Stripe patterns and the eikonal equation

TL;DR

This paper introduces a new formulation of the eikonal equation using projection fields, providing existence and uniqueness results, and applies it to analyze stripe patterns in physical systems like block copolymers and liquid crystals.

Contribution

It develops a novel projection-based formulation of the eikonal equation, establishing existence, uniqueness, and geometric characterization of solutions in specific domains.

Findings

Solutions exist only for tubular neighborhoods of regular closed curves.

The formulation is well-suited for analyzing stripe patterns in rotationally invariant systems.

Provides a geometric description comparable to classical solutions.

Abstract

We study a new formulation for the eikonal equation |grad u| =1 on a bounded subset of R^2. Considering a field P of orthogonal projections onto 1-dimensional subspaces, with divergence bounded in L^2, we prove existence and uniqueness for solutions of the equation P div P=0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only if the domain is a tubular neighbourhood of a regular closed curve. This formulation provides a useful approach to the analysis of stripe patterns. It is specifically suited to systems where the physical properties of the pattern are invariant under rotation over 180 degrees, such as systems of block copolymers or liquid crystals.