An SBV relaxation of the Cross-Newell energy for modeling stripe patterns and their defects

TL;DR

This paper introduces a novel relaxation of the Cross-Newell energy using SBV functions to model stripe patterns with defects, combining topological, analytic, and numerical methods to better understand pattern formation.

Contribution

It presents a new SBV relaxation of the Cross-Newell energy, providing a mathematical framework for modeling stripe patterns with defects and their topological features.

Findings

Energy minimizers exhibit defect structures similar to experimental observations

The SBV relaxation captures multi-valued phase functions and gauge symmetries

Numerical solutions support the theoretical model's validity

Abstract

We investigate stripe patterns formation far from threshold using a combination of topological, analytic, and numerical methods. We first give a definition of the mathematical structure of `multi-valued' phase functions that are needed for describing layered structures or stripe patterns containing defects. This definition yields insight into the appropriate `gauge symmetries' of patterns, and leads to the formulation of variational problems, in the class of special functions with bounded variation, to model patterns with defects. We then discuss approaches to discretize and numerically solve these variational problems. These energy minimizing solutions support defects having the same character as seen in experiments.