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

Xianzhe Dai; Shoo Seto; Guofang Wei

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arXiv · 1803.01115

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

Xianzhe Dai·Shoo Seto·Guofang Wei

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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 $\mathbb S^n$ is $\geq 3 \frac{\pi^2}{D^2}$ when $n \geq 3$. 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].

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Xianzhe Dai, Shoo Seto, Guofang Wei. 2018-03-03. Fundamental gap estimate for convex domains on sphere -- the case $n=2$. https://arxiv.org/abs/1803.01115

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