# On nodal domains in Euclidean balls

@inproceedings{Helffer2015OnND, title={On nodal domains in Euclidean balls}, author={Bernard Helffer and Mikael Persson Sundqvist}, year={2015} }

A. Pleijel (1956) has proved that in the case of the Laplacian with Dirichlet condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues when the dimension is ≥ 2. Recently Polterovich extended the result to the Neumann problem in two dimensions in the case when the boundary is piecewise analytic. A question coming from the theory of spectral minimal partitions has motivated the analysis of the cases when one has equality…

## 29 Citations

### Dirichlet eigenfunctions in the cube , sharpening the Courant nodal inequality

- Mathematics
- 2019

This paper is devoted to the refined analysis of Courant’s theorem for the Dirichlet Laplacian in a bounded open set. Starting from the work by Å. Pleijel in 1956, many papers have investigated in…

### Pleijel’s nodal domain theorem for Neumann and Robin eigenfunctions

- MathematicsAnnales de l'Institut Fourier
- 2019

In this paper, 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 the case of an open, bounded and connected…

### Courant-sharp eigenvalues of the three-dimensional square torus

- Mathematics
- 2015

In this paper, we determine, in the case of the Laplacian on the at three-dimensional torus ( R=Z) 3 , all the eigenvalues having an eigenfunction which satises the Courant nodal domains theorem with…

### Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip

- MathematicsPortugaliae Mathematica
- 2021

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was…

### Pleijel’s theorem for Schrödinger operators with radial potentials

- Physics, Mathematics
- 2016

In 1956, Pleijel gave his celebrated theorem showing that the inequality in Courant’s theorem on the number of nodal domains is strict for large eigenvalues of the Laplacian. This was a consequence…

### Stability of spectral partitions and the Dirichlet-to-Neumann map

- MathematicsCalculus of Variations and Partial Differential Equations
- 2022

The oscillation of a Laplacian eigenfunction gives a great deal of information about the manifold on which it is defined. This oscillation can be encoded in the nodal deficiency, an important…

### Pleijel's theorem for Schr\"odinger operators

- Mathematics
- 2021

We are concerned in this paper with the real eigenfunctions of Schrödinger operators. We prove an asymptotic upper bound for the number of their nodal domains, which implies in particular that the…

### Upper bounds for Courant-sharp Neumann and Robin eigenvalues

- MathematicsBulletin de la Société mathématique de France
- 2020

We consider the eigenvalues of the Laplacian on an open, bounded, connected set in $\mathbb{R}^n$ with $C^2$ boundary, with a Neumann boundary condition or a Robin boundary condition. We obtain upper…

### Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders

- MathematicsProceedings of the American Mathematical Society
- 2020

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was…

### Isoperimetric relations between Dirichlet and Neumann eigenvalues

- Mathematics
- 2019

Inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian have received much attention in the literature, but open problems abound. Here, we study the number of Neumann eigenvalues…

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