Dirichlet Eigenfunctions of the Square Membrane: Courant’s Property, and A. Stern’s and Å. Pleijel’s Analyses

DOI10.1007/978-3-319-17443-3_6zbMath1390.35209arXiv1402.6054OpenAlexW3104967885MaRDI QIDQ3447482

Bernard Helffer, Pierre H. Bérard

Publication date: 27 October 2015

Published in: Analysis and Geometry (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1402.6054

zbMATH Keywords

nodal domainsnodal linesCourant theorem

Mathematics Subject Classification ID

Spectral problems; spectral geometry; scattering theory on manifolds (58J50) Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs (35B05) Asymptotic distributions of eigenvalues in context of PDEs (35P20)

Related Items (14)

On the number of nodal domains of toral eigenfunctions ⋮ Courant-sharp eigenvalues of Neumann 2-rep-tiles ⋮ Tensor solutions in irregular domains: eigenvalue problems ⋮ Some nodal properties of the quantum harmonic oscillator and other Schrödinger operators in ℝ² ⋮ Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle ⋮ Courant-sharp Robin eigenvalues for the square: the case with small Robin parameter ⋮ The weak Pleijel theorem with geometric control ⋮ On the number of Courant-sharp Dirichlet eigenvalues ⋮ Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders ⋮ A. Stern's analysis of the nodal sets of some families of spherical harmonics revisited ⋮ Non-universality of the Nazarov-Sodin constant ⋮ Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip ⋮ Courant-sharp eigenvalues of the three-dimensional square torus ⋮ Courant-sharp Robin eigenvalues for the square and other planar domains

Cites Work

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• A. Stern's analysis of the nodal sets of some families of spherical harmonics revisited
• Nodal domains and spectral minimal partitions
• On the number of nodal domains of spherical harmonics
• A review on large 𝑘 minimal spectral 𝑘-partitions and Pleijel’s Theorem
• Remarks on courant's nodal line theorem
• A generalization of Courant's nodal domain theorem.
• Inégalités isopérimétriques et applications
• On the mininum number of domains in which the nodal lines of spherical harmonics divide the sphere

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