Distance Functions and Almost Global Solutions of Eikonal Equations

摘要

Consider a function u defined onn, except, perhaps, on a closed set of potentialsingularities.Suppose that u solves the eikonal equation Du = 1in the pointwise sense onn, where Du denotes the gradient of u and ·is a norm onn with the dual norm.For a class of norms which includes the standard p-norms on n, 1 < p <, we show that if has Hausdorff 1-measure zero and n ≥ 2, then u is either affine or a"cone unction"that is, a function of the formux = a ±x z.