American Institute of Mathematical Sciences

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

Dimensional reduction for supremal functionals

 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.
Citation: Jean-François Babadjian, Francesca Prinari, Elvira Zappale. Dimensional reduction for supremal functionals. Discrete & Continuous Dynamical Systems - A, 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. 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. Google Scholar [3] O. Alvarez and E. N. Barron, Homogenization in $L^\infty$,, J. Diff. Eq., 183 (2002), 132. Google Scholar [4] G. Aronsson, Minimization problems for the functional sup$_x F(x,f(x),f'(x))$,, Ark. Mat., 6 (1965), 33. 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. Google Scholar [6] G. Aronsson, Minimization problems for the functional sup$_xF(x,f(x),f'(x))$, III,, Ark. Mat., 7 (1969), 509. Google Scholar [7] G. Aronsson, Extension of functions satisfying Lipschitz conditions,, Ark. Mat., 6 (1967), 551. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [16] E. N. Barron and W. Liu, Calculus of variation in $L^\infty$,, Appl. Math. Optim., 35 (1997), 237. doi: 10.1007/s002459900047. Google Scholar [17] A. Braides, "$\Gamma$-Convergence for Beginners,", Oxford Lecture Series in Mathematics and its Applications, 22 (2002). Google Scholar [18] A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals,", Oxford Lectures Series in Mathematics and its Applications, 12 (1998). Google Scholar [19] A. Braides and I. Fonseca, Brittle thin films,, Appl. Math. Optim., 44 (2001), 299. 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. Google Scholar [21] A. Briani and F. Prinari, A representation result for $\Gamma$-limit of supremal functionals,, J. Nonlinear Convex Anal., 4 (2003), 245. 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. 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. 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 (1989). Google Scholar [25] P. Cardaliaguet and F. Prinari, Supremal representation of $L^\infty$ functionals,, App. Math. Optim., 52 (2005), 129. 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. 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 (1993). Google Scholar [29] G. Dal Maso and P. Longo, $\Gamma$-limits of obstacles,, Ann. Mat. Pura Appl. (4), 128 (1981), 1. doi: 10.1007/BF01789466. Google Scholar [30] G. Dal Maso and L. Modica, A general theory of variational functionals,, in, (1981), 1980. 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. 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. 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. 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. 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. Google Scholar [36] F. Prinari, Semicontinuity and relaxation of $L^\infty$-functionals,, Adv. Calc. Var., 2 (2009), 43. 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. 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. 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. Google Scholar [3] O. Alvarez and E. N. Barron, Homogenization in $L^\infty$,, J. Diff. Eq., 183 (2002), 132. Google Scholar [4] G. Aronsson, Minimization problems for the functional sup$_x F(x,f(x),f'(x))$,, Ark. Mat., 6 (1965), 33. 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. Google Scholar [6] G. Aronsson, Minimization problems for the functional sup$_xF(x,f(x),f'(x))$, III,, Ark. Mat., 7 (1969), 509. Google Scholar [7] G. Aronsson, Extension of functions satisfying Lipschitz conditions,, Ark. Mat., 6 (1967), 551. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [16] E. N. Barron and W. Liu, Calculus of variation in $L^\infty$,, Appl. Math. Optim., 35 (1997), 237. doi: 10.1007/s002459900047. Google Scholar [17] A. Braides, "$\Gamma$-Convergence for Beginners,", Oxford Lecture Series in Mathematics and its Applications, 22 (2002). Google Scholar [18] A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals,", Oxford Lectures Series in Mathematics and its Applications, 12 (1998). Google Scholar [19] A. Braides and I. Fonseca, Brittle thin films,, Appl. Math. Optim., 44 (2001), 299. 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. Google Scholar [21] A. Briani and F. Prinari, A representation result for $\Gamma$-limit of supremal functionals,, J. Nonlinear Convex Anal., 4 (2003), 245. 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. 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. 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 (1989). Google Scholar [25] P. Cardaliaguet and F. Prinari, Supremal representation of $L^\infty$ functionals,, App. Math. Optim., 52 (2005), 129. 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. 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 (1993). Google Scholar [29] G. Dal Maso and P. Longo, $\Gamma$-limits of obstacles,, Ann. Mat. Pura Appl. (4), 128 (1981), 1. doi: 10.1007/BF01789466. Google Scholar [30] G. Dal Maso and L. Modica, A general theory of variational functionals,, in, (1981), 1980. 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. 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. 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. 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. 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. Google Scholar [36] F. Prinari, Semicontinuity and relaxation of $L^\infty$-functionals,, Adv. Calc. Var., 2 (2009), 43. 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. doi: 10.1090/S0002-9947-1961-0138018-9. Google Scholar
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