Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 126, pp. 1-15.

Title: Spectral properties of non-local uniformly-elliptic operators

Authors: Fordyce A. Davidson (Univ. of Dundee, Dundee, UK)
         Niall Dodds         (Univ. of Dundee, Dundee, UK)

Abstract: 
  In this paper we consider the spectral properties of a class of
 non-local uniformly elliptic  operators, which arise from the study
 of  non-local uniformly elliptic partial differential equations.
 Such equations arise naturally in the study of
 a variety of physical and biological systems with examples
 ranging  from Ohmic heating to population dynamics.
 The operators studied here are  bounded perturbations of
 linear (local) differential operators, and the non-local perturbation
 is in the form of an integral term.
 We study the eigenvalues, the multiplicities of these eigenvalues,
 and the existence of corresponding positive eigenfunctions.
 It is shown here that the spectral properties of these non-local
 operators can differ considerably  from those of their local counterpart.
 However, we show that under suitable hypotheses, there still exists
 a principal eigenvalue of these operators.
 
Submitted July 6, 2006. Published October 11, 2006.
Math Subject Classifications: 34P05, 35P10, 47A75, 47G20.
Key Words: Non-local; uniformly elliptic; eigenvalues; multiplicities.