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

Xianzhe Dai; Shoo Seto; Guofang Wei

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

Differential Geometry 2018-03-06 v1 Analysis of PDEs Spectral Theory

Authors: Xianzhe Dai , Shoo Seto , Guofang Wei

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].

Keywords

laplacian eigenvalues dirichlet boundary convex domain sphere fundamental gap asymptotic expansion

Cite

@article{arxiv.1803.01115,
  title  = {Fundamental gap estimate for convex domains on sphere -- the case $n=2$},
  author = {Xianzhe Dai and Shoo Seto and Guofang Wei},
  journal= {arXiv preprint arXiv:1803.01115},
  year   = {2018}
}

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19 pages

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

Abstract

Keywords

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