We study a new formulation for the Eikonal equation |∇u| = 1 on a bounded subset of R^2. 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 pat- terns. 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.
Stripe patterns and the eikonal equation
VENERONI, MARCO
2012-01-01
Abstract
We study a new formulation for the Eikonal equation |∇u| = 1 on a bounded subset of R^2. 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 pat- terns. 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
