On the Krein-Rutman theorem and beyond

Claudia Fonte Sanchez (1) , Pierre Gabriel (2) , Stéphane Mischler (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 UVSQ - Université de Versailles Saint-Quentin-en-Yvelines

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, intro-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.

Domaines: Théorie spectrale [math.SP]; Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

Claudia Fonte Sanchez, Pierre Gabriel, Stéphane Mischler. On the Krein-Rutman theorem and beyond. 2023. ⟨hal-04093201v1⟩