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Generalized principal eigenvalues of space-time periodic, weakly coupled, cooperative, parabolic systems

Léo Girardin, Idriss Mazari

Abstract

This paper is concerned with generalizations of the notion of principal eigenvalue in the context of space-time periodic cooperative systems. When the spatial domain is the whole space, the Krein-Rutman theorem cannot be applied and this leads to more sophisticated constructions and to the notion of generalized principal eigenvalues. These are not unique in general and we focus on a one-parameter family corresponding to principal eigenfunctions that are space-time periodic multiplicative perturbations of exponentials of the space variable. Besides existence and uniqueness properties of such principal eigenpairs, we also prove various dependence and optimization results illustrating how known results in the scalar setting can, or cannot, be extended to the vector setting. We especially prove an optimization property on minimizers and maximizers among mutation operators valued in the set of bistochastic matrices that is, to the best of our knowledge, new.

Generalized principal eigenvalues of space-time periodic, weakly coupled, cooperative, parabolic systems

Abstract

This paper is concerned with generalizations of the notion of principal eigenvalue in the context of space-time periodic cooperative systems. When the spatial domain is the whole space, the Krein-Rutman theorem cannot be applied and this leads to more sophisticated constructions and to the notion of generalized principal eigenvalues. These are not unique in general and we focus on a one-parameter family corresponding to principal eigenfunctions that are space-time periodic multiplicative perturbations of exponentials of the space variable. Besides existence and uniqueness properties of such principal eigenpairs, we also prove various dependence and optimization results illustrating how known results in the scalar setting can, or cannot, be extended to the vector setting. We especially prove an optimization property on minimizers and maximizers among mutation operators valued in the set of bistochastic matrices that is, to the best of our knowledge, new.

Paper Structure

This paper contains 38 sections, 61 theorems, 454 equations.

Table of Contents

  1. Introduction
  2. Scope of the paper
  3. Motivations
  4. Notations
  5. Generalized principal eigenvalues in space-time periodic media
  6. Results
  7. Existence and characterization of generalized principal eigenpairs
  8. Monotonic or convex dependence with respect to the coefficients
  9. Asymptotic dependence with respect to the coefficients
  10. Dependence on the space or time frequencies
  11. Formulas and estimates in special cases
  12. Optimization
  13. Extension to systems with a coupling default
  14. Relation with KPP-type semilinear systems
  15. Preliminaries
  16. ...and 23 more sections

Key Result

Theorem 1.1

The generalized principal eigenvalues $\lambda_1$ and $\lambda_1'$ are well-defined real numbers related to the family $\left(\lambda_{1,z}\right)_{z\in\mathbb{R}^n}$: The maximum is uniquely achieved. Consequently, $\lambda_1'\leq \lambda_1$, $\bm{\mathbf{u}}_0$ is a generalized principal eigenfunction associated with $\lambda_1'$ and there exists a unique $z^\star\in\mathbb{R}^n$ such that $\te

Theorems & Definitions (138)

  • Definition 1.1
  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • ...and 128 more