Calculus of Variations and Geometric Measure Theory

A. Braides - A. Cancedda - V. Chiadò Piat

Homogenization of metrics in oscillating manifolds

created by braidesa on 06 Nov 2015
modified on 19 May 2017


Published Paper

Inserted: 6 nov 2015
Last Updated: 19 may 2017

Journal: ESAIM: Control Optim. Calc. Var.
Volume: 23
Pages: 889-912
Year: 2017
Doi: 10.1051/cocv/2016018


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 ${\mathbb R}^{m+m'}$ described by a constraint $(x_{m+1},\ldots,x_{m'})= \delta \, \varphi(x_1/\varepsilon,\ldots, x_m/\varepsilon)$ where $\varepsilon$ is the period of the oscillation, $\delta$ its amplitude and $\varphi$ its profile. The interesting case is $\varepsilon<\!<\delta<\!<1$, in which the limit of the energies is described by a Finsler metric on ${\mathbb R}^{m}$ 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 ${\mathbb R}^{3}$.