Differentiability of effective fronts in the continuous setting in two dimensions

TL;DR

This paper proves that the boundary of the effective front in two-dimensional periodic media with continuous coefficients is differentiable at irrational points, revealing a fundamental property of effective fronts in the PDE setting.

Contribution

It establishes the differentiability of the effective front boundary at irrational points for continuous metrics, a novel result in the PDE theory context.

Findings

Effective front boundary is differentiable at irrational points.

Polygonal effective fronts must be centrally symmetric with rational vertices.

First nontrivial property of effective fronts in the continuous setting.

Abstract

We study the effective front associated with first-order front propagations in two dimensions ($n=2$) in the periodic setting with continuous coefficients. Our main result says that that the boundary of the effective front is differentiable at every irrational point. Equivalently, the stable norm associated with a continuous $\mathbb{Z}^2$-periodic Riemannian metric is differentiable at irrational points. This conclusion was obtained decades ago for smooth metrics ([3,5]). To the best of our knowledge, our result provides the first nontrivial property of the effective fronts in the continuous setting, which is the standard assumption in the PDE theory. Combining with the sufficiency result in [12], our result implies that for continuous coefficients, a polygon could be an effective front if and only if it is centrally symmetric with rational vertices and nonempty interior.