We show that equality in Courant's nodal domain theorem can only be reached for a finite number of eigenvalues of the Neumann Laplacian, in an open, bounded, and connected subset of R-n with a C-1,C-1 boundary, when n >= 2. This result is analogous to the theorem proved by Pleijel in 1956 for the Dirichlet Laplacian. We also show that the argument and the result extend to a class of Robin boundary conditions.
PLEIJEL'S NODAL DOMAIN THEOREM FOR NEUMANN AND ROBIN EIGENFUNCTIONS
Léna, Corentin
2019
Abstract
We show that equality in Courant's nodal domain theorem can only be reached for a finite number of eigenvalues of the Neumann Laplacian, in an open, bounded, and connected subset of R-n with a C-1,C-1 boundary, when n >= 2. This result is analogous to the theorem proved by Pleijel in 1956 for the Dirichlet Laplacian. We also show that the argument and the result extend to a class of Robin boundary conditions.
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