# American Institute of Mathematical Sciences

November  2016, 10(4): 1111-1139. doi: 10.3934/ipi.2016034

## Location of eigenvalues for the wave equation with dissipative boundary conditions

 1 Université de Bordeaux, Institut de Mathématiques de Bordeaux, 351, Cours de la Libération, 33405 Talence

Received  June 2015 Revised  March 2016 Published  October 2016

We examine the location of the eigenvalues of the generator $G$ of a semi-group $V(t) = e^{tG},\: t \geq 0,$ related to the wave equation in an unbounded domain $\Omega \subset \mathbb{R}^d$ with dissipative boundary condition $\partial_{\nu}u - \gamma(x) \partial_t u = 0$ on $\Gamma = \partial \Omega.$ We study two cases: $(A): \: 0 < \gamma(x) < 1,\: \forall x \in \Gamma$ and $(B):\: 1 < \gamma(x), \: \forall x \in \Gamma.$ We prove that for every $0 < \epsilon \ll 1,$ the eigenvalues of $G$ in the case $(A)$ lie in the region $\Lambda_{\epsilon} = \{ z \in \mathbb{C}:\: |Re z | \leq C_{\epsilon} (|Im z|^{\frac{1}{2} + \epsilon} + 1), \: Re z < 0\},$ while in the case $(B)$ for every $0 < \epsilon \ll 1$ and every $N \in \mathbb{N}$ the eigenvalues lie in $\Lambda_{\epsilon} \cup {\mathcal R}_N,$ where ${\mathcal R}_N = \{z \in \mathbb{C}:\: |Im z| \leq C_N (|Re z| + 1)^{-N},\: Re z < 0\}.$
Citation: Vesselin Petkov. Location of eigenvalues for the wave equation with dissipative boundary conditions. Inverse Problems and Imaging, 2016, 10 (4) : 1111-1139. doi: 10.3934/ipi.2016034
##### References:
 [1] F. Cardoso, G. Popov and G. Vodev, Asymptotic of the number of resonances in the transmission problem, Comm. PDE, 26 (2001), 1811-1859. doi: 10.1081/PDE-100107460. [2] F. Colombini, V. Petkov and J. Rauch, Spectral problems for non elliptic symmetric systems with dissipative boundary conditions, J. Funct. Anal., 267 (2014), 1637-1661. doi: 10.1016/j.jfa.2014.06.018. [3] M. Dimassi and J. Sjöstrand, Spectral Asymptotics in Semi-Classical Limits, London Mathematical Society, Lecture Notes Series, 268, Cambridge University Press, 1999. doi: 10.1017/CBO9780511662195. [4] V. Georgiev and Ja. Arnaoudov, Inverse scattering problem for dissipative wave equation, Mat. App. Comput., 9 (1990), 59-78. [5] P. Lax and R. Phillips, Scattering Theory, $2^{nd}$ edition, Academic Press, New York, 1989. [6] P. Lax and R. Phillips, Scattering theory for dissipative systems, J. Funct. Anal., 14 (1973), 172-235. doi: 10.1016/0022-1236(73)90049-9. [7] A. Majda, Disappearing solutions for the dissipative wave equation, Indiana Univ. Math. J., 24 (1975), 1119-1133. [8] A. Majda, The location of the spectrum for the dissipative acoustic operator, Indiana Univ. Math. J., 25 (1976), 973-987. doi: 10.1512/iumj.1976.25.25077. [9] A. Majda, High frequency asymptotics for the scattering matrix and the inverse problem of acoustical scattering, Comm. Pure Appl. Math., 29 (1976), 261-291. doi: 10.1002/cpa.3160290303. [10] R. Melrose, Polynomial bound on the distribution of poles in scattering by an obstacle, Journées Equations aux Dérivées partielles, (1984), 1-8. doi: 10.5802/jedp.285. [11] R. Melrose and M. Taylor, Boundary Problems for Wave Equations with Glancing and Gliding Rays,, Available from , (). [12] F. Olver, Asymptotics and Special Functions, Academic Press,New York, London, 1974. [13] V. Petkov, Scattering problems for symmetric systems with dissipative boundary conditions, in Studies in Phase Space Analysis and Applications to PDEs, Progress in Nonlinear Differential Equations and their Applications, Birkhauser, 84 (2013), 337-353. doi: 10.1007/978-1-4614-6348-1_15. [14] V. Petkov and G. Vodev, Asymptotics of the number of the interior transmission eigenvalues,, Journal of Spectral Theory, (). [15] G. Popov and G. Vodev, Resonances near the real axis for transparent obstacles, Commun. Math. Phys., 207 (1999), 411-438. doi: 10.1007/s002200050731. [16] J. Sjöstrand and G. Vodev, Asymptotics of the number of Rayleigh resonances, Math. Ann., 309 (1997), 287-306. doi: 10.1007/s002080050113. [17] J. Sjöstrand, Weyl law for semi-classical resonances with randomly perturbed potentials, Mémoire de SMF, 136 (2014), vi+144 pp. [18] B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon & Breach Science Publishers, New York, 1989. [19] G. Vodev, Transmission eigenvalue-free regions}, Commun. Math. Phys., 336 (2015), 1141-1166. doi: 10.1007/s00220-015-2311-2. [20] G. Vodev, Transmission eigenvalues for strictly concave domains, Math. Ann., 366 (2016), 301-336. doi: 10.1007/s00208-015-1329-2.

