Spectral Properties of Integral Operators in Problems of Interface Dynamics and Metastability
1998, v.4, №1, 27-112
In this paper we study some integral operators that are obtained by linearizations of a non-local evolution equation for a non-conserved order parameter describing the phase of a fluid. We prove a Perron - Frobenius theorem by showing that there is an isolated, simple, maximal eigenvalue larger than 1 with a positive eigenvector and that the rest of the spectrum is strictly contained in the unit ball. Such properties are responsible for the existence of invariant, attractive, unstable one-dimensional manifolds under the full, non linear evolution. This part of the analysis and the application to interface dynamics and metastability will be carried out in separate papers.
Keywords: Perron - Frobenius theorem,interfaces,stationary fronts,instantons