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In mathematics, the infinity Laplace operator is a 2nd-order partial differential operator, commonly abbreviated Δ ∞ {\displaystyle \Delta _{\infty }}. It is alternately defined by
or
The first version avoids the singularity which occurs when the gradient vanishes, while the second version is homogeneous of order zero in the gradient. Verbally, the second version is the second derivative in the direction of the gradient. In the case of the infinity Laplace equation Δ ∞ u = 0 {\displaystyle \Delta _{\infty }u=0} , the two definitions are equivalent.
While the equation involves second derivatives, usually solutions are not twice differentiable, as evidenced by the well-known Aronsson solution u = | x | 4 / 3 − | y | 4 / 3 {\displaystyle u=|x|^{4/3}-|y|^{4/3}}. For this reason the correct notion of solutions is that given by the viscosity solutions.