On the Fredholm Property of the Stokes Operator in a Layer-Like Domain

Abstract

The Stokes problem is studied in the domain $\Omega \subset {\mathbb{R}}^3$ coinciding with the layer $\Pi =x=(y,z):y=(y_1,y_2)\in (0,1)$ outside some ball. It is shown that the operator of such problem is of Fredholm type; this operator is defined on a certain weighted function space ${\mathcal{D}}_{\beta }^l(\Omega ­$) with norm determined by a stepwise anisotropic distribution of weight factors (the direction of $z$ is distinguished). The smoothness exponent $l$ is allowed to be a positive integer, and the weight exponent $\beta$ is an arbitrary real number except for the integer set $\mathbb{Z}$ where the Fredholm property is lost. Dimensions of the kernel and cokernel of the operator are calculated in dependence of $\beta$. It turns out that, at any admissible $\beta$, the operator index does not vanish. Based on the generalized Green formula, asymptotic conditions at infinity are imposed to provide the problem with index zero.