Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 122, pp. 1-4.

Title: A remark on $C^2$ infinity-harmonic functions

Author: Yifeng Yu (Univ. of Texas, Austin, Texas, USA)

Abstract: 
 In this paper, we prove that any nonconstant, $C^2$ solution
 of the infinity Laplacian equation $u_{x_i}u_{x_j}u_{x_ix_j}=0$
 can not have interior critical points. This result was first
 proved by Aronsson [2] in two dimensions. When the solution
 is $C^4$,  Evans [6]  established a Harnack inequality
 for $|Du|$,  which  implies that non-constant $C^4$ solutions
 have no interior critical points for any dimension.
 Our method is strongly motivated by the work in [6].
 
Submitted June 15, 2006. Published October 6, 2006.
Math Subject Classifications: 35B38.
Key Words: Infinity Laplacian equation; infinity harmonic function;  
           viscosity solutions.