18 citations · 90 across the 16 of their papers we have counts for
16 papers
One can't hear orientability of surfaces
Pierre Bérard, David L. Webb
The main result of this paper is that one cannot hear orientability of a surface with boundary. More precisely, we construct two isospectral flat surfaces with boundary with the sa…
An Euler-type formula for partitions of the Möbius strip
Pierre Bérard, Bernard Helffer
The purpose of this note is to prove an Euler-type formula for partitions of the Möbius strip. This formula was introduced in our joint paper with R.~Kiwan, "Courant-sharp property…
Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip
Pierre Bérard, Bernard Helffer, Rola Kiwan
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…
Sturm's theorem on the zeros of sums of eigenfunctions: Gelfand's strategy implemented
Pierre Bérard, Bernard Helffer
In the second section "Courant-Gelfand theorem" of his last published paper (Topological properties of eigenoscillations in mathematical physics, Proc. Steklov Institute Math. 273…
Level sets of certain Neumann eigenfunctions under deformation of Lipschitz domains. Application to the Extended Courant Property
Pierre Bérard, Bernard Helffer
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 combi…
On Courant's nodal domain property for linear combinations of eigenfunctions, Part II
Pierre Bérard, Bernard Helffer
Generalizing Courant's nodal domain theorem, the "Extended Courant property" is the statement that a linear combination of the first eigenfunctions has at most nodal domain…
Sturm's theorem on zeros of linear combinations of eigenfunctions
Pierre Bérard, Bernard Helffer
Motivated by recent questions about the extension of Courant's nodal domain theorem, we revisit a theorem published by C. Sturm in 1836, which deals with zeros of linear combinatio…
The weak Pleijel theorem with geometric control
Pierre Bérard, Bernard Helffer
Let , be a bounded open set, and denote by , the eigenvalues of the Dirichlet Laplacian arranged in nondecreasing order, with mul…
On the number of nodal domains of the 2D isotropic quantum harmonic oscillator -- an extension of results of A. Stern --
Pierre Bérard, Bernard Helffer
In the case of the sphere and the square, Antonie Stern (1925) claimed in her PhD thesis the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contai…
Dirichlet eigenfunctions of the square membrane: Courant's property, and A. Stern's and Å. Pleijel's analyses
Pierre Bérard, Bernard Helffer
In this paper, we revisit Courant's nodal domain theorem for the Dirichlet eigenfunctions of a square membrane, and the analyses of A. Stern and Å. Pleijel.
Spectral positivity and Riemannian coverings
Pierre Bérard, Philippe Castillon
Let be a complete non-compact Riemannian manifold. We consider operators of the form , where is the non-negative Laplacian associated with the metric , an…
Inverse spectral positivity for surfaces
Pierre Bérard, Philippe Castillon
Let be a complete non-compact Riemannian surface. We consider operators of the form , where is the non-negative Laplacian, the Gaussian curvature, a…
Stability Properties of Rotational Catenoids in the Heisenberg Groups
Pierre Bérard, Marcos P. Cavalcante
In this paper, we determine the maximally stable, rotationally invariant domains on the catenoids $\cC_a$ (minimal surfaces invariant by rotations) in the Heisenberg group with a l…
Lindelöf's theorem for catenoids revisited
Pierre Bérard, Ricardo Sa Earp
In this paper we study the maximal stable domains on minimal catenoids in Euclidean and hyperbolic spaces and in . We in particular investigate whether half-vertical…
Examples of -hypersurfaces in and geometric applications
Pierre Bérard, Ricardo Sa Earp
In this paper we describe all rotation -hypersurfaces in and use them as barriers to prove existence and characterization of certain vertical -graphs and to gi…
Minimal hypersurfaces in $\HH^n \times \R$, total curvature and index
Pierre Bérard, Ricardo Sa Earp
In this paper, we consider minimal hypersurfaces in the product space . We begin by studying examples of rotation hypersurfaces and hypersurfaces in…