On generalized principal eigenvalues of nonlocal operators with a drift * - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Nonlinear Analysis: Theory, Methods and Applications Année : 2019

On generalized principal eigenvalues of nonlocal operators with a drift *

Jérôme Coville
François Hamel

Résumé

This article is concerned with the following spectral problem: to find a positive function ϕ ∈ C 1 (Ω) and λ ∈ R such that q(x)ϕ (x) + ˆ Ω J(x, y)ϕ(y) dy + a(x)ϕ(x) + λϕ(x) = 0 for x ∈ Ω, where Ω ⊂ R is a non-empty domain (open interval), possibly unbounded, J is a positive continuous kernel, and a and q are continuous coefficients. Such a spectral problem naturally arises in the study of nonlocal population dynamics models defined in a space-time varying environment encoding the influence of a climate change through a spatial shift of the coefficient. In such models, working directly in a moving frame that matches the spatial shift leads to consider a problem where the dispersal of the population is modeled by a nonlocal operator with a drift term. Assuming that the drift q is a positive function, for rather general assumptions on J and a, we prove the existence of a principal eigenpair (λ p , ϕ p) and derive some of its main properties. In particular, we prove that λ p (Ω) = lim R→+∞ λ p (Ω R), where Ω R = Ω ∩ (−R, R) and λ p (Ω R) corresponds to the principal eigenvalue of the truncation operator defined in Ω R. The proofs especially rely on the derivation of a new Harnack type inequality for positive solutions of such problems.
Fichier principal
Vignette du fichier
ch.pdf (419.3 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01966709 , version 1 (29-12-2018)

Identifiants

Citer

Jérôme Coville, François Hamel. On generalized principal eigenvalues of nonlocal operators with a drift *. Nonlinear Analysis: Theory, Methods and Applications, 2019, 193, pp.111569. ⟨10.1016/j.na.2019.07.002⟩. ⟨hal-01966709⟩
140 Consultations
174 Téléchargements

Altmetric

Partager

Gmail Mastodon Facebook X LinkedIn More