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arXiv · 2401.09801

Is the Faber-Krahn inequality true for the Stokes operator?

Antoine Henrot·Idriss Mazari-Fouquer·Yannick Privat
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Abstract

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

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Antoine Henrot, Idriss Mazari-Fouquer, Yannick Privat. 2024-01-18. Is the Faber-Krahn inequality true for the Stokes operator?. https://arxiv.org/abs/2401.09801

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