Non-oriented solutions of the eikonal equation

TL;DR

This paper introduces a novel formulation of the eikonal equation using projection fields instead of gradients, proving existence and uniqueness of solutions under specific geometric conditions, with applications to pattern analysis in physical systems.

Contribution

It presents a new projection-based formulation of the eikonal equation, establishing existence, uniqueness, and geometric characterization of solutions, applicable to pattern-invariant physical systems.

Findings

Solutions exist only for domains around regular closed curves.

The formulation is effective for analyzing stripe patterns.

Existence and uniqueness are proven under L^2 divergence conditions.

Abstract

We study a new formulation for the eikonal equation |grad u| =1 on a bounded subset of R^2. Instead of a vector field grad u, we consider a field P of orthogonal projections on 1-dimensional subspaces, with div P 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. The idea of the proof is to apply a generalized method of characteristics introduced in Jabin, Otto, Perthame, "Line-energy Ginzburg-Landau models: zero-energy states", Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), to a suitable vector field m satisfying P = m \otimes m. 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.