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On the principal eigenvalue of elliptic operators in ℝ N and applications

Henry Berestycki;  Luca Rossi — 2006

Journal of the European Mathematical Society

Two generalizations of the notion of principal eigenvalue for elliptic operators in ℝ N are examined in this paper. We prove several results comparing these two eigenvalues in various settings: general operators in dimension one; self-adjoint operators; and “limit periodic” operators. These results apply to questions of existence and uniqueness for some semilinear problems in the whole space. We also indicate several outstanding open problems and formulate some conjectures.

The speed of propagation for KPP type problems. I: Periodic framework

Henry Berestycki;  François Hamel;  Nikolai Nadirashvili — 2005

Journal of the European Mathematical Society

This paper is devoted to some nonlinear propagation phenomena in periodic and more general domains, for reaction-diffusion equations with Kolmogorov–Petrovsky–Piskunov (KPP) type nonlinearities. The case of periodic domains with periodic underlying excitable media is a follow-up of the article [7]. It is proved that the minimal speed of pulsating fronts is given by a variational formula involving linear eigenvalue problems. Some consequences concerning the influence of the geometry of the domain,...

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