On unbounded solutions of ergodic problems for non-local Hamilton–Jacobi equations. (March 2019)

• Record Type: Journal Article
• Authors: Brändle, Cristina; Chasseigne, Emmanuel
• Is Part Of: Nonlinear analysis. Volume 180(2019)
• Page Start: 94
• Page End: 128
• Publication Date: 2019-03
• Subjects: primary 45K05 49L25 35B50 35J60; Viscosity solutions -- Non-local equation -- Ergodic problem; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.09.015
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study an ergodic problem associated to a non-local Hamilton–Jacobi equation defined on the whole space λ − L [ u ] ( x ) + | D u ( x ) | m = f ( x ) and determine whether (unbounded) solutions exist or not. We prove that there is a threshold growth of the function f, that separates existence and non-existence of solutions, a phenomenon that does not appear in the local version of the problem. Moreover, we show that there exists a critical ergodic constant, λ ∗, such that the ergodic problem has solutions for λ ⩽ λ ∗ and such that the only solution bounded from below, which is unique up to an additive constant, is the one associated to λ ∗ .