Robustness for a Liouville type theorem in exterior domains

TL;DR

This paper examines the stability of a Liouville type theorem for reaction-diffusion equations in exterior domains, showing it holds under smooth domain perturbations but not under rougher ones.

Contribution

It demonstrates the robustness of the Liouville theorem under $C^{2,eta}$ perturbations of the domain, extending previous geometric conditions.

Findings

Liouville theorem remains valid under smooth perturbations

The theorem fails for less smooth domain perturbations

Highlights importance of domain regularity in PDE properties

Abstract

We are interested in the robustness of a Liouville type theorem for a reaction diffusion equation in exterior domains. Indeed H. Berestycki, F. Hamel and H. Matano (2009) proved such a result as soon as the domain satisfies some geometric properties. We investigate here whether their result holds for perturbations of the domain. We prove that as soon as our perturbation is close to the initial domain in the $C^{2,\alpha}$ topology the result remains true while it does not if the perturbation is not smooth enough.