Singularities of Solutions of the Eikonal Equation

Abstract

The structure of singularities of solutions u(·) of the eikonal equation ∣∇u∣ = 1, u∣∂Ω = 0 on a domain Ω ⊂ ℝⁿ is studied. To this end, we consider smooth hypersurfaces that play the role of level surfaces of a possible solution. The singularities of the distance function are considered for such surfaces, since it is known that locally, up to a constant and sign, a smooth solution of the eikonal equation can be represented in the form f (x) = ρ(x, Ω), where ρ is the distance from a point to a set. In a finite-dimensional space, the singular set of a nonempty closed subset M is defined as the closure of the set of nonuniqueness points of the metric projection onto the set M. In the present paper, we describe the C¹ -hypersurfaces in ℝⁿ representing solutions to the eikonal equations for which the singular set is a subspace of any finite dimension in ℝⁿ.