Related papers: Courant-sharp eigenvalues of Neumann 2-rep-tiles
In this paper, we determine, in the case of the Laplacian on the flat three-dimensional torus $(\mathbb{R}/\mathbb{Z})^3$, all the eigenvalues having an eigenfunction which satisfies the Courant nodal domains theorem with equality…
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 bounds for those eigenvalues that have a…
In this paper, we determine, in the case of the Laplacian on the flat two-dimensional torus (R/Z) 2 , all the eigenvalues having an eigenfunction which satisfies Courant's theorem with equality (Courant-sharp situation). Following the…
We address the question of determining the eigenvalues $\lambda\_n$ (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 $n$…
We study the structure of eigenfunctions of the Laplacian on quantum graphs, with a particular focus on Morse eigenfunctions via nodal and Neumann domains. Building on Courant-type arguments, we establish upper bounds for the number of…
{\AA} 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 number of eigenvalues. We identify five…
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 motivated by the analysis of minimal spectral…
We obtain upper bounds on the number of nodal domains of Laplace eigenfunctions on chain domains with Neumann boundary conditions. The chain domains consist of a family of planar domains, with piecewise smooth boundary, that are joined by…
\AA. 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 $\geq…
In this paper, we successfully establish a Courant-type nodal domain theorem for both the Dirichlet eigenvalue problem and the closed eigenvalue problem of the Witten-Laplacian. Moreover, we also characterize the properties of the nodal…
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 motivated by the analysis of minimal spectral…
A Laplacian eigenfunction on a two-dimensional manifold dictates some natural partitions of the manifold; the most apparent one being the well studied nodal domain partition. An alternative partition is revealed by considering a set of…
According to Courant's theorem, an eigenfunction as\-sociated with the $n$-th eigenvalue $\lambda\_n$ has at most $n$ nodal domains. A footnote in the book of Courant and Hilbert, states that the same assertion is true for any linear…
This paper is devoted to the refine analysis of Courant's theorem for the Dirichlet Laplacian. Many papers (and some of them quite recent) have investigated in which cases this inequality in Courant's theorem is an equality: Pleijel,…
In this paper, we prove that the Extended Courant Property fails to be true for certain smooth, strictly convex domains with Neumann boundary condition: there exists a linear combination of a second and a first Neumann eigenfunctions, with…
A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains (a.k.a. a Morse--Smale complex). This partition is generated by gradient flow lines of the eigenfunction, which bound the…
We prove sharp upper bounds for the first and second non-trivial eigenvalues of the Neumann Laplacian in two classes of domains: parallelograms and domains of constant width. This gives in particular a new proof of an isoperimetric…
We study the nodal deficiency of pairs of Neumann eigenfunctions defined over symmetric dumbbell domains. As the width of the connecting neck shrinks, these eigenfunctions converge to Neumann eigenfunctions defined over the ends of the…
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 (Documenta Mathematica, 2018, Vol.…
The paper addresses the the number of nodal domains for eigenfunctions of Schr\"{o}dinger operators with Dirichlet boundary conditions in bounded domains. In dimension one, the $n$th eigenfunction has $n$ nodal domains. The Courant Theorem…