Singular perturbations for abstract elliptic operators and applications

Abstract

Dirichlet problem for parameter depended elliptic differential-operator equation with variable coefficients in smooth domains is studied. The uniform maximal regularity, Fredholmness and the positivity of this problem in vector-valued Lp-spaces are obtained. It is proven that the corresponding differential operator is positive and is a generator of an analytic semigroup. In application, the maximal regularity properties of Cauchy problem for abstract parabolic equation and anisotropic elliptic equations with small parameters are established.