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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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Sesquilinear Forms in Hilbert Spaces

Sesquilinear Forms in Hilbert Spaces

Chapter:
(p.169) 4 Sesquilinear Forms in Hilbert Spaces
Source:
Spectral Theory and Differential Operators
Author(s):

D. E. Edmunds

W. D. Evans

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198812050.003.0004

The centre-pieces of this chapter are the Lax–Milgram Theorem and the existence of weak or variational solutions to problems involving sesquilinear forms. An important application is to Kato’s First Representation Theorem, which associates a unique m-sectorial operator with a closed, densely defined sesquilinear form, thus extending the Friedrichs extension for a lower bounded symmetric operator. Stampacchia’s generalization of the Lax–Milgram Theorem to variational inequalities is also discussed.

Keywords:   Lax–Milgram Theorem, Kato’s First Representation Theorem, Friedrichs extension, variational inequality, Stampacchia’s variational inequality

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