Work

Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator

Year: 2025

Type: article

Abstract: Given a compact manifold \mathcal{M} with boundary of dimension n\geq 3 and any integers K and N , we show that there exists a metric on \mathcal{M} for which the first K nonconstant eigenfunctions of the Dirichlet-to-Neumann map on \partial\mathcal{M} have at least N nodal components. This provides a negative answer to the question of whether the number of nodal domains of Dirichlet-to-Neumann eigenfunctions satisfies a Courant-type bound, which has been featured in recent surveys by Girouard and Polterovich (2017) and by Colbois, Girouard, Gordon and Sher (2024).

Source: Journal of the European Mathematical Society

DOI: 10.4171/jems/1722

Authors: Alberto Enciso, Angela Pistoia, Luigi Provenzano

Institutions: Consejo Superior de Investigaciones Científicas, Sapienza University of Rome

Language: English

Topic: Spectral Theory in Mathematical Physics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: Istituto Nazionale di Alta Matematica "Francesco Severi", European Commission, Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni, Agencia Estatal de Investigación