黎曼流形上线性弹性中的两项谱渐近
谱理论
2024-01-02 v6 数学物理
偏微分方程分析
微分几何
math.MP
摘要
在本注记中，通过阐释 \cite{Liu-21} 中所用的两种关键方法并给出若干评注，我们表明 \cite{Liu-21} 中定理1.1的证明是基于强连续半群与伪微分算子理论的严格证明。Matteo Capoferri、Leonid Friedlander、Michael Levitin 与 Dmitri Vassiliev 在 \cite{CaFrLeVa-22} 中对论文 \cite{Liu-21} 给出的所有评注与评论均不正确。\cite{CaFrLeVa-22} 中所谓的“数值反例”对弹性特征值计数函数的两项渐近而言是无用的例子。显然，\cite{Liu-21} 的结论与证明完全正确。
引用
@article{arxiv.2208.02679,
title = {Two-term spectral asymptotics in linear elasticity on a Riemannian manifold},
author = {Genqian Liu},
journal= {arXiv preprint arXiv:2208.02679},
year = {2024}
}
备注
16 pages. In order to coincide with my original paper, the denotes $x$ and $y$ are exchanged from line -2 to line -10 line on p.2. Also in equalities (1.11) and (1.12), the star is put over y on p.3. More explanations are given for fundamental solution $mathbf{K}^+(t,x,y)$ with Neumann (i.e., free) boundary condition