Electron. J. Diff. Eqns., Vol. 2006(2006), No. 122, pp. 1-4.

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

Yifeng Yu

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.

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Yifeng Yu
Department of Mathematics
University of Texas
Austin, TX 78712, USA
email: yifengyu@math.utexas.edu

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