Plane-like minimizers and differentiability of the stable norm

TL;DR

This paper studies the smoothness and convexity of the stable norm in periodic surface tension problems, showing differentiability in irrational directions and linking it to foliation properties of minimizers.

Contribution

It establishes the differentiability of the stable norm in irrational directions and characterizes when it is differentiable in rational directions based on foliation by plane-like minimizers.

Findings

Stable norm is always differentiable in totally irrational directions.

Differentiability in rational directions depends on foliation by plane-like minimizers.

Discusses conditions for uniqueness of homogenization correctors.

Abstract

In this paper we investigate the strict convexity and the differentiability properties of the stable norm, which corresponds to the homogenized surface tension for a periodic perimeter homogenization problem (in a regular and uniformly elliptic case). We prove that it is always differentiable in totally irrational directions, while in other directions, it is differentiable if and only if the corresponding plane-like minimizers satisfying a strong Birkhoff property foliate the torus. We also discuss the issue of the uniqueness of the correctors for the corresponding homogenization problem.