Non-oriented solutions of the eikonal equation
Mark A. Peletier, Marco Veneroni · Comptes Rendus Mathématique · 2010
We study a new formulation for the eikonal equation | ∇ u | = 1 on a bounded subset of R 2 . Instead of a vector field ∇ u , we consider a field P of orthogonal projections on one-dimensional subspaces, with div P ∈ L 2 . We prove that solutions of this equation propagate direction as in the classical eikonal equation. We also show that solutions exist if and only if the domain is a tubular neighborhood of a regular closed curve.