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Two-term spectral asymptotics in linear elasticity on a Riemannian manifold

Genqian Liu

Abstract

In this note, by explaining two key methods that were employed in \cite{Liu-21} and by giving some remarks, we show that the proof of Theorem 1.1 in \cite{Liu-21} is a rigorous proof based on theory of strongly continuous semigroups and pseudodifferential operators. All remarks and comments to paper \cite{Liu-21}, which were given by Matteo Capoferri, Leonid Friedlander, Michael Levitin and Dmitri Vassiliev in \cite{CaFrLeVa-22}, are incorrect. The so-called "numerical counter-examples" in \cite{CaFrLeVa-22} are useless examples for the two-term asymptotics of the counting functions of the elastic eigenvalues. Clearly, the conclusion and the proof of \cite{Liu-21} are completely correct.

Two-term spectral asymptotics in linear elasticity on a Riemannian manifold

Abstract

In this note, by explaining two key methods that were employed in \cite{Liu-21} and by giving some remarks, we show that the proof of Theorem 1.1 in \cite{Liu-21} is a rigorous proof based on theory of strongly continuous semigroups and pseudodifferential operators. All remarks and comments to paper \cite{Liu-21}, which were given by Matteo Capoferri, Leonid Friedlander, Michael Levitin and Dmitri Vassiliev in \cite{CaFrLeVa-22}, are incorrect. The so-called "numerical counter-examples" in \cite{CaFrLeVa-22} are useless examples for the two-term asymptotics of the counting functions of the elastic eigenvalues. Clearly, the conclusion and the proof of \cite{Liu-21} are completely correct.
Paper Structure (3 sections, 2 theorems, 73 equations)

This paper contains 3 sections, 2 theorems, 73 equations.

Table of Contents

  1. Two key methods
  2. Several remarks
  3. Some responses

Key Result

Theorem 1.1

Let $(\Omega,g)$ be a compact smooth Riemannian manifold of dimension $n$ with smooth boundary $\partial \Omega$, and let $0< \tau_1^-< \tau_2^- \le \tau^-_3\le \cdots \le \tau_k^- \le \cdots$ (respectively, $0\le \tau_1^+ < \tau_2^+ \le \tau_3^+ \le \cdots \le \tau_k^+ \le \cdots$) be the eigenvalu Here ${\hbox{Vol}}\,(\Omega)$ denotes the $n$-dimensional volume of $\Omega$, ${\hbox{Vol}}\, (\par

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4