On the Fredholm property of the Stokes operator in a layer-like domain

The Stokes problem is studied in the domain Omega subset of R-3 coinciding with the layer Pi = {x = (y, z) : y = (y(1), y(2)) is an element of R-2, z is an element of (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 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 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.