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May  2012, 32(5): 1503-1535. doi: 10.3934/dcds.2012.32.1503

Dimensional reduction for supremal functionals

Jean-François Babadjian 1, , Francesca Prinari 2, and  Elvira Zappale 3,

1. 

Université Pierre et Marie Curie – Paris 6, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
2. 

Dipartimento di Matematica, Università di Ferrara, Via Machiavelli, 35, 44100, Ferrara, Italy
3. 

DIEII, Università degli Studi di Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy

Received  November 2010 Revised  May 2011 Published  January 2012

A 3D-2D dimensional reduction analysis for supremal functionals is performed in the realm of $\Gamma^*$-convergence. We show that the limit functional still admits a supremal representation, and we provide a precise identification of its density in some particular cases. Our results rely on an abstract representation theorem for the $\Gamma^*$-limit of a family of supremal functionals.
Keywords: Supremal functionals, level convexity, homogenization., dimension reduction, $\Gamma$-convergence.
Mathematics Subject Classification: 49J45, 74K99, 74Q9.
Citation: Jean-François Babadjian, Francesca Prinari, Elvira Zappale. Dimensional reduction for supremal functionals. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1503-1535. doi: 10.3934/dcds.2012.32.1503
References:
[1]

E. Acerbi, G. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string, J. Elasticity, 25 (1991), 137-148. doi: 10.1007/BF00042462.  Google Scholar

[2]

E. Acerbi, G. Buttazzo and F. Prinari, The class of functionals which can be represented by a supremum, J. Convex Anal., 9 (2002), 225-236.  Google Scholar

[3]

O. Alvarez and E. N. Barron, Homogenization in $L^\infty$, J. Diff. Eq., 183 (2002), 132-164.  Google Scholar

[4]

G. Aronsson, Minimization problems for the functional sup$_x F(x,f(x),f'(x))$, Ark. Mat., 6 (1965), 33-53. doi: 10.1007/BF02591326.  Google Scholar

[5]

G. Aronsson, Minimization problems for the functional sup$_x F(x,f(x),f'(x))$, II, Ark. Mat., 6 (1966), 409-431.  Google Scholar

[6]

G. Aronsson, Minimization problems for the functional sup$_xF(x,f(x),f'(x))$, III, Ark. Mat., 7 (1969), 509-512.  Google Scholar

[7]

G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561. doi: 10.1007/BF02591928.  Google Scholar

[8]

J.-F. Babadjian and M. Baía, 3D-2D analysis of a thin film with periodic microstructure, Proc. Roy. Soc. Ed. Sect. A, 136 (2006), 223-243. doi: 10.1017/S0308210500004534.  Google Scholar

[9]

J.-F. Babadjian and M. Baía, Multiscale nonconvex relaxation and application to thin films, Asympt. Anal., 48 (2006), 173-218.  Google Scholar

[10]

J.-F. Babadjian and G. A. Francfort, Spatial heterogeneity in 3D-2D dimensional reduction, ESAIM Cont. Optim. Calc. Var., 11 (2005), 139-160. doi: 10.1051/cocv:2004031.  Google Scholar

[11]

J.-F. Babadjian, E. Zappale and H. Zorgati, Dimensional reduction for energies with linear growth involving the bending moment, J. Math. Pures Appl. (9), 90 (2008), 520-549. doi: 10.1016/j.matpur.2008.07.003.  Google Scholar

[12]

M Baía and I. Fonseca, The limit behavior of a family of variational multiscale problems, Indiana Univ. Math. J., 56 (2007), 1-50. doi: 10.1512/iumj.2007.56.2869.  Google Scholar

[13]

E. N. Barron, P. Cardaliaguet and R. R. Jensen, Radon-Nikodym theorem in $L^\infty$, Appl. Math. Optim., 42 (2000), 103-126. doi: 10.1007/s002450010006.  Google Scholar

[14]

E. N. Barron, R. R. Jensen and C. Y. Wang, Lower semicontinuity of $L^\infty$ functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 495-517.  Google Scholar

[15]

E. N. Barron, R. R. Jensen and C. Y. Wang, The Euler equation and absolute minimizers of $L^\infty$ functionals, Arch. Rational Mech. Anal., 157 (2001), 225-283.  Google Scholar

[16]

E. N. Barron and W. Liu, Calculus of variation in $L^\infty$, Appl. Math. Optim., 35 (1997), 237-263. doi: 10.1007/s002459900047.  Google Scholar

[17]

A. Braides, "$\Gamma$-Convergence for Beginners," Oxford Lecture Series in Mathematics and its Applications, 22, Oxford University Press, Oxford, 2002.  Google Scholar

[18]

A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals," Oxford Lectures Series in Mathematics and its Applications, 12, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[19]

A. Braides and I. Fonseca, Brittle thin films, Appl. Math. Optim., 44 (2001), 299-323. doi: 10.1007/s00245-001-0022-x.  Google Scholar

[20]

