There are several non-equivalent definitions of the fractional \(\infty\)-Laplacian. The authors study the initial value problem \[ \begin{cases} \begin{alignedat}{3} \frac{\partial}{\partial t} u(x,t) & = \Delta^s_\infty u(x,t),\qquad & x\in\mathbb{R}^n, &\ t>0,\\ u(x,0) &= u_0(x), & x\in\mathbb{R}^n & \end{alignedat} \end{cases} \] using the definition \[ \Delta^s_\infty \phi(x)= C_s \sup_{|y|=1}\inf_{|\widetilde{y}|=1} \int^\infty_0\frac{\phi(x+\eta y)+\phi(x-\eta\widetilde{y})- 2\phi(x)}{\eta^{1+2s}}d\eta \] by C. Bjorland et al. [Adv. Math. 230, No. 4–6, 1859–1894 (2012; Zbl 1252.35099)]. They first rewrite the definition in a practical way, discriminating the critical points \((\nabla\phi= 0)\). The classical formula \[ \phi\left(x+\varepsilon\frac{\nabla\phi(x)}{|\nabla\phi(x)|}\right)+\phi\left(x-\varepsilon\frac{\nabla\phi(x)}{|\nabla\phi(x)|}\right)-2\phi(x)=\varepsilon^2|\nabla\phi(x)|^{-2}\Delta_\infty\phi(x)+O(\varepsilon^4) \] is lurking. (Here \(\Delta_\infty\phi= \sum_{i,j=1}^n \frac{\partial\phi}{\partial x_i}\frac{\partial\phi}{\partial x_j}\frac{\partial^2\phi}{\partial x_i\partial x_j}\).)
This paper contains substantial results:

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Viscosity solutions are defined and provided with an existence theory.

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For classical solutions, uniqueness is established.

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A so-called Harnack estimate is obtained for viscosity solutions.

A delicate situation emerges. The uniqueness of viscosity solutions is yet lacking, while the existence result for classical solutions is restricted to some expedient special cases. In an ingenious way, the authors use a special kind of classical solutions that are directly related to the one-dimensional fractional heat equation \[ \frac{\partial v(x,t)}{\partial t}+(-\partial_{xx})^s v(x,t)=0. \] These special solutions are then compared with the viscosity solutions. In this way, Harnack estimates are reached for viscosity solutions.
An approximation scheme plays a central rôle.