Two-term spectral asymptotics in linear elasticity on a Riemannian manifold
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916077080936448 |
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| author | Liu, Genqian |
| author_facet | Liu, Genqian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_02679 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Two-term spectral asymptotics in linear elasticity on a Riemannian manifold Liu, Genqian Spectral Theory Mathematical Physics Analysis of PDEs Differential Geometry 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. |
| title | Two-term spectral asymptotics in linear elasticity on a Riemannian manifold |
| topic | Spectral Theory Mathematical Physics Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2208.02679 |