A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J., 49 (2000), 1367-1404.  Google Scholar

[21]

A. Briani and F. Prinari, A representation result for $\Gamma$-limit of supremal functionals, J. Nonlinear Convex Anal., 4 (2003), 245-268.  Google Scholar

[22]

A. Briani, A. Garroni and F. Prinari, Homogenization of $L^\infty$ functionals, Math. Models and Methods in Applied Sciences, 14 (2004), 1761-1784. doi: 10.1142/S0218202504003817.  Google Scholar

[23]

G. Buttazzo and G. Dal Maso, Integral representation and relaxation of local functionals, Nonlinear Anal., 9 (1985), 515-532. doi: 10.1016/0362-546X(85)90038-0.  Google Scholar

[24]

G. Buttazzo, "Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations," Pitman Research Notes in Mathematics Series, 207, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.  Google Scholar

[25]

P. Cardaliaguet and F. Prinari, Supremal representation of $L^\infty$ functionals, App. Math. Optim., 52 (2005), 129-141. doi: 10.1007/s00245-005-0821-6.  Google Scholar

[26]

, P. Cardaliaguet and F. Prinari,, Unpublished., ().   Google Scholar

[27]

T. Champion, L. De Pascale and F. Prinari, Semicontinuity and absolute minimizers for supremal functionals, ESAIM Control Optim. Calc. Var., 10 (2004), 14-27. doi: 10.1051/cocv:2003036.  Google Scholar

[28]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993.  Google Scholar

[29]

G. Dal Maso and P. Longo, $\Gamma$-limits of obstacles, Ann. Mat. Pura Appl. (4), 128 (1981), 1-50. doi: 10.1007/BF01789466.  Google Scholar

[30]

G. Dal Maso and L. Modica, A general theory of variational functionals, in "Topics in Functional Analysis," 1980-81, Quaderni, Scuola Norm. Sup. Pisa, Pisa, (1981), 149-221.  Google Scholar

[31]

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.  Google Scholar

[32]

A. Garroni, M. Ponsiglione and F. Prinari, Positively 1-homogeneous supremal functionals to difference quotients: Relaxation and $\Gamma$-convergence, Calc. Var. Partial Differential Equations, 27 (2006), 397-420.  Google Scholar

[33]

A. Garroni, V. Nesi and M. Ponsiglione, Dielectric breakdown: Optimal bounds, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2317-2335. doi: 10.1098/rspa.2001.0803.  Google Scholar

[34]

H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. (9), 74 (1995), 549-578.  Google Scholar

[35]

F. Prinari, Relaxation and $\Gamma$-convergence of supremal functionals, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 9 (2006), 101-132.  Google Scholar

[36]

F. Prinari, Semicontinuity and relaxation of $L^\infty$-functionals, Adv. Calc. Var., 2 (2009), 43-71. doi: 10.1515/ACV.2009.003.  Google Scholar

[37]

J. Serrin, On the definition and the properties of certain variational integrals, Trans. Amer. Math. Soc., 101 (1961), 139-167. doi: 10.1090/S0002-9947-1961-0138018-9.  Google Scholar

show all references

References:
[1]

E. Acerbi, G. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string, J. Elasticity, 25 (1991), 137-148. doi: 10.1007/BF00042462.  Google Scholar

[2]

E. Acerbi, G. Buttazzo and F. Prinari, The class of functionals which can be represented by a supremum, J. Convex Anal., 9 (2002), 225-236.  Google Scholar

[3]

O. Alvarez and E. N. Barron, Homogenization in $L^\infty$, J. Diff. Eq., 183 (2002), 132-164.  Google Scholar

[4]

G. Aronsson, Minimization problems for the functional sup$_x F(x,f(x),f'(x))$, Ark. Mat., 6 (1965), 33-53. doi: 10.1007/BF02591326.  Google Scholar

[5]

G. Aronsson, Minimization problems for the functional sup$_x F(x,f(x),f'(x))$, II, Ark. Mat., 6 (1966), 409-431.  Google Scholar

[6]

G. Aronsson, Minimization problems for the functional sup$_xF(x,f(x),f'(x))$, III, Ark. Mat., 7 (1969), 509-512.  Google Scholar

[7]

G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561. doi: 10.1007/BF02591928.  Google Scholar

[8]

J.-F. Babadjian and M. Baía, 3D-2D analysis of a thin film with periodic microstructure, Proc. Roy. Soc. Ed. Sect. A, 136 (2006), 223-243. doi: 10.1017/S0308210500004534.  Google Scholar

[9]

J.-F. Babadjian and M. Baía, Multiscale nonconvex relaxation and application to thin films, Asympt. Anal., 48 (2006), 173-218.  Google Scholar

[10]

J.-F. Babadjian and G. A. Francfort, Spatial heterogeneity in 3D-2D dimensional reduction, ESAIM Cont. Optim. Calc. Var., 11 (2005), 139-160. doi: 10.1051/cocv:2004031.  Google Scholar

