in Sobolev Spaces. J. Geom. Anal.4(1994) 59-90.  Zbl0808.46046
  23. 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).  
  24. P. Pedregal, Parametrized measures and variational principles. Birkhäuser Verlag, Basel, Progr. in Nonlinear Differential Equations Appl. 30 (1997).  

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