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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 |
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] | |
[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] | |
[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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