Skip to main content Accessibility help
×
Home

Homogenization of variational problems in manifold valued Sobolev spaces

  • Jean-François Babadjian (a1) and Vincent Millot (a2)

Abstract

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].

Copyright

References

Hide All
[1] Alicandro, R. and Leone, C., 3D-2D asymptotic analysis for micromagnetic energies. ESAIM: COCV 6 (2001) 489498.
[2] 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.
[3] Babadjian, J.-F. and Millot, V., Homogenization of variational problems in manifold valued $BV$ -spaces. Calc. Var. Part. Diff. Eq. 36 (2009) 747.
[4] Béthuel, F., The approximation problem for Sobolev maps between two manifolds. Acta Math. 167 (1991) 153206.
[5] Béthuel, F. and Zheng, X., Density of smooth functions between two manifolds in Sobolev spaces. J. Funct. Anal. 80 (1988) 6075.
[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.
[7] Braides, A., Homogenization of some almost periodic coercive functional. Rend. Accad. Naz. Sci. XL 103 (1985) 313322.
[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).
[9] Braides, A., Defranceschi, A. and Vitali, E., Homogenization of free discontinuity problems. Arch. Rational Mech. Anal. 135 (1996) 297356.
[10] Brézis, H., Coron, J.M. and Lieb, E.H., Harmonic maps with defects. Comm. Math. Phys. 107 (1986) 649705.
[11] B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag (1989).
[12] Dacorogna, B., Fonseca, I., Malý, J. and Trivisa, K., Manifold constrained variational problems. Calc. Var. Part. Diff. Eq. 9 (1999) 185206.
[13] G. Dal Maso, An Introdution to Γ-convergence. Birkhäuser, Boston (1993).
[14] I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels. Dunod, Gauthiers-Villars, Paris (1974).
[15] Fonseca, I. and Müller, S., Quasiconvex integrands and lower semicontinuity in L 1. SIAM J. Math. Anal. 23 (1992) 10811098.
[16] 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.
[17] Fonseca, I., Müller, S. and Pedregal, P., Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736756.
[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).
[19] Giaquinta, M., Modica, L. and Mucci, D., The relaxed Dirichlet energy of manifold constrained mappings. Adv. Calc. Var. 1 (2008) 151.
[20] Marcellini, P., Periodic solutions and homogenization of nonlinear variational problems. Ann. Mat. Pura Appl. (4) 117 (1978) 139152.
[21] Müller, S., Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Rational Mech. Anal. 99 (1987) 189212.

Keywords

Related content

Powered by UNSILO

Homogenization of variational problems in manifold valued Sobolev spaces

  • Jean-François Babadjian (a1) and Vincent Millot (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.