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 \(\mathbb {R}^2\) ${\mathbb{R}}^2$ , but does not hold in \(\mathbb {R}^3\) ${\mathbb{R}}^3$ . The multiplicity of the first eigenvalue of the Dirichlet–Stokes operator in the ball in \(\mathbb {R}^3\) ${\mathbb{R}}^3$ plays a crucial role in the proof of that claim.