Abstract
We consider the Dirichlet eigenvalue problem on a polytope. We use the Rellich identity to obtain an explicit formula expressing the Dirichlet eigenvalue in terms of the Neumann data on the faces of the polytope of the corresponding eigenfunction. The formula is particularly simple for polytopes admitting an inscribed ball tangent to all the faces. Our result could be viewed as a generalization of similar identities for simplices recently found by Christianson (Equidistribution of Neumann data mass on simplices and a simple inverse problem, ArXiv e-prints, 2017, Equidistribution of Neumann data mass on triangles. ArXiv e-prints, 2017).
Résumé
Le problème aux valeurs propres de Dirichlet est considéré sur un polytope. L’identité de Rellich est utilisée pour obtenir une formule exprimant la valeur propre de Dirichlet en terme de l’information de Neumann sur les faces du polytope de la fonction propre correspondante. Cette formule est particulièrement simple pour un polytope ayant une balle inscrite tangente à toutes ses faces. Notre résultat peut être vu comme une généralisation d’une identité similaire pour les simplexes trouvée par Christianson (Equidistribution of Neumann data mass on simplices and a simple inverse problem, ArXiv e-prints, 2017, Equidistribution of Neumann data mass on triangles. ArXiv e-prints, 2017).
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References
Christianson, H.: Equidistribution of Neumann data mass on simplices and a simple inverse problem. (2017). arXiv:1704.02048
Christianson, H.: Equidistribution of Neumann data mass on triangles. (2017). arXiv:1701.02793
Hassell, A., Tao, T.: Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions. Math. Res. Lett. 9(3), 289–305 (2002)
Rellich, F.: Darstellung der Eigenwerte von \(\varDelta \) \(u+\lambda u=0\) durch ein Randintegral. Mathematische Zeitschrift 46, 635–636 (1940)
Acknowledgements
This paper is part of a MSc thesis written under the supervision of Iosif Polterovich. I would like to thank him for bringing to my attention the Rellich identity, which greatly simplified my proof.
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Métras, A. Rellich–Christianson type identities for the Neumann data mass of Dirichlet eigenfunctions on polytopes. Ann. Math. Québec 42, 95–99 (2018). https://doi.org/10.1007/s40316-017-0096-8
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DOI: https://doi.org/10.1007/s40316-017-0096-8