On the Fredholm Property of the Stokes Operator in a Layer-Like Domain
С. А. Назаров, Konstantin Pileckas · Zeitschrift für Analysis und ihre Anwendungen · 2001
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^l_{\beta} (\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.