On unbounded solutions of ergodic problems for non-local Hamilton-Jacobi equations Articles uri icon

publication date

  • March 2019

start page

  • 94

end page

  • 128

volume

  • 180

International Standard Serial Number (ISSN)

  • 0362-546X

Electronic International Standard Serial Number (EISSN)

  • 1873-5215

abstract

  • We study an ergodic problem associated to a non-local Hamilton–Jacobi equation defined on the whole space λ−L[u](x)+|Du(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∗.

keywords

  • viscosity solutions; non-local equation; ergodic problem