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ON Lp-SPECTRA AND ESSENTIAL SPECTRA OF SECOND-ORDER ELLIPTIC OPERATORS

Published online by Cambridge University Press:  20 August 2001

VITALI LISKEVICH  and
HENDRIK VOGT
VITALI LISKEVICH
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, v.liskevich@bristol.ac.uk
HENDRIK VOGT
Affiliation:
Institut für Analysis Fachrichtung Mathematik, Technische Universität, Dresden, 01062, Dresden, Germany, vogt@math.tu-dresden.de
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Abstract

In this paper we study the problem of $L^p$-independence of the spectrum of second-order elliptic operators with measurable coefficients. A new technique is developed to treat the problem in cases when the kernel of the corresponding semigroup does not satisfy upper bounds of Gaussian type and the semigroup itself exists only in a certain interval of the $L^p$-scale. This allows the authors to treat second-order elliptic operators of divergence type with singular coefficients in the main part and singular lower-order terms. A criterion of $L^p$-independence obtained in the paper applies also to certain classes of higher-order elliptic operators. The problem of stability of the essential spectrum is also studied. It is shown, under some mild conditions on the coefficients of the elliptic operators, that its essential spectrum is the same as the one of the Laplacian. 1991 Mathematics Subject Classification: 35P99, 35B25, 35J15.

Keywords

$L^p$-spectrumessential spectrumelliptic operators
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 80 , Issue 3 , May 2000 , pp. 590 - 610
Copyright
1999 London Mathematical Society

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