论文标题
重新审视球体和平面域的特征值乘数上的上限
Upper bounds on eigenvalue multiplicities for spheres and plane domains revisited
论文作者
论文摘要
我们重新审视了1999年出现的两篇论文：[1] M. Hoffmann-Ostenhof，T。Hoffmann-Ostenhof和N. Nadirashvili。在表面上的拉普拉斯人的特征值。安。全球肛门。地理。 17（1999）43--48。 [2] T. Hoffmann-Ostenhof，P。Michor和N. Nadirashvili。固定膜的特征值多样性的边界。地理。功能。肛门。 9（1999）1169--1188。这些论文的主要结果是，Riemannian Surface $ m $的$ k $ th eigenvalue的多重性$ \ mathrm {mult}（λ_k（m））$从上面的$（2k-3）$限制为$ k \ ge 3 $。在[1]中，$ m $是球体同构的。在[2]中，$ m $是一个具有Dirichlet边界条件的平面域。在这两种情况下，特征值的起始标签为$ 1 $。 [1,2]中给出的证据不是很详细，并且通常依赖于节点集的图形或特殊配置。该专着的目的是为上述上限提供详细的一般证明，并将结果扩展到罗宾边界条件。当$ m $对球体同构时，我们提供了一个完整的证明，证明任何$ k \ ge 3 $ $ \ mathrm {mult}（λ_k）\ le（2k-3）$，通过介绍和仔细研究组合类型和一些特定特定特定特定特定eigenfunctions的结合域的组合类型和仔细研究。当$ m $是一个平面域时，我们会考虑Dirichlet和Robin边界条件，并且还研究了某些特定特征功能的结合域的组合类型和标记。我们证明了不平等$ \ mathrm {mult}（λ_k）\ le（2k-2）$ for General $ c^{\ infty} $有界域和所有$ k \ ge 3 $。我们证明了不等式的$ \ mathrm {mult}（λ_k）\ le（2k-3）$ for $ k \ ge 3 $在域仅连接域的附加假设下。
We revisit two papers which appeared in 1999: [1] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and N. Nadirashvili. On the multiplicity of eigenvalues of the Laplacian on surfaces. Ann. Global Anal. Geom. 17 (1999) 43--48. [2] T. Hoffmann-Ostenhof, P. Michor, and N. Nadirashvili. Bounds on the multiplicity of eigenvalues for fixed membranes. Geom. Funct. Anal. 9 (1999) 1169--1188. The main result of these papers is that the multiplicity $\mathrm{mult}(λ_k(M))$ of the $k$th eigenvalue of the Riemannian surface $M$ is bounded from above by $(2k-3)$ provided that $k \ge 3$. In [1], $M$ is homeomorphic to a sphere. In [2], $M$ is a plane domain with Dirichlet boundary condition. In both cases, the starting label of eigenvalues is $1$. The proofs given in [1,2] are not very detailed, and often rely on figures or special configurations of nodal sets. The purpose of this monograph is to provide detailed general proofs for the above upper bounds and to extend the results to Robin boundary conditions. When $M$ is homeomorphic to a sphere, we provide a complete proof that $\mathrm{mult}(λ_k) \le (2k-3)$ for any $k\ge 3$, by introducing and carefully studying the combinatorial type and a labeling of the nodal domains of some particular eigenfunctions. When $M$ is a plane domain, we consider Dirichlet and Robin boundary conditions and we also study the combinatorial types and a labeling of the nodal domains of some particular eigenfunctions. We prove the inequality $\mathrm{mult}(λ_k) \le (2k-2)$ for general $C^{\infty}$ bounded domains and all $k \ge 3$. We prove the inequality $\mathrm{mult}(λ_k) \le (2k-3)$ for $k \ge 3$ under the additional assumption that the domain is simply connected.
