Spectral properties of non-local uniformly-elliptic operators

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    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.
    Original languageEnglish
    Pages (from-to)1-15
    Number of pages15
    JournalElectronic Journal of Differential Equations
    Volume2006
    Issue number126
    Publication statusPublished - Oct 2006

    Keywords

    • Uniformly elliptic
    • Eigenvalues
    • Multiplicities

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