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Spectral Theory and Differential Operators$
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David Edmunds and Des Evans

Print publication date: 2018

Print ISBN-13: 9780198812050

Published to Oxford Scholarship Online: September 2018

DOI: 10.1093/oso/9780198812050.001.0001

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Capacity and Compactness Criteria

Capacity and Compactness Criteria

Chapter:
(p.395) 8 Capacity and Compactness Criteria
Source:
Spectral Theory and Differential Operators
Author(s):

D. E. Edmunds

W. D. Evans

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198812050.003.0008

In this chapter, necessary and sufficient conditions are derived for the Poincaré inequality to hold, for the embedding of W01,p(Ω) in Lp(Ω‎) to be compact, and for a self-adjoint realization of − aijDiDj + q to have a wholly discrete spectrum when q is real and bounded below. The results are proved using a method of Maz’ya.

Keywords:   Poincaré inequality, discrete spectrum, capacity, method of Maz’ya

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