SINGULAR PERTURBATIONS FOR ABSTRACT ELLIPTIC OPERATORS AND APPLICATIONS

SINGULAR PERTURBATIONS FOR ABSTRACT ELLIPTIC OPERATORS AND APPLICATIONS

Veli Shakhmurov    

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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.

Keywords

Boundary value problems, differential-operator equations, Banach-valued function spaces, operator-valued multipliers, interpolation of Banach spaces, semigroup of operators.