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 and 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.

[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.

[3]

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

[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.

[5]

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

[6]

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

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[19]

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

[20]

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

[21]

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

[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.

[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.

[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.

[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.

[26]

, P. Cardaliaguet and F. Prinari, Unpublished.

[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.

[28]

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

[29]

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

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[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.

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.

[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.

[3]

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

[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.

[5]

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

[6]

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

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[19]

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

[20]

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

[21]

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

[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.

[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.

[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.

[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.

[26]

, P. Cardaliaguet and F. Prinari, Unpublished.

[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.

[28]

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

[29]

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

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[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.

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