Electronic Journal of Differential Equations,
Conference 06 (2001), pp. 123-129.

Title: A remark on infinity harmonic functions.

Authors: Michael G. Crandall (Univ. of California, Santa Barbara, CA, USA)
  L. C. Evans (Univ. of California, Berkeley, CA, USA)
 
Abstract: 
  A  real-valued function $u$ is said to be {\it infinity harmonic} if it
  solves  the nonlinear degenerate elliptic equation
  $-\sum_{i,j=1}^nu_{x_1}u_{x_j}u_{x_ix_j}=0$
  in the viscosity sense.  This
  is equivalent to the requirement that $u$ enjoys comparison with cones,
  an elementary notion explained below.   Perhaps the primary open problem
  concerning infinity harmonic functions is to determine whether or not
  they are continuously differentiable.  Results in this note reduce the
  problem of whether or not a function $u$ which enjoys comparison with
  cones has a derivative at a point $x_0$ in its domain to determining
  whether or not  maximum points of $u$ relative to spheres centered
  at $x_0$ have a limiting direction as the radius shrinks to zero.

Published January 8, 2001.
Math Subject Classifications: 35D10, 35J60, 35J70.
Key Words: infnity Laplacian; degenerate elliptic; regularity;
  fully nonlinear.