Spherical caps do not always maximize Neumann eigenvalues on the sphere

2025/03/19 by Bucur, Dorin, Laugesen, Richard S., Martinet, Eloi +1
#35P15 #58J50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2503.15385

Abstract

We prove the existence of an open set Ω⊂\mathbbS² for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large areas and contrasts with results by Bandle and later authors proving maximality of the disk under additional topological or geometric conditions, thereby revealing such conditions to be necessary.

Related

• Sharp inequalities for Neumann eigenvalues on the sphere
• Eigenvalue problems and free boundary minimal surfaces in spherical caps
• Asymptotic location and shape of the optimal favorable region in a Neumann spectral problem
• Maximization of the second Laplacian Eigenvalue on the Sphere
• An isoperimetric inequality for Laplace eigenvalues on the sphere