[11]

J.-F. Babadjian, E. Zappale and H. Zorgati, Dimensional reduction for energies with linear growth involving the bending moment, J. Math. Pures Appl. (9), 90 (2008), 520-549. doi: 10.1016/j.matpur.2008.07.003.  Google Scholar

[12]

M Baía and I. Fonseca, The limit behavior of a family of variational multiscale problems, Indiana Univ. Math. J., 56 (2007), 1-50. doi: 10.1512/iumj.2007.56.2869.  Google Scholar

[13]

E. N. Barron, P. Cardaliaguet and R. R. Jensen, Radon-Nikodym theorem in $L^\infty$, Appl. Math. Optim., 42 (2000), 103-126. doi: 10.1007/s002450010006.  Google Scholar

[14]

E. N. Barron, R. R. Jensen and C. Y. Wang, Lower semicontinuity of $L^\infty$ functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 495-517.  Google Scholar

[15]

E. N. Barron, R. R. Jensen and C. Y. Wang, The Euler equation and absolute minimizers of $L^\infty$ functionals, Arch. Rational Mech. Anal., 157 (2001), 225-283.  Google Scholar

[16]

E. N. Barron and W. Liu, Calculus of variation in $L^\infty$, Appl. Math. Optim., 35 (1997), 237-263. doi: 10.1007/s002459900047.  Google Scholar

[17]

A. Braides, "$\Gamma$-Convergence for Beginners," Oxford Lecture Series in Mathematics and its Applications, 22, Oxford University Press, Oxford, 2002.  Google Scholar

[18]

A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals," Oxford Lectures Series in Mathematics and its Applications, 12, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[19]

A. Braides and I. Fonseca, Brittle thin films, Appl. Math. Optim., 44 (2001), 299-323. doi: 10.1007/s00245-001-0022-x.  Google Scholar

[20]

A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J., 49 (2000), 1367-1404.  Google Scholar

[21]

A. Briani and F. Prinari, A representation result for $\Gamma$-limit of supremal functionals, J. Nonlinear Convex Anal., 4 (2003), 245-268.  Google Scholar

[22]

A. Briani, A. Garroni and F. Prinari, Homogenization of $L^\infty$ functionals, Math. Models and Methods in Applied Sciences, 14 (2004), 1761-1784. doi: 10.1142/S0218202504003817.  Google Scholar

[23]

G. Buttazzo and G. Dal Maso, Integral representation and relaxation of local functionals, Nonlinear Anal., 9 (1985), 515-532. doi: 10.1016/0362-546X(85)90038-0.  Google Scholar

[24]

G. Buttazzo, "Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations," Pitman Research Notes in Mathematics Series, 207, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.  Google Scholar

[25]

P. Cardaliaguet and F. Prinari, Supremal representation of $L^\infty$ functionals, App. Math. Optim., 52 (2005), 129-141. doi: 10.1007/s00245-005-0821-6.  Google Scholar

[26]

, P. Cardaliaguet and F. Prinari,, Unpublished., ().   Google Scholar

[27]

T. Champion, L. De Pascale and F. Prinari, Semicontinuity and absolute minimizers for supremal functionals, ESAIM Control Optim. Calc. Var., 10 (2004), 14-27. doi: 10.1051/cocv:2003036.  Google Scholar

[28]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993.  Google Scholar

[29]

G. Dal Maso and P. Longo, $\Gamma$-limits of obstacles, Ann. Mat. Pura Appl. (4), 128 (1981), 1-50. doi: 10.1007/BF01789466.  Google Scholar

[30]

G. Dal Maso and L. Modica, A general theory of variational functionals, in "Topics in Functional Analysis," 1980-81, Quaderni, Scuola Norm. Sup. Pisa, Pisa, (1981), 149-221.  Google Scholar

[31]

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.  Google Scholar

[32]

A. Garroni, M. Ponsiglione and F. Prinari, Positively 1-homogeneous supremal functionals to difference quotients: Relaxation and $\Gamma$-convergence, Calc. Var. Partial Differential Equations, 27 (2006), 397-420.  Google Scholar

[33]

A. Garroni, V. Nesi and M. Ponsiglione, Dielectric breakdown: Optimal bounds, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2317-2335. doi: 10.1098/rspa.2001.0803.  Google Scholar

[34]

H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. (9), 74 (1995), 549-578.  Google Scholar

[35]

F. Prinari, Relaxation and $\Gamma$-convergence of supremal functionals, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 9 (2006), 101-132.  Google Scholar

[36]

F. Prinari, Semicontinuity and relaxation of $L^\infty$-functionals, Adv. Calc. Var., 2 (2009), 43-71. doi: 10.1515/ACV.2009.003.  Google Scholar

[37]

J. Serrin, On the definition and the properties of certain variational integrals, Trans. Amer. Math. Soc., 101 (1961), 139-167. doi: 10.1090/S0002-9947-1961-0138018-9.  Google Scholar

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