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# Generalizations of Pleijel's nodal domain

GFM seminar
FCUL, C6, room 6.2.33
2017-04-04 14:00 .. 15:00
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by Corentin Léna (Università degli Studi di Torino)

Courant's nodal domain theorem tells us that an eigenfunction associated with the $$k^{th}$$ eigenvalue of the Laplacian has at most $$k$$ nodal domains. A. Pleijel showed in 1956 that for a given planar domain, eigenfunctions satisfying a Dirichlet boundary condition reach equality only for a finite number of $$k$$. We will study a generalization of this theorem to Robin-type boundary conditions, including the Neumann one, in any dimension. We will also consider the sharper results that can be obtained for particular domains.