We study a new formulation for the eikonal equation |grad u| =1 on a bounded
subset of R^2. Instead of a vector field grad u, we consider a field P of
orthogonal projections on 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. The idea of the proof is to apply a generalized method of
characteristics introduced in Jabin, Otto, Perthame, "Line-energy
Ginzburg-Landau models: zero-energy states", Ann. Sc. Norm. Super. Pisa Cl.
Sci. (5) 1 (2002), to a suitable vector field m satisfying P = m \otimes m.
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.Comment: 14 pages, 4 figures, submitte
