USA-Chile Workshop on Nonlinear Analysis,
Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 123-129.

A remark on infinity harmonic functions

Michael G. Crandall & L. C. Evans

Abstract:
A real-valued function $u$ is said to be 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.

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Michael G. Crandall
Department of Mathematics
University of California
Santa Barbara, CA, 93106-3080, USA
e-mail: crandall@math.ucsb.edu

Lawrence Craig Evans
Department of Mathematics
University of California, Berkeley
Berkeley, CA 94720-0001, USA
e-mail: evans@math.berkeley.edu

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