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

TL;DR

This paper clarifies and defends the rigorous proof of two-term spectral asymptotics in linear elasticity on Riemannian manifolds, addressing and correcting previous critiques and numerical counterexamples.

Contribution

It provides a detailed explanation and validation of the proof method used in Liu-21, countering incorrect remarks and numerical counterexamples from prior critiques.

Findings

The proof in Liu-21 is rigorous and based on semigroup and pseudodifferential operator theory.

Previous critiques and numerical counterexamples are invalid.

The two-term asymptotics for elastic eigenvalues are confirmed as correct.

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.