In [2], the number of nodal domains for the eigenfunctions of the Neumann or Robin Laplacian is bounded from above using the classical (Euclidean) Faber-Krahn inequality. However, in an arbitrary Riemannian manifold, this inequality might not hold. We supply the missing arguments in two dimensions and outline a modification of the method, which preserves most of the results, in n dimensions.
Corrections "Upper bounds for Courant-sharp Neumann and Robin eigenvalues"
LENA, Corentin
;
2026
Abstract
In [2], the number of nodal domains for the eigenfunctions of the Neumann or Robin Laplacian is bounded from above using the classical (Euclidean) Faber-Krahn inequality. However, in an arbitrary Riemannian manifold, this inequality might not hold. We supply the missing arguments in two dimensions and outline a modification of the method, which preserves most of the results, in n dimensions.
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