# Courant-Sharp Eigenvalues for the Equilateral Torus, and for the Equilateral Triangle

@article{Brard2015CourantSharpEF,
title={Courant-Sharp Eigenvalues for the Equilateral Torus, and for the Equilateral Triangle},
author={Pierre H. B{\'e}rard and Bernard Helffer},
journal={Letters in Mathematical Physics},
year={2015},
volume={106},
pages={1729-1789}
}
• Published 28 February 2015
• Mathematics, Physics
• Letters in Mathematical Physics
We address the question of determining the eigenvalues $${\lambda_{n}}$$λn (listed in nondecreasing order, with multiplicities) for which Courant’s nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with $${n}$$n nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), we prove that the only Courant-sharp eigenvalues of the flat equilateral torus are the first and second, and that the only Courant-sharp Dirichlet eigenvalues of…
Courant-sharp property for eigenfunctions of the Klein bottle
• Mathematics
• 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
Courant-sharp eigenvalues of Neumann 2-rep-tiles
• Physics, Mathematics
• 2015
We find the Courant-sharp Neumann eigenvalues of the Laplacian on some 2-rep-tile domains. In $$\mathbb {R}^{2}$$R2, the domains we consider are the isosceles right triangle and the rectangle with
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 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 eigenvalues of a two-dimensional torus
Abstract In this note, we determine, in the case of the Laplacian on the flat two-dimensional torus ( R / Z ) 2 , all the eigenvalues having an eigenfunction that satisfies Courant's theorem with
Courant-sharp eigenvalues of a two-dimensional torus
In this paper, we determine, in the case of the Laplacian on the at two-dimensional torus ( R=Z) 2 , all the eigenvalues having an eigenfunction which satises Courant’s theorem with equality
The weak Pleijel theorem with geometric control
• Mathematics, Physics
• 2015
Let $\Omega\subset \mathbb R^d\,, d\geq 2$, be a bounded open set, and denote by $\lambda\_j(\Omega), j\geq 1$, the eigenvalues of the Dirichlet Laplacian arranged in nondecreasing order, with
Dirichlet eigenfunctions in the cube , sharpening the Courant nodal inequality
• 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
Dirichlet eigenfunctions in the cube, sharpening the Courant nodal inequality
• Mathematics
• 2017
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
ON COURANT’S NODAL DOMAIN PROPERTY FOR LINEAR COMBINATIONS OF EIGENFUNCTIONS PART II
Generalizing Courant’s nodal domain theorem, the “Extended Courant property” is the statement that a linear combination of the first n eigenfunctions has at most n nodal domains. In a previous paper

## References

SHOWING 1-10 OF 22 REFERENCES
Courant-sharp eigenvalues of a two-dimensional torus
Abstract In this note, we determine, in the case of the Laplacian on the flat two-dimensional torus ( R / Z ) 2 , all the eigenvalues having an eigenfunction that satisfies Courant's theorem with
Courant-sharp eigenvalues of a two-dimensional torus
In this paper, we determine, in the case of the Laplacian on the at two-dimensional torus ( R=Z) 2 , all the eigenvalues having an eigenfunction which satises Courant’s theorem with equality
Eigenstructure of the equilateral triangle
Lamé's formulas for the eigenvalues and eigenfunctions of the Laplacian on an equilateral triangle under Dirichlet and Neumann boundary conditions are herein extended to the Robin boundary condition.
The Eigenvalues of an Equilateral Triangle
Let D be an equilateral triangle of side 1. We consider solutions of $\Delta u + \lambda u = 0$ in D with either the boundary condition $u = 0$ or ${{\partial u} / {\partial n }} = 0$. Let \$n(\lambda
Nodal domains of a non-separable problem—the right-angled isosceles triangle
• Mathematics, Physics
• 2011
We study the nodal set of eigenfunctions of the Laplace operator on the right-angled isosceles triangle. A local analysis of the nodal pattern provides an algorithm for computing the number νn of
Asymptotic solution of eigenvalue problems
• Physics
• 1960
Abstract A method is presented for the construction of asymptotic formulas for the large eigenvalues and the corresponding eigenfunctions of boundary value problems for partial differential
NODAL DOMAINS IN THE SQUARE—THE NEUMANN CASE
• Mathematics
• 2014
˚ 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
Remarks on the boundary set of spectral equipartitions
• Mathematics, Medicine
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
• 2014
A lower bound for the length of the boundary set of a partition in the two-dimensional situation is proved and estimates involving the cardinality of the partition are considered.
Nodal and spectral minimal partitions -- The state of the art in 2015 --
• Mathematics
• 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
A note on Courant sharp eigenvalues of the Neumann right-angled isosceles triangle
• Mathematics
• 2015
We determine the Courant sharp eigenvalues of the right-angled isosceles triangle with Neumann boundary conditions.