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arXiv · 1805.02382

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

Cristina Brändle·Emmanuel Chasseigne
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Abstract

We study an ergodic problem associated to a non-local Hamilton-Jacobi equation defined on the whole space $λ-\mathcal{L}[u](x)+|Du(x)|^m=f(x)$ 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 $λ\leq λ_*$ and such that the only solution bounded from below, which is unique up to an additive constant, is the one associated to $λ_*$.

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Cristina Brändle, Emmanuel Chasseigne. 2018-05-07. On unbounded solutions of ergodic problems for non-local Hamilton-Jacobi equations. https://arxiv.org/abs/1805.02382

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