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.