Homogenization of metrics in oscillating manifolds

TL;DR

This paper studies the limiting behavior of energies of curves constrained to oscillating manifolds, revealing a homogenized Finsler metric that captures multi-scale oscillation effects.

Contribution

It introduces a new homogenization framework for energies on oscillating manifolds, deriving explicit formulas for the limit metrics involving convexification and multi-scale analysis.

Findings

Limit energies are described by a Finsler metric on R^m.

All symmetric Finsler metrics larger than Euclidean can be obtained as limits.

The formulas combine homogenization and convexification processes.

Abstract

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 $R^{m+m'}$ described by a constraint $(x_{m+1},\ldots,x_{m'})= \delta \, \varphi(x_1/\epsilon,\ldots, x_m/\epsilon)$ 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 $R^{3}$