Some results on the Stokes eigenvalue problem under Navier boundary conditions

We study the Stokes eigenvalue problem under Navier boundary conditions in 2D or 3D bounded domains with connected boundary of class $ C^1 $. Differently from the Dirichlet boundary conditions, zero may be the least eigenvalue. We fully characterize the domains where this happens and we show that the ball is the unique domain where the zero eigenvalue is not simple. We apply these results to show the validity/failure of a suitable Poincaré inequality. We then consider the general version of the problem in any space dimension with $ n\geq2 $, characterizing the kernel of the strain tensor for solenoidal vector fields with homogeneous normal trace. We conclude analyzing some similarities and differences with the Laplacian eigenvalue problem.

This is based on a joint work with Filippo Gazzola, Politecnico di Milano.