An infinity-harmonic function is a solution of 
, where 
is the unnormalized infinity
Laplacian. For a continuous function,
the viscosity interpretation requires 
whenever a twice continuously differentiable
test function 
touches 
from above at 
, and the reverse inequality for a test touching from below.
Tests whose gradient is zero are included. Affine functions are examples.
Infinity-Harmonic Function
See also
Hölder Condition, Infinity Laplacian, Viscosity SolutionExplore with Wolfram|Alpha
References
Lindqvist, P. Notes on the Infinity Laplace Equation. Cham, Switzerland: Springer, 2016. https://doi.org/10.1007/978-3-319-31532-4.Cite this as:
Weisstein, Eric W. "Infinity-Harmonic Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Infinity-HarmonicFunction.html