On an evolutionary equation involving the nonlocal infinity Laplacian

Abstract

We study the evolutionary equation $$\frac{\partial u}{\partial t} = (\mathcal{L}_{\infty})^{\alpha}u \quad \text{in} \quad \Omega,$$ where $(\mathcal{L}_{\infty})^{\alpha}u$ denotes a nonlocal infinity Laplacian acting on the function $u$, $0<\alpha \leq 1$ 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 $0<\alpha<1$.