Mixed boundary value problems for the Stokes system

Abstract

We prove the well-posedness of the mixed problem for the Stokes system in a class of Lipschitz domains in R n {\mathbb {R}}^n , n ≥ 3 n\geq 3 . The strategy is to reduce the original problem to a boundary integral equation, and we establish certain new Rellich-type estimates which imply that the intervening boundary integral operator is semi-Fredholm. We then prove that its index is zero by performing a homotopic deformation of it onto an operator related to the Lamé system, which has recently been shown to be invertible.