Homogenization of metrics in oscillating manifolds

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 $(x_{m+1},\ldots,x_{m'})= δ\, φ(x_1/ε,\ldots, x_m/ε)$ where $ε$ is the period of the oscillation, $δ$ its amplitude and $φ$ its profile. The interesting case is $ε<\!<δ<\!<1$, 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 $\{φ=c\}$. 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 $R^{3}$