A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space

Benguria, Rafael D.; Linde, Helmut

Duke Math. J., Tome 136 (2007) no. 1, p. 245-279 / Harvested from Project Euclid

Résumé

Let $\Omega$ be some domain in the hyperbolic space $\mathbb{H}^n$ (with $n\ge 2$ ), and let $S_1$ be a geodesic ball that has the same first Dirichlet eigenvalue as $\Omega$ . We prove the Payne-Pólya-Weinberger (PPW) conjecture for $\mathbb{H}^n$ , namely, that the second Dirichlet eigenvalue on $\Omega$ is smaller than or equal to the second Dirichlet eigenvalue on $S_1$ . We also prove that the ratio of the first two eigenvalues on geodesic balls is a decreasing function of the radius

Publié le : 2007-11-01
Classification: 58J50, 35P15, 49R50

@article{1192715420,
     author = {Benguria, Rafael D. and Linde, Helmut},
     title = {A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space},
     journal = {Duke Math. J.},
     volume = {136},
     number = {1},
     year = {2007},
     pages = { 245-279},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1192715420}
}

Benguria, Rafael D.; Linde, Helmut. A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space. Duke Math. J., Tome 136 (2007) no. 1, pp.  245-279. http://gdmltest.u-ga.fr/item/1192715420/