黎曼流形中Dirichlet Laplacian的区域变形和特征值
Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold
摘要：
对于实解析黎曼流形$M$的任何有界正则域$Ω$，我们用$λ_{k}(Ω)$表示$Ω$的Dirichlet Laplacian的第$k$个特征值.本文考虑$λ_k$和作为$M$中固定体积域集合上的泛函.我们介绍和调查的临界域的自然概念，这个功能。特别地，我们得到了一个域是临界的，局部最小化或局部最大化的$λ_k$的充要条件。这些结果依赖于Hadamard型变分公式，我们建立在这个一般的设置。
英文摘要：
For any bounded regular domain $Ω$ of a real analytic Riemannian manifold $M$, we denote by $λ_{k}(Ω)$ the $k$-th eigenvalue of the Dirichlet Laplacian of $Ω$. In this paper, we consider $λ_k$ and as a functional upon the set of domains of fixed volume in $M$. We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain necessary and sufficient conditions for a domain to be critical, locally minimizing or locally maximizing for $λ_k$. These results rely on Hadamard type variational formulae that we establish in this general setting.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
