Eigenvalues and resonances of dissipative acoustic operator for strictly convex obstacles
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913660281028608 |
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| author | Petkov, Vesselin |
| author_facet | Petkov, Vesselin |
| contents | We examine the wave equation in the exterior of a strictly convex bounded domain $K$ with dissipative boundary condition $\partial_ν u - γ(x) \partial_t u = 0$ on the boundary $Γ$ and $0 < γ(x) <1, \:\forall x \in Γ.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \: t \geq 0.$ The poles $λ$ of the meromorphic incoming resolvent $(G - λ)^{-1}: \:{ \mathcal H}_{comp} \rightarrow {\mathcal D}_{loc}$ are eigenvalues of G if ${\rm Re}\: λ< 0$ and incoming resonances if ${\rm Re}\: λ> 0$. We obtain sharper results for the location of the eigenvalues of $G$ and incoming resonances in $Λ= \{λ\in \mathbb C:\: |{\rm Re}\: λ| \leq C_2(1 + |{\rm Im}\: λ|)^{-2},\: |{\rm Im}\: λ| \geq A_2 > 1\}$ and we prove a Weyl formula for their asymptotic. For $K = \{x \in {\mathbb R}^3:\:|x| \leq 1\}$ and $γ$ constant we show that $G$ has no eigenvalues so the Weyl formula concerns only the incoming resonances. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_01192 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Eigenvalues and resonances of dissipative acoustic operator for strictly convex obstacles Petkov, Vesselin Analysis of PDEs Mathematical Physics 35P20, 35P25, 47A40, 58J50 We examine the wave equation in the exterior of a strictly convex bounded domain $K$ with dissipative boundary condition $\partial_ν u - γ(x) \partial_t u = 0$ on the boundary $Γ$ and $0 < γ(x) <1, \:\forall x \in Γ.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \: t \geq 0.$ The poles $λ$ of the meromorphic incoming resolvent $(G - λ)^{-1}: \:{ \mathcal H}_{comp} \rightarrow {\mathcal D}_{loc}$ are eigenvalues of G if ${\rm Re}\: λ< 0$ and incoming resonances if ${\rm Re}\: λ> 0$. We obtain sharper results for the location of the eigenvalues of $G$ and incoming resonances in $Λ= \{λ\in \mathbb C:\: |{\rm Re}\: λ| \leq C_2(1 + |{\rm Im}\: λ|)^{-2},\: |{\rm Im}\: λ| \geq A_2 > 1\}$ and we prove a Weyl formula for their asymptotic. For $K = \{x \in {\mathbb R}^3:\:|x| \leq 1\}$ and $γ$ constant we show that $G$ has no eigenvalues so the Weyl formula concerns only the incoming resonances. |
| title | Eigenvalues and resonances of dissipative acoustic operator for strictly convex obstacles |
| topic | Analysis of PDEs Mathematical Physics 35P20, 35P25, 47A40, 58J50 |
| url | https://arxiv.org/abs/2310.01192 |