Annali di Matematica Pura ed Applicata

, Volume 146, Issue 1, pp 65–96 | Cite as

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

  • Jacques Simon


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.


Banach Space Optimal Parameter Nonlinear Boundary Compactness Method Compact Imbed 
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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1985

Authors and Affiliations

  • Jacques Simon
    • 1
  1. 1.Laboratoire d'Analyse Numérique(U. A. 189)Université Pierre et Marie CurieParis Cedex 05

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