Fundamental gap estimate for convex domains on sphere -- the case n=2

2018/03/03 by Dai, Xianzhe, Seto, Shoo, Wei, Guofang · 3 citations
#35P15 #53C35 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1803.01115

Abstract

In [SWW16, HW17] it is shown that the difference of the first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter D of sphere \mathbb Sⁿ is ≥ 3 (π²)/(D²) when n ≥ 3. We prove the same result when n=2. In fact our proof works for all dimension. We also give an asymptotic expansion of the first and second Dirichlet eigenvalues of the model in [SWW16].

Cited by

• Constant potentials do not minimise the fundamental gap on convex domains in hyperbolic space
• Gradient bounds for viscosity solutions to certain elliptic equations
• Sharp Fundamental Gap Estimate on Convex Domains in Gaussian Spaces

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