The Kernel of the Strain Tensor for Solenoidal Vector Fields with Homogeneous Normal Trace

For solenoidal vector fields with homogeneous normal trace, we study the kernel of the strain tensor in any space dimension \(n\ge 2\) . We characterize the open bounded domains \(\varOmega \subset \mathbb {R}^n\) with connected boundary \(\partial \varOmega \in C^{1}\) for which the kernel is nontrivial. This happens for domains having suitable rotational symmetries, and the maximal kernel dimension is achieved for balls. We also exhibit some nonsmooth domains for which the kernel is nontrivial. Finally, we introduce a related Stokes eigenvalue problem, and we discuss the relationship between our results and the classical Faber–Krahn inequality.