 
Summary: DISTANCE FUNCTIONS AND ALMOST GLOBAL SOLUTIONS OF EIKONAL
EQUATIONS
LUIS A. CAFFARELLI
MICHAEL G. CRANDALL
Abstract. 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 pnorms on Rn, 1 < p < , we show that if S has Hausdorff 1measure 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 .
1. Introduction
The function
u(x) = x, p for x Rn
,
where p Rn
and x, p is the Euclidean innerproduct of x and p, satisfies Du = p and Du = p on Rn
.
Here Du is the gradient of u and  ·  is the Euclidean norm. If p = 1, then u C1
(Rn
) and Du = 1
everywhere. If z Rn
