 
Summary: UNIQUENESS OF HARMONIC FUNCTIONS AND THE EIKONAL EQUATION
MICHAEL G. CRANDALL, GUNNAR GUNNARSSON, & PEIYONG WANG
Abstract. Comparison results are obtained between infinity subharmonic and infinity superharmonic func
tions defined on unbounded domains. The primary new tool employed is an approximation of infinity subhar
monic functions that allows one to assume that gradients are bounded away from zero. This approximation
also demystifies the proof in the case of a bounded domain. A second, quite different, topic is also taken up.
This is the uniqueness of absolutely minimizing functions with respect to other norms besides the Euclidean,
norms that correspond to comparison results for partial differential equations which are quite discontinuous.
Introduction
In this paper, we take up two topics related to the uniqueness of solutions of the Dirichlet problem for
the  Laplace equation u = 0, where the "Laplacian" is given on smooth functions by
u =
n
i,j=1
uxi
uxj
uxixj
.
The first topic is the study of the uniqueness of solutions of Dirichlet problems for the Laplace equation in
unbounded domains, for example, exterior domains. To explain, let U be an open subset of Rn
