Stability of extremal domains for the first eigenvalue of the Laplacian operator
Differential Geometry
2024-07-30 v2 Analysis of PDEs
Abstract
In this paper, we compute the second variation of the first Dirichlet eigenvalue on extremal domains in general Riemannian manifolds and establish a criterion for stability. We classify the stable extremal domains in the 2-sphere and higher-dimensional spheres when the boundary is minimal. Additionally, we establish topological bounds for stable domains in a general compact Riemannian surface, assuming either nonnegative total Gaussian curvature or small volume.
Keywords
Cite
@article{arxiv.2406.13628,
title = {Stability of extremal domains for the first eigenvalue of the Laplacian operator},
author = {Marcos P. Cavalcante and Ivaldo Nunes},
journal= {arXiv preprint arXiv:2406.13628},
year = {2024}
}
Comments
This new version includes a new theorem (Theorem 1.3) and additional references. 23 pages. Comments and feedback are welcome!