Ergodic pairs for fractional Hamilton-Jacobi equations on bounded domains: large solutions

TL;DR

This paper investigates the existence and properties of ergodic solutions to fractional Hamilton-Jacobi equations with censored fractional Laplacian on bounded domains, focusing on blow-up behavior and ergodic constants.

Contribution

It extends the theory of ergodic problems to nonlocal fractional operators with boundary blow-up solutions, using the vanishing discount method.

Findings

Existence of ergodic pairs with boundary blow-up solutions.

Characterization of the ergodic constant and blow-up rates.

Qualitative properties of solutions for fractional Hamilton-Jacobi equations.

Abstract

In this article, we study the ergodic problem associated to viscous Hamilton-Jacobi equation where the diffusion is governed by the censored fractional Laplacian, a nonlocal elliptic operator restricted to a bounded domain $\Omega \subset \mathbb{R}^N$. We restrict ourselves to the case in which the nonlinear gradient term has a scaling less or equal than the fractional order of the diffusion. In similarity to its second-order counterpart, we provide existence of ergodic pairs involving solutions that blow-up on $\partial \Omega$. We use the celebrated vanishing discount method, where the analysis of the approximated solutions have its own interest, leading to qualitative properties for the ergodic problem such as precise blow-up rates for the solution and characterization of the ergodic constant. The main difficulties arise from the state-dependency of the operator, from which the arguments of the local case based on well-known invariance properties of the Laplacian are not longer at disposal.