Homogenization of variational problems in manifold valued Sobolev spaces

Babadjian, Jean-François; Millot, Vincent

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].

Classification : 74Q05, 49J45, 49Q20
Mots clés : homogenization, Γ-convergence, manifold valued maps

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

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

[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

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

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

[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

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

[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

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

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

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

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

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

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

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

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

[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

[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

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

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

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