Homogenization of metrics in oscillating manifolds

Braides, Andrea; Cancedda, Andrea; Piat, Valeria Chiadò

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 889-912.

We consider energies defined as the Dirichlet integral of curves taking values in fast-oscillating manifolds converging to a linear subspace. We model such manifolds as subsets of $R^{m+m\text{'}}$ described by a constraint $(x_{m+1},...,x_{m\text{'}})=\delta \varphi (x_1/\epsilon ,...,x_m/\epsilon )$ where $\epsilon$ is the period of the oscillation, $\delta$ its amplitude and $\varphi$ its profile. The interesting case is $\epsilon <<\delta <<1$, in which the limit of the energies is described by a Finsler metric on $R^m$ which is defined by optimizing the contribution of oscillations on each level set $\{\varphi =c\}$. The formulas describing the limit mix homogenization and convexification processes, highlighting a multi-scale behaviour of optimal sequences. We apply these formulas to show that we may obtain all (homogeneous) symmetric Finsler metrics larger than the Euclidean metric as limits in the case of oscillating surfaces in $R^3$.

Classification : 35B27, 49J45, 58B20
Mots clés : Homogenization, oscillating manifolds, Finsler metrics

N. Ansini, A. Braides and V. Chiadò Piat, Homogenization of periodic multi-dimensional structures. Boll. Unione Mat. Ital. 2 (1999) 735–758. | MR | Zbl

J.F. Babadjian and V. Millot, Homogenization of variational problems in manifold valued BV-spaces. Calc. Var. Partial Differ. Eq. 36 (2009) 7–47. | DOI | MR | Zbl

J.F. Babadjian and V. Millot, Homogenization of variational problems in manifold valued Sobolev spaces. ESAIM: COCV 16 (2010) 833–855. | Numdam | MR | Zbl

A. Braides, $\Gamma$-convergence for Beginners. Oxford University Press, Oxford (2002). | MR | Zbl

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998). | MR | Zbl

A. Braides and V. Chiadò Piat, Non convex homogenization problems for singular structures. Netw. Heterog. Media 3 (2008) 489–508. | DOI | MR | Zbl

A. Braides, G. Buttazzo and I. Fragalà, Riemannian approximation of Finsler metrics. Asymptot. Anal. 31 (2002) 177–187. | MR | Zbl

A. Braides, A handbook of $\Gamma$-convergence. In Vol. $3$ of Handbook of Differential Equations. Stationary Partial Differential Equations, edited by M. Chipot and P. Quittner. Elsevier (2006) 101–213. | Zbl

G. Bouchitté and I. Fragalà, Homogenization of thin structures by two-scale method with respect to measures. SIAM J. Math. Anal. 32 (2001) 1198–1226. | DOI | MR | Zbl

D. Burago, Periodic metrics, In Seminar on Dynamical Systems, edited by S. Kuksin, V. Lazutkin and J. Pöschel. Vol. 12 of Progress in Nonlinear Differential Equations and Their Applications Birkhäuser, Basel (1994) 90–95. | Zbl

G. Dal Maso, An Introduction to $\Gamma$-convergence. Birkhauser, Boston (1993). | MR | Zbl

A. Davini, On the relaxation of a class of functionals defined on Riemannian distances. J. Convex Anal. 12 (2005) 113–130. | MR | Zbl

A.L. Piatnitski and V.V. Zhikov, Homogenization of random singular structures and random measures. Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006) 23–74. Translation in Izv. Math. 70 (2006) 19–67. | MR | Zbl

A.P. Rybalko, Homogenization of harmonic 1-forms on pseudo Riemannian manifolds of complex microstructure. Mat. Fiz. Anal. Geom. 11 (2004) 249–257. | MR | Zbl

V.V. Zhikov, Averaging of problems in the theory of elasticity on singular structures. Dokl. Akad. Nauk 380 (2001) 741–745. | MR | Zbl