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

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 se...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations
Main Authors: Henrot, Antoine, Mazari, Idriss, Privat, Yannick
Format: Journal Article
Language:English
Published: Springer Verlag 18.01.2024
Subjects:
Analysis of PDEs
Mathematics
Optimization and Control
Spectral optimisation
Shape derivation
Stokes operator
Faber-Krahn inequality
Shape semi-differentiation
ISSN:0944-2669, 1432-0835
Online Access:Get full text
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Summary: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.
ISSN:0944-2669
1432-0835
DOI:10.48550/arXiv.2401.09801