Anisotropic Differential Operators with Parameters and Applications

Abstract

In the paper, we study the boundary-value problems for parameter-dependent anisotropic differential-operator equations with variable coefficients. Several conditions for the uniform separability and Fredholmness in Banach-valued L _p -spaces are given. Sharp uniform estimates for the resolvent are established. It follows from these estimates that the indicated operator is uniformly positive. Moreover, it is also the generator of an analytic semigroup. As an application, the maximal regularity properties of the parameter-dependent abstract parabolic problem and infinite systems of parabolic equations are established in mixed L _p -spaces.