On the nodal line of the second eigenfunction of the Laplacian in $\mathbf{R}^2$

Melas, Antonios D.

doi:10.4310/jdg/1214447811

Cited by 66 publications

(52 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…When Ω is a plane convex domain, this nodal line hits the boundary of Ω at exactly two points, see Melas [24], or Alessandrini [1]. For general simply connected plane domains Ω, it is still a conjecture, named after Larry Payne, the "Payne conjecture".…”

Section: General Resultsmentioning

confidence: 99%

Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions

Henrot

Oudet

2003

Archive for Rational Mechanics and Analysis

View full text Add to dashboard Cite

Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditionsAntoine HENROT, Edouard OUDET AbstractIn this paper, we are interested in the minimization of the second eigenvalue of the Laplacian with Dirichlet boundary conditions amongst convex plane domains with given area. The natural candidate to be the optimum was the "stadium", convex hull of two identical tangent disks. We refute this conjecture. Nevertheless, we prove existence of a minimzer. We also study some qualitative properties of the minimizer (regularity, geometric properties).

show abstract

Section: General Resultsmentioning

confidence: 99%

Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions

Henrot

Oudet

2003

Archive for Rational Mechanics and Analysis

View full text Add to dashboard Cite

show abstract

“…One direction along which much work has been developed over the last three decades centres around a conjecture of Payne's from 1967 [49], which states that the nodal set of a second eigenfunction of problem (1.1) for any planar domain cannot consist of a closed curve. This conjecture can be extended in an obvious way to higher dimensions as follows: The most general result obtained so far was given by Melas in 1992 [47], who showed that Conjecture 1 holds in the case of planar convex domains (cf also [2]). This followed a string of results by several authors under some additional symmetry restrictions on the convex domain [50,45,52,13].…”

Section: Introductionmentioning

confidence: 99%

Location of the nodal set for thin curved tubes

Freitas¹,

Krejčiřı́k²

2008

Indiana Univ. Math. J.

View full text Add to dashboard Cite

The Dirichlet Laplacian in curved tubes of arbitrary constant cross-section rotating together with the Tang frame along a bounded curve in Euclidean spaces of arbitrary dimension is investigated in the limit when the volume of the cross-section diminishes. We show that spectral properties of the Laplacian are, in this limit, approximated well by those of the sum of the Dirichlet Laplacian in the cross-section and a one-dimensional Schrödinger operator whose potential is expressed solely in terms of the first curvature of the reference curve. In particular, we establish the convergence of eigenvalues, the uniform convergence of eigenfunctions and locate the nodal set of the Dirichlet Laplacian in the tube near nodal points of the one-dimensional Schrödinger operator. As a consequence, we prove the "nodal-line conjecture" for a class of non-convex and possibly multiply connected domains. The results are based on a perturbation theory developed for Schrödinger-type operators in a straight tube of diminishing cross-section.

show abstract

“…A proof of the theorem is embedded in a paper by S.-Y. Cheng [16]. For purposes of keeping our paper self contained, we describe a more direct proof.…”

Section: Corollaries (I) a Single Nodal Intersects A Smooth Boundarymentioning

confidence: 99%

Nodal and other properties of the second eigenfunction of the Laplacian in the plane

Liboff

2005

Quart. Appl. Math.

View full text Add to dashboard Cite

Abstract. Rules are established for the intersection of nodals at a boundary in the plane relevant to the second eigenfunction of the Laplacian. Employing these results together with regularity theorems related to odd reflection of solutions of the Helmholtz equation, as well as a variation of C.S. Lin's analysis, the following theorem is revisited: The nodal curve of the second eigenstate of the Laplacian for bounded convex domains in the plane, with Dirichlet boundary conditions, is a simple curve that intersects the boundary in two distinct points. Application is made to the regular convex polygons with C n , n ≥ 2, symmetry and to convex billiards with smooth boundaries.

show abstract

On the nodal line of the second eigenfunction of the Laplacian in $\mathbf{R}^2$

Cited by 66 publications

References 9 publications

Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions

Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions

Location of the nodal set for thin curved tubes

Nodal and other properties of the second eigenfunction of the Laplacian in the plane

Contact Info

Product

Resources

About