Fordyce A. Davidson, Niall Dodds
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.
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Fordyce A. Davidson Division of Mathematics, University of Dundee Dundee, DD1 4HN, United Kingdom Tel. 01382 384692, fax 01382 385516 email: fdavidso@maths.dundee.ac.uk |
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Niall Dodds Division of Mathematics, University of Dundee Dundee, DD1 4HN, United Kingdom Tel. 01382 384471, fax 01382 385516 email: ndodds@maths.dundee.ac.uk |
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