ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS

Abstract

The weighted Sobolev-Lions type spaces W-p,gamma(l) (Omega; E) boolean AND L-p,L-gamma (Omega; E-0) are studied, where E-0, E are two Banach spaces and E-0 is continuously and densely embedded on E. A new concept of capacity of region Omega is an element of R-n in Wp,gamma(l)(Omega; E-0, E) is introduced. Several conditions in terms of capacity of region Omega and interpolations of E-0 and E are found such that ensure the continuity and compactness of embedding operators. In, particular, the most regular class of interpolation spaces E-alpha between E-0 and E, depending of alpha and l, are found such that mixed differential operators D-alpha are bounded and compact from W-p,gamma(l)(Omega; E-0, E) to E-alpha-valued L-p,L-gamma spaces. In applications, the maximal regularity for differential-operator equations with parameters are studied.