Is the Faber-Krahn inequality true for the Stokes operator? - Equations aux dérivées partielles
Article Dans Une Revue Calculus of Variations and Partial Differential Equations Année : 2024

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

Résumé

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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Dates et versions

hal-04402162 , version 1 (18-01-2024)
hal-04402162 , version 2 (31-08-2024)

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  • HAL Id : hal-04402162 , version 2

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Antoine Henrot, Idriss Mazari, Yannick Privat. Is the Faber-Krahn inequality true for the Stokes operator?. Calculus of Variations and Partial Differential Equations, In press. ⟨hal-04402162v2⟩
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