Mark A. Peletier - Dept. of Mathematics and Computer Science and Institute for Complex Molecular Systems, Technische Universiteit Eindhoven, PO Box 513, 5600 MB Eindhoven, Netherlands (email)
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.
Keywords: Eikonal equation, orientable vector fields, pattern formation, Gamma-convergence, block copolymers.
Received: April 2009; Revised: December 2009; Published: February 2011.