Remarks on the 3D Stokes eigenvalue problem under Navier boundary conditions

We study the Stokes eigenvalue problem under Navier boundary conditions in C-1,C-1-domains Omega subset of R-3. 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, it has multiplicity three. We apply these results to show the validity/failure of a suitable Poincare-type inequality. The proofs are obtained by combining analytic and geometric arguments.

Automated summary

This study investigates the Stokes eigenvalue problem under Navier boundary conditions in C-1,C-1-domains Omega subset of R-3, finding that zero may be the least eigenvalue in this scenario. The analysis fully characterizes the domains where this occurs, and shows that the ball is the only domain where the zero eigenvalue is not simple, having a multiplicity of three. These results are then applied to demonstrate the validity or failure of a suitable Poincare-type inequality, with proofs obtained through a combination of analytic and geometric arguments.