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.

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