We study a new formulation for the eikonal equation |¿u| = 1 on a bounded
subset of R2. Considering a field P of orthogonal projections onto 1-dimensional
subspaces, with div P ¿ L2, 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.

Original language | English |
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Publisher | arXiv.org |
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Number of pages | 9 |
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Publication status | Published - 2009 |
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Name | arXiv.org [math.AP] |
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Volume | 0904.0731 |
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