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Compact sets in the spaceL p (O,T; B)

Compact sets in the spaceL p (O,T; B) A characterization of compact sets in Lp (0, T; B) is given, where 1⩽P⩾∞ and B is a Banach space. For the existence of solutions in nonlinear boundary value problems by the compactness method, the point is to obtain compactness in a space Lp (0,T; B) from estimates with values in some spaces X, Y or B where X⊂B⊂Y with compact imbedding X→B. Using the present characterization for this kind of situations, sufficient conditions for compactness are given with optimal parameters. As an example, it is proved that if {fn} is bounded in Lq(0,T; B) and in L loc 1 (0, T; X) and if {∂fn/∂t} is bounded in L loc 1 (0, T; Y) then {fn} is relatively compact in Lp(0,T; B), ∀p<q. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annali di Matematica Pura ed Applicata (1923 -) Springer Journals

Compact sets in the spaceL p (O,T; B)

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References (18)

Publisher
Springer Journals
Copyright
Copyright © 1985 by Fondazione Annali di Matematica Pura ed Applicata
Subject
Mathematics; Mathematics, general
ISSN
0373-3114
eISSN
1618-1891
DOI
10.1007/BF01762360
Publisher site
See Article on Publisher Site

Abstract

A characterization of compact sets in Lp (0, T; B) is given, where 1⩽P⩾∞ and B is a Banach space. For the existence of solutions in nonlinear boundary value problems by the compactness method, the point is to obtain compactness in a space Lp (0,T; B) from estimates with values in some spaces X, Y or B where X⊂B⊂Y with compact imbedding X→B. Using the present characterization for this kind of situations, sufficient conditions for compactness are given with optimal parameters. As an example, it is proved that if {fn} is bounded in Lq(0,T; B) and in L loc 1 (0, T; X) and if {∂fn/∂t} is bounded in L loc 1 (0, T; Y) then {fn} is relatively compact in Lp(0,T; B), ∀p<q.

Journal

Annali di Matematica Pura ed Applicata (1923 -)Springer Journals

Published: May 8, 2005

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