Homogenization of variational problems in manifold valued Sobolev spaces
ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 4, pp. 833-855.

Homogenization of integral functionals is studied under the constraint that admissible maps have to take their values into a given smooth manifold. The notion of tangential homogenization is defined by analogy with the tangential quasiconvexity introduced by Dacorogna et al. [Calc. Var. Part. Diff. Eq. 9 (1999) 185-206]. For energies with superlinear or linear growth, a Γ-convergence result is established in Sobolev spaces, the homogenization problem in the space of functions of bounded variation being the object of [Babadjian and Millot, Calc. Var. Part. Diff. Eq. 36 (2009) 7-47].

DOI : https://doi.org/10.1051/cocv/2009025
Classification : 74Q05,  49J45,  49Q20
Mots clés : homogenization, Γ-convergence, manifold valued maps
@article{COCV_2010__16_4_833_0,
author = {Babadjian, Jean-Fran\c{c}ois and Millot, Vincent},
title = {Homogenization of variational problems in manifold valued Sobolev spaces},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {833--855},
publisher = {EDP-Sciences},
volume = {16},
number = {4},
year = {2010},
doi = {10.1051/cocv/2009025},
mrnumber = {2744153},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2009025/}
}
Babadjian, Jean-François; Millot, Vincent. Homogenization of variational problems in manifold valued Sobolev spaces. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 4, pp. 833-855. doi : 10.1051/cocv/2009025. http://www.numdam.org/articles/10.1051/cocv/2009025/

[1] R. Alicandro and C. Leone, 3D-2D asymptotic analysis for micromagnetic energies. ESAIM: COCV 6 (2001) 489-498. | Numdam | Zbl 0989.35009

[2] L. Ambrosio and G. Dal Maso, On the relaxation in $BV\left(\Omega ;{ℝ}^{m}\right)$ of quasiconvex integrals. J. Funct. Anal. 109 (1992) 76-97. | Zbl 0769.49009

[3] J.-F. Babadjian and V. Millot, Homogenization of variational problems in manifold valued $BV$-spaces. Calc. Var. Part. Diff. Eq. 36 (2009) 7-47. | Zbl 1169.74037

[4] F. Béthuel, The approximation problem for Sobolev maps between two manifolds. Acta Math. 167 (1991) 153-206. | Zbl 0756.46017

[5] F. Béthuel and X. Zheng, Density of smooth functions between two manifolds in Sobolev spaces. J. Funct. Anal. 80 (1988) 60-75. | Zbl 0657.46027

[6] F. Béthuel, H. Brézis and J.M. Coron, Relaxed energies for harmonic maps, in Variational methods, Paris (1988), H. Berestycki, J.M. Coron and I. Ekeland Eds., Progress in Nonlinear Differential Equations and Their Applications 4, Birkhäuser, Boston (1990) 37-52. | Zbl 0793.58011

[7] A. Braides, Homogenization of some almost periodic coercive functional. Rend. Accad. Naz. Sci. XL 103 (1985) 313-322. | Zbl 0582.49014

[8] A. Braides and A. Defranceschi, Homogenization of multiple integrals, Oxford Lecture Series in Mathematics and its Applications 12. Oxford University Press, New York (1998). | Zbl 0911.49010

[9] A. Braides, A. Defranceschi and E. Vitali, Homogenization of free discontinuity problems. Arch. Rational Mech. Anal. 135 (1996) 297-356. | Zbl 0924.35015

[10] H. Brézis, J.M. Coron and E.H. Lieb, Harmonic maps with defects. Comm. Math. Phys. 107 (1986) 649-705. | Zbl 0608.58016

[11] B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag (1989). | Zbl 1140.49001

[12] B. Dacorogna, I. Fonseca, J. Malý and K. Trivisa, Manifold constrained variational problems. Calc. Var. Part. Diff. Eq. 9 (1999) 185-206. | Zbl 0935.49006

[13] G. Dal Maso, An Introdution to Γ-convergence. Birkhäuser, Boston (1993). | Zbl 0816.49001

[14] I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels. Dunod, Gauthiers-Villars, Paris (1974). | Zbl 0281.49001

[15] I. Fonseca and S. Müller, Quasiconvex integrands and lower semicontinuity in L1. SIAM J. Math. Anal. 23 (1992) 1081-1098. | Zbl 0764.49012

[16] I. Fonseca and S. Müller, Relaxation of quasiconvex functionals in $BV\left(\Omega ;{ℝ}^{p}\right)$ for integrands $f\left(x,u,\nabla u\right)$. Arch. Rational Mech. Anal. 123 (1993) 1-49. | Zbl 0788.49039

[17] I. Fonseca, S. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736-756. | Zbl 0920.49009

[18] M. Giaquinta, L. Modica and J. Souček, Cartesian currents in the calculus of variations, Modern surveys in Mathematics 37-38. Springer-Verlag, Berlin (1998). | Zbl 0914.49002

[19] M. Giaquinta, L. Modica and D. Mucci, The relaxed Dirichlet energy of manifold constrained mappings. Adv. Calc. Var. 1 (2008) 1-51. | Zbl 1157.49043

[20] P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems. Ann. Mat. Pura Appl. (4) 117 (1978) 139-152. | Zbl 0395.49007

[21] S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Rational Mech. Anal. 99 (1987) 189-212. | Zbl 0629.73009