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Hadamard variational formula for eigenvalues of the Stokes equations and its application. (English) Zbl 1367.35122

Summary: Based on the explicit representation of the Hadamard variational formula [the first and fourth author, Far East J. Math. Sci. (FJMS) 98, No. 6, 713–739 (2015; Zbl 1334.76041)] for eigenvalues of the Stokes equations, we investigate the geometry of the domain in \(\mathbb R^3\). It turns out that if the first variation of some eigenvalue of the Stokes equations for all volume preserving perturbations vanishes, then the domain is necessarily diffeomorphic to the 2-dimensional torus \(T^2\).

MSC:

35Q35 PDEs in connection with fluid mechanics
35C07 Traveling wave solutions
49R05 Variational methods for eigenvalues of operators

Citations:

Zbl 1334.76041
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References:

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