Distance Functions and Almost Global Solutions of Eikonal Equations

Consider a function u defined on Rn, except, perhaps, on a closed set of potential singularities S. Suppose that u solves the eikonal equation ‖Du‖ = 1 in the pointwise sense on Rn \S, where Du denotes the gradient of u and ‖ · ‖ is a norm on Rn with the dual norm ‖ · ‖∗. For a class of norms which includes the standard p-norms on Rn, 1 < p < ∞, we show that if S has Hausdorff 1-measure zero and n ≥ 2, then u is either affine or a “cone function,” that is, a function of the form u(x) = a± ‖x− z‖∗.