Displaying similar documents to “Two Methods of Solution of the Three-Dimensional Inverse Nodal Problem.”

Localization effects for eigenfunctions near to the edge of a thin domain

Serguei A. Nazarov (2002)

Mathematica Bohemica

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It is proved that the first eigenfunction of the mixed boundary-value problem for the Laplacian in a thin domain Ω h is localized either at the whole lateral surface Γ h of the domain, or at a point of Γ h , while the eigenfunction decays exponentially inside Ω h . Other effects, attributed to the high-frequency range of the spectrum, are discussed for eigenfunctions of the mixed boundary-value and Neumann problems, too.

On the nodal set of the second eigenfunction of the laplacian in symmetric domains in R N

Lucio Damascelli (2000)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We present a simple proof of the fact that if Ω is a bounded domain in R N , N 2 , which is convex and symmetric with respect to k orthogonal directions, 1 k N , then the nodal sets of the eigenfunctions of the laplacian corresponding to the eigenvalues λ 2 , , λ k + 1 must intersect the boundary. This result was proved by Payne in the case N = 2 for the second eigenfunction, and by other authors in the case of convex domains in the plane, again for the second eigenfunction.