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##### References:
 [1] F. Cardoso, G. Popov and G. Vodev, Asymptotic of the number of resonances in the transmission problem, Comm. PDE, 26 (2001), 1811-1859. doi: 10.1081/PDE-100107460. [2] F. Colombini, V. Petkov and J. Rauch, Spectral problems for non elliptic symmetric systems with dissipative boundary conditions, J. Funct. Anal., 267 (2014), 1637-1661. doi: 10.1016/j.jfa.2014.06.018. [3] M. Dimassi and J. Sjöstrand, Spectral Asymptotics in Semi-Classical Limits, London Mathematical Society, Lecture Notes Series, 268, Cambridge University Press, 1999. doi: 10.1017/CBO9780511662195. [4] V. Georgiev and Ja. Arnaoudov, Inverse scattering problem for dissipative wave equation, Mat. App. Comput., 9 (1990), 59-78. [5] P. Lax and R. Phillips, Scattering Theory, $2^{nd}$ edition, Academic Press, New York, 1989. [6] P. Lax and R. Phillips, Scattering theory for dissipative systems, J. Funct. Anal., 14 (1973), 172-235. doi: 10.1016/0022-1236(73)90049-9. [7] A. Majda, Disappearing solutions for the dissipative wave equation, Indiana Univ. Math. J., 24 (1975), 1119-1133. [8] A. Majda, The location of the spectrum for the dissipative acoustic operator, Indiana Univ. Math. J., 25 (1976), 973-987. doi: 10.1512/iumj.1976.25.25077. [9] A. Majda, High frequency asymptotics for the scattering matrix and the inverse problem of acoustical scattering, Comm. Pure Appl. Math., 29 (1976), 261-291. doi: 10.1002/cpa.3160290303. [10] R. Melrose, Polynomial bound on the distribution of poles in scattering by an obstacle, Journées Equations aux Dérivées partielles, (1984), 1-8. doi: 10.5802/jedp.285. [11] R. Melrose and M. Taylor, Boundary Problems for Wave Equations with Glancing and Gliding Rays,, Available from , (). [12] F. Olver, Asymptotics and Special Functions, Academic Press,New York, London, 1974. [13] V. Petkov, Scattering problems for symmetric systems with dissipative boundary conditions, in Studies in Phase Space Analysis and Applications to PDEs, Progress in Nonlinear Differential Equations and their Applications, Birkhauser, 84 (2013), 337-353. doi: 10.1007/978-1-4614-6348-1_15. [14] V. Petkov and G. Vodev, Asymptotics of the number of the interior transmission eigenvalues,, Journal of Spectral Theory, (). [15] G. Popov and G. Vodev, Resonances near the real axis for transparent obstacles, Commun. Math. Phys., 207 (1999), 411-438. doi: 10.1007/s002200050731. [16] J. Sjöstrand and G. Vodev, Asymptotics of the number of Rayleigh resonances, Math. Ann., 309 (1997), 287-306. doi: 10.1007/s002080050113. [17] J. Sjöstrand, Weyl law for semi-classical resonances with randomly perturbed potentials, Mémoire de SMF, 136 (2014), vi+144 pp. [18] B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon & Breach Science Publishers, New York, 1989. [19] G. Vodev, Transmission eigenvalue-free regions}, Commun. Math. Phys., 336 (2015), 1141-1166. doi: 10.1007/s00220-015-2311-2. [20] G. Vodev, Transmission eigenvalues for strictly concave domains, Math. Ann., 366 (2016), 301-336. doi: 10.1007/s00208-015-1329-2.
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