Partial Differential Equations Non-oriented Solutions of the Eikonal Equation

We study a new formulation for the eikonal equation |∇u| = 1 on a bounded subset of R. Instead of a vector field ∇u, we consider a field P of orthogonal projections on 1-dimensional subspaces, with div P ∈ L. 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 neighbourhood of a regular closed curve. Résumé Nous étudions une nouvelle formulation de l’équation eikonale |∇u| = 1 sur un sous-ensemble borné de R. Au lieu d’un champ de vecteurs ∇u, nous considérons un champ P de projections orthogonales sur les sous-espaces de dimension 1, avec div P ∈ L. Nous montrons que les solutions de cette équation propagent la direction comme dans l’équation eikonale classique. Nous montrons aussi que les solutions existent si et seulement si le domaine est un voisinage tubulaire d’une courbe régulière fermée. 1. Stripe patterns and the eikonal equation Many pattern-forming systems produce parallel stripes, both straight and curved. In this note we report on a new mathematical description of curved striped patterns. We recently studied the behaviour of a stripe-forming energy, and investigated a limit process in which the stripe width tends to zero [4]. In that limit the stripes not only become thin, but also uniform in width, and the stripe pattern comes to resemble the level sets of a solution of the eikonal equation. The rigorous version of this statement, in the 0 [email protected] 0 [email protected] Preprint submitted to the Académie des sciences May 22, 2010 form of a Gamma-convergence result, gives rise to a new formulation of the eikonal equation, in which the directionality of the stripes is represented, rather than by vector fields, by line fields, which capture direction only up to a sign (Figure 1 (right)). Figure 1. Stripe patterns represented by level sets of functions u with |∇u| = 1 (left, arrows are the gradient ∇u) or by unoriented line fields (right). The line field is represented by a projection, which for the purposes of this paper we define to be a matrix P that can be written in terms of a unit vector m as P = m⊗m. Note that both m and −m give rise to the same projection P ; this is the unsigned nature of a line field. The projection-valued eikonal equation is as follows. Let Ω be an open subset of R2. Find P ∈ L ∞(R2; R2×2), with P = 0 a.e. in R2\Ω, such that P 2 = P, rank(P ) = 1, P is symmetric, a.e. in Ω, (1a) div P ∈ L(R; R) (1b) P divP = 0 a.e. in Ω. (1c) Here div P is defined as the vector-valued function whose i-th component is given by (div P )i := �2 j=1 ∂xj Pij . The first line encodes the property that P (x) is a projection, in the sense above, at a.e. x ∈ Ω. Given the regularity provided by (1b), the final condition (1c) is the eikonal equation itself, as a calculation for a smooth unit-length vector field m(x) shows. Indeed, we have 0 = P div P = m(m · (m div m +∇m · m)) = m div m + m(m ·∇m · m) = m div m, (2) where the final equality follows from differentiating the identity |m|2 = 1. A solution vector field m therefore is divergence-free, implying that its rotation over 90 degrees is a gradient ∇u; from |m| = 1 follows the classical eikonal equation |∇u| = 1. Property (1b) also prevents the normal component of P from jumping across ∂Ω, and it implies that P · n = 0 in the sense of traces on ∂Ω. The exponent 2 in (1b) is critical in the following sense. Natural possibilities for singularities in a line field are jump discontinuities (‘grain boundaries’) and target patterns. At a grain boundary the jump in P causes div P to have a line singularity, comparable to the one-dimensional Hausdorff measure; condition (1b) clearly excludes that possibility. For a target pattern the curvature κ of the stripes scales as 1/r, where r is the distance to the center; then � κ p is locally finite for p < 2, and diverges logarithmically for p = 2. The cases p < 2 and p ≥ 2 therefore distinguish between whether target patterns are admissible (p < 2) or not.