On the Krein-Rutman theorem and beyond

Claudia Fonte Sanchez; Pierre Gabriel; Stéphane Mischler

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arXiv · 2305.06652

On the Krein-Rutman theorem and beyond

Claudia Fonte Sanchez·Pierre Gabriel·Stéphane Mischler

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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.

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Claudia Fonte Sanchez, Pierre Gabriel, Stéphane Mischler. 2023-05-11. On the Krein-Rutman theorem and beyond. https://arxiv.org/abs/2305.06652

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