On an evolutionary equation involving the nonlocal infinity Laplacian

Abstract

We study the evolutionary equation $$\frac{\partial u}{\partial t}=({\mathcal{L}}_{\infty }{)}^{\alpha }u\text{in}\Omega ,$$ where $({\mathcal{L}}_{\infty }{)}^{\alpha }u$ denotes a nonlocal infinity Laplacian acting on the function $u$, $0<\alpha \le 1$ and $\Omega$ is a bounded open set in ${\mathbb{R}}^{n+1}$. We prove existence and uniqueness using Perron's method for the Dirichlet problem when $\Omega$ is a cylinder and $0<\alpha <1$.

26 pages