Stripe patterns and the Eikonal equation

Mark A. Peletier; Marco Veneroni

doi:10.3934/dcdss.2012.5.183

Article Contents

2012, Volume 5, Issue 1: 183-189. Doi: 10.3934/dcdss.2012.5.183

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Stripe patterns and the Eikonal equation

Mark A. Peletier^1, and
Marco Veneroni^2,

1.
Dept. of Mathematics and Computer Science and Institute for Complex Molecular Systems, Technische Universiteit Eindhoven, PO Box 513, 5600 MB Eindhoven
2.
Technische Universität Dortmund, Fakultät für Mathematik, Lehrstuhl I, Vogelpothsweg 87, 44227 Dortmund

Received: March 31, 2009

Revised: November 30, 2009

Published: January 2012

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Abstract

We study a new formulation for the Eikonal equation $|\nabla u| =1$ on a bounded subset of $\R^2$. Considering a field $P$ of orthogonal projections onto $1$-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.
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.

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Mathematics Subject Classification: 35L65, 35B65.

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References

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