Dimensional reduction for supremal functionals

Previous Article
Stable manifolds with optimal regularity for difference equations
DCDS Home
This Issue
Next Article
Front tracking approximations for slow erosion

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

Abstract
Related Papers

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 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.

[1]	Lorenza D'Elia. $ \Gamma $-convergence of quadratic functionals with non uniformly elliptic conductivity matrices. Networks and Heterogeneous Media, 2022, 17 (1) : 15-45. doi: 10.3934/nhm.2021022
[2]	Tasnim Fatima, Ekeoma Ijioma, Toshiyuki Ogawa, Adrian Muntean. Homogenization and dimension reduction of filtration combustion in heterogeneous thin layers. Networks and Heterogeneous Media, 2014, 9 (4) : 709-737. doi: 10.3934/nhm.2014.9.709
[3]	David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1335-1343. doi: 10.3934/cpaa.2008.7.1335
[4]	Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. Homogenization of variational functionals with nonstandard growth in perforated domains. Networks and Heterogeneous Media, 2010, 5 (2) : 189-215. doi: 10.3934/nhm.2010.5.189
[5]	Julián Fernández Bonder, Analía Silva, Juan F. Spedaletti. Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2125-2140. doi: 10.3934/dcds.2020355
[6]	Gianni Dal Maso. Ennio De Giorgi and $\mathbf\Gamma$-convergence. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1017-1021. doi: 10.3934/dcds.2011.31.1017
[7]	Alexander Mielke. Deriving amplitude equations via evolutionary $\Gamma$-convergence. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2679-2700. doi: 10.3934/dcds.2015.35.2679
[8]	Ning Zhang, Qiang Wu. Online learning for supervised dimension reduction. Mathematical Foundations of Computing, 2019, 2 (2) : 95-106. doi: 10.3934/mfc.2019008
[9]	Lyudmila Grigoryeva, Juan-Pablo Ortega. Dimension reduction in recurrent networks by canonicalization. Journal of Geometric Mechanics, 2021, 13 (4) : 647-677. doi: 10.3934/jgm.2021028
[10]	Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations and Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067
[11]	Hartmut Schwetlick, Daniel C. Sutton, Johannes Zimmer. On the $\Gamma$-limit for a non-uniformly bounded sequence of two-phase metric functionals. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 411-426. doi: 10.3934/dcds.2015.35.411
[12]	Sylvia Serfaty. Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1427-1451. doi: 10.3934/dcds.2011.31.1427
[13]	Antonio De Rosa, Domenico Angelo La Manna. A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2101-2116. doi: 10.3934/cpaa.2021059
[14]	Martin Kružík, Ulisse Stefanelli, Chiara Zanini. Quasistatic evolution of magnetoelastic plates via dimension reduction. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5999-6013. doi: 10.3934/dcds.2015.35.5999
[15]	Jean Louis Woukeng. $\sum $-convergence and reiterated homogenization of nonlinear parabolic operators. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1753-1789. doi: 10.3934/cpaa.2010.9.1753
[16]	Jie Zhao. Convergence rates for elliptic reiterated homogenization problems. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2787-2795. doi: 10.3934/cpaa.2013.12.2787
[17]	Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503
[18]	Frédéric Legoll, William Minvielle. Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 1-27. doi: 10.3934/dcdss.2015.8.1
[19]	Florian Méhats, Christof Sparber. Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5097-5118. doi: 10.3934/dcds.2016021
[20]	Nikolai Dokuchaev. Dimension reduction and Mutual Fund Theorem in maximin setting for bond market. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1039-1053. doi: 10.3934/dcdsb.2011.16.1039

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Jean-François Babadjian Francesca Prinari Elvira Zappale
on Google Scholar
Jean-François Babadjian Francesca Prinari Elvira Zappale

[Back to Top]