Abstract
We study the Stokes eigenvalue problem under Navier boundary conditions in C^{1,1}-domains Omega subset mathbb {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 Poincaré-type inequality. The proofs are obtained by combining analytic and geometric arguments.
Highlights
Let Ω ⊂ R3 be a bounded C1,1-domain
To be complemented with some boundary conditions; here, ∈ R is the eigenvalue, u ∈ R3 is the eigenvector, while ∇p stands for the gradient of the pressure p of the fluid
(1) states that u is an eigenvector, up to the addition of a gradient. This can be functionally characterized in the framework of the Helmholtz–Weyl decomposition [11, 21], see (5)
Summary
Let Ω ⊂ R3 be a bounded C1,1-domain. The Stokes eigenvalue problem may be written as−Δu + ∇p = u in Ω , ∇⋅u=0 in Ω , (1)to be complemented with some boundary conditions; here, ∈ R is the eigenvalue, u ∈ R3 is the eigenvector (representing the velocity of a fluid in the context of Navier–Stokes equations), while ∇p stands for the gradient of the pressure p of the fluid. Keywords Stokes eigenvalue problem · Navier boundary conditions In the zero-friction case, the homogeneous Navier boundary conditions read u ⋅ = ( u ⋅ ) ⋅ = 0 on Ω , (4)
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