光滑有界域上旋度算子的谱几何
谱理论
2025-05-30 v2 偏微分方程分析
摘要
我们证明三维欧几里得空间中一般光滑有界域上旋度算子的谱由单重特征值组成。证明中的主要新工具是一个关于域扰动下旋度特征值变化的公式，这与狄利克雷边界条件下拉普拉斯特征值变化的 Hadamard 公式类似。作为该变分公式的另一个应用，我们简化了已知必要条件的推导——即在给定体积的域中最小化第一旋度特征值泛函的域必须满足的条件，并推导了域在更高阶特征值泛函取极值时类似的必要条件。
引用
@article{arxiv.2502.13067,
title = {Spectral geometry of the curl operator on smoothly bounded domains},
author = {Josef Greilhuber and Willi Kepplinger},
journal= {arXiv preprint arXiv:2502.13067},
year = {2025}
}
备注
For version 2, the proof of the Hadamard rule (Theorem 4.1) has been significantly shortened and streamlined, mostly by switching from singular/simplicial to de Rham cohomology. Additionally more care was taken in spelling out the meaning of using complex versus real Lagrangean boundary conditions. Otherwise many typos were fixed. 25 pages, 1 figure