# Weyl formula for the eigenvalues of the dissipative acoustic operator

Weyl formula for the eigenvalues of the dissipative acoustic operator We study the wave equation in the exterior of a bounded domain K with dissipative boundary condition ∂νu-γ(x)∂tu=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial _{\nu } u - \gamma (x) \partial _t u = 0$$\end{document} on the boundary Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma$$\end{document} and γ(x)>0.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\gamma (x) > 0.$$\end{document} The solutions are described by a contraction semi-group V(t)=etG,t≥0.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$V(t) = e^{tG}, \, t \ge 0.$$\end{document} The eigenvalues λk\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda _k$$\end{document} of G with Reλk<0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{ Re }\,\lambda _k < 0$$\end{document} yield asymptotically disappearing solutions u(t,x)=eλktf(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$u(t, x) = e^{\lambda _k t} f(x)$$\end{document} having exponentially decreasing global energy. We establish a Weyl formula for these eigenvalues in the case minx∈Γγ(x)>1.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\min _{x\in \Gamma } \gamma (x) > 1.$$\end{document} For strictly convex obstacles K, this formula concerns all eigenvalues of G. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

# Weyl formula for the eigenvalues of the dissipative acoustic operator

, Volume 9 (1) – Mar 1, 2022
18 pages

/lp/springer-journals/weyl-formula-for-the-eigenvalues-of-the-dissipative-acoustic-operator-YLziVaaekv
Publisher
Springer Journals
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021
eISSN
2197-9847
DOI
10.1007/s40687-021-00301-3
Publisher site
See Article on Publisher Site

### Abstract

We study the wave equation in the exterior of a bounded domain K with dissipative boundary condition ∂νu-γ(x)∂tu=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial _{\nu } u - \gamma (x) \partial _t u = 0$$\end{document} on the boundary Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma$$\end{document} and γ(x)>0.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\gamma (x) > 0.$$\end{document} The solutions are described by a contraction semi-group V(t)=etG,t≥0.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$V(t) = e^{tG}, \, t \ge 0.$$\end{document} The eigenvalues λk\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda _k$$\end{document} of G with Reλk<0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{ Re }\,\lambda _k < 0$$\end{document} yield asymptotically disappearing solutions u(t,x)=eλktf(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$u(t, x) = e^{\lambda _k t} f(x)$$\end{document} having exponentially decreasing global energy. We establish a Weyl formula for these eigenvalues in the case minx∈Γγ(x)>1.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\min _{x\in \Gamma } \gamma (x) > 1.$$\end{document} For strictly convex obstacles K, this formula concerns all eigenvalues of G.

### Journal

Research in the Mathematical SciencesSpringer Journals

Published: Mar 1, 2022

Keywords: Dissipative boundary conditions; Eigenvalues asymptotics; 35P20; 35P25; 47A40; 58J50

### References

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