Compact sets in the space Lp(O, T; B)

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 Lloc1(0, T; X) and if {∂fn/∂t} is bounded in Lloc1(0, T; Y) then {fn} is relatively compact in Lp(0, T; B), ∀p