Is the Faber-Krahn inequality true for the Stokes operator?

The goal of this paper is to investigate the minimisation of the first eigenvalue of the (vectorial) incompressible Dirichlet-Stokes operator. After providing an existence result, we investigate optimality conditions and we prove the following surprising result: while the ball satisfies first and second-order optimality conditions in dimension 2, it does not in dimension 3, so that the Faber-Krahn inequality for the Stokes operator is probably true in ℝ^2, but does not hold in ℝ^3. The multiplicity of the first eigenvalue of the Dirichlet-Stokes operator in the ball in ℝ^3 plays a crucial role in the proof of that claim.