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
is said to be * infinity harmonic * if it
solves the nonlinear degenerate elliptic equation

in the viscosity sense. This
is equivalent to the requirement that
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
which enjoys comparison with cones has a derivative at a point
in its domain to determining whether or not maximum points of
relative to spheres centered at
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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