On the Krein-Rutman theorem and beyond
Abstract
In this work, we revisit the Krein-Rutman theory for semigroups of positive operators in a Banach lattice framework and we provide some very general, efficient and handy results with constructive estimates about: the existence of a solution to the first eigentriplet problem; the geometry of the principal eigenvalue problem; the asymptotic stability of the first eigenvector with possible constructive rate of convergence. This abstract theory is motivated and illustrated by several examples of differential, integro-differential and integral operators. In particular, we revisit the first eigenvalue problem and the asymptotic stability of the first eigenvector for: some parabolic equations in a bounded domain and in the whole space; some transport equations in a bounded or unbounded domain, including some growth-fragmentation models and some kinetic models; the kinetic Fokker-Planck equation in the torus and in the whole space; some mutation-selection models.
Keywords
Cite
@article{arxiv.2305.06652,
title = {On the Krein-Rutman theorem and beyond},
author = {Claudia Fonte Sanchez and Pierre Gabriel and Stéphane Mischler},
journal= {arXiv preprint arXiv:2305.06652},
year = {2025}
}