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Homogenization of variational problemsin manifold valued Sobolev spaces

Published online by Cambridge University Press:  31 July 2009

Jean-François Babadjian  and
Vincent Millot
Jean-François Babadjian
Affiliation:
CMAP, UMR 7641, École polytechnique, 91128 Palaiseau, France. [email protected]
Vincent Millot
Affiliation:
Université Paris Diderot – Paris 7, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France. [email protected]
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Abstract

Homogenization of integral functionals is studiedunder the constraint that admissible maps have to take their valuesinto a given smooth manifold. The notion of tangentialhomogenization is defined by analogy with the tangentialquasiconvexity introduced by Dacorogna et al. [Calc. Var. Part. Diff. Eq. 9 (1999) 185–206]. For energies with superlinear or linear growth, aΓ-convergence result is established in Sobolev spaces, thehomogenization problem in the space of functions of boundedvariation being the object of [Babadjian and Millot, Calc. Var. Part. Diff. Eq.36 (2009) 7–47].

Keywords

HomogenizationΓ-convergencemanifold valued maps
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 4 , October 2010 , pp. 833 - 855
Copyright
© EDP Sciences, SMAI, 2009

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References

Alicandro, R. and Leone, C., 3D-2D asymptotic analysis for micromagnetic energies. ESAIM: COCV 6 (2001) 489–498. CrossRef
L. Ambrosio and G. Dal Maso, On the relaxation in $BV(\Omega;\mathbb{R}^m)$ of quasiconvex integrals. J. Funct. Anal. 109 (1992) 76–97.
Babadjian, J.-F. and Millot, V., Homogenization of variational problems in manifold valued $BV$ -spaces. Calc. Var. Part. Diff. Eq. 36 (2009) 7–47. CrossRef
Béthuel, F., The approximation problem for Sobolev maps between two manifolds. Acta Math. 167 (1991) 153–206. CrossRef
Béthuel, F. and Zheng, X., Density of smooth functions between two manifolds in Sobolev spaces. J. Funct. Anal. 80 (1988) 60–75. CrossRef
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.
Braides, A., Homogenization of some almost periodic coercive functional. Rend. Accad. Naz. Sci. XL 103 (1985) 313–322.
A. Braides and A. Defranceschi, Homogenization of multiple integrals, Oxford Lecture Series in Mathematics and its Applications 12. Oxford University Press, New York (1998).
Braides, A., Defranceschi, A. and Vitali, E., Homogenization of free discontinuity problems. Arch. Rational Mech. Anal. 135 (1996) 297–356. CrossRef
Brézis, H., Coron, J.M. and Lieb, E.H., Harmonic maps with defects. Comm. Math. Phys. 107 (1986) 649–705. CrossRef
B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag (1989).
Dacorogna, B., Fonseca, I., Malý, J. and Trivisa, K., Manifold constrained variational problems. Calc. Var. Part. Diff. Eq. 9 (1999) 185–206. CrossRef
G. Dal Maso, An Introdution to Γ-convergence. Birkhäuser, Boston (1993).
I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels. Dunod, Gauthiers-Villars, Paris (1974).
Fonseca, I. and Müller, S., Quasiconvex integrands and lower semicontinuity in L 1. SIAM J. Math. Anal. 23 (1992) 1081–1098. CrossRef
I. Fonseca and S. Müller, Relaxation of quasiconvex functionals in $BV(\Omega;\mathbb{R}^p)$ for integrands $f(x,u,\nabla u)$ . Arch. Rational Mech. Anal. 123 (1993) 1–49.
Fonseca, I., Müller, S. and Pedregal, P., Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736–756. CrossRef
M. Giaquinta, L. Modica and J. Souček, Cartesian currents in the calculus of variations, Modern surveys in Mathematics 37-38. Springer-Verlag, Berlin (1998).
Giaquinta, M., Modica, L. and Mucci, D., The relaxed Dirichlet energy of manifold constrained mappings. Adv. Calc. Var. 1 (2008) 1–51. CrossRef
Marcellini, P., Periodic solutions and homogenization of nonlinear variational problems. Ann. Mat. Pura Appl. (4) 117 (1978) 139–152. CrossRef
Müller, S., Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Rational Mech. Anal. 99 (1987) 189–212. CrossRef