Generic simplicity of the eigenvalues of the Stokes system in two space dimensions

Abstract

We analyze the multiplicity of the eigenvalues for the Stokes operator in a bounded domain of $\mathbb R^2$ with Dirichlet boundary conditions. We prove that, generically with respect to the domain, all the eigenvalues are simple. In other words, given a bounded domain of $\mathbb R^2,$ we prove the existence of arbitrarily small deformations of its boundary such that the spectrum of the Stokes operator in the deformed domain is simple. We prove that this can be achieved by means of deformations which leave invariant an arbitrarily large subset of the boundary. The proof combines Baire's lemma and shape differentiation. However, in contrast with the situation one encounters when dealing with scalar elliptic eigenvalue problems, the problem is reduced to a unique continuation question that may not be solved by means of Holmgrem's uniqueness theorem. We show however that this unique continuation property holds generically with respect to the domain and that this fact suffices to prove the generic simplicity of the spectrum.