A remark on C2 infinity-harmonic functions

Abstract

In this paper, we prove that any nonconstant, C2 solution of the infinity Laplacian equation uₓiuₓj uₓiₓj = 0 can not have interior critical points. This result was first proved by Aronsson [2] in two dimensions. When the solution is C4, Evans [6] established a Harnack inequality for |Du|, which implies that non-constant C4 solutions have no interior critical points for any dimension. Our method is strongly motivated by the work in [6].