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… 
View PDF on arXiv
29 Citations
Highly Influential Citations
1
Background Citations
9
Methods Citations
2
Results Citations
1

Figures and Tables from this paper

29 Citations

Dirichlet eigenfunctions in the cube , sharpening the Courant nodal inequality

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

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

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

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

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

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

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

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

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

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…

References

SHOWING 1-10 OF 32 REFERENCES

Dirichlet eigenfunctions in the cube, sharpening the Courant nodal inequality

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 byAbyA. Pleijel in 1956, many papers have investigated in…

NODAL DOMAINS IN THE SQUARE—THE NEUMANN CASE

˚ A. Pleijel has proved that in the case of the Laplacian on the square with Neumann condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite…

Liouville theorems and 1-dimensional symmetry for solutions of an elliptic system modelling phase separation

On spectral minimal partitions : the case of the sphere

In continuation of previous work, we analyze the properties of spectral minimal partitions and focus in this paper our analysis on the case of the sphere. We prove that a minimal 3-partition for the…

Nodal and spectral minimal partitions -- The state of the art in 2015 --

In this article, we propose a state of the art concerning the nodal and spectral minimal partitions. First we focus on the nodal partitions and give some examples of Courant sharp cases. Then we are…

Pleijel's nodal domain theorem for free membranes

We prove an analogue of Pleijel's nodal domain theorem for piecewise analytic planar domains with Neumann boundary conditions. This confirms a conjecture made by Pleijel in 1956. The proof is a…

Counting Nodal Lines Which Touch the Boundary of an Analytic Domain

We consider the zeros on the boundary $\partial \Omega$ of a Neumann eigenfunction $\phi_{\lambda}$ of a real analytic plane domain $\Omega$. We prove that the number of its boundary zeros is $O…

A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions

*The first author's work was partially supported by FONDECYT (Chile), project 0132-88, and by a Summer Research Fellowship provided by the Research Council of the University of Missouri-Columbia. He…

Asymptotic Behaviour of the Spectral Function

Let M be a closed n-dimensional manifold on which there is given a smooth positive density dx and let A be an elliptic, self-adjoint operator of degree m on M such that a m (x, ξ) > 0 for ξ ≠ 0. We…

Nodal Domains and Spectral Minimal Partitions