# 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 & Imaging, 2016, 10 (4) : 1111-1139. doi: 10.3934/ipi.2016034
##### References:
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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.  doi: 10.1081/PDE-100107460.  Google Scholar [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.  doi: 10.1016/j.jfa.2014.06.018.  Google Scholar [3] M. Dimassi and J. Sjöstrand, Spectral Asymptotics in Semi-Classical Limits,, London Mathematical Society, 268 (1999).  doi: 10.1017/CBO9780511662195.  Google Scholar [4] V. Georgiev and Ja. Arnaoudov, Inverse scattering problem for dissipative wave equation,, Mat. App. Comput., 9 (1990), 59.   Google Scholar [5] P. Lax and R. Phillips, Scattering Theory,, $2^{nd}$ edition, (1989).   Google Scholar [6] P. Lax and R. Phillips, Scattering theory for dissipative systems,, J. Funct. Anal., 14 (1973), 172.  doi: 10.1016/0022-1236(73)90049-9.  Google Scholar [7] A. Majda, Disappearing solutions for the dissipative wave equation,, Indiana Univ. Math. J., 24 (1975), 1119.   Google Scholar [8] A. Majda, The location of the spectrum for the dissipative acoustic operator,, Indiana Univ. Math. J., 25 (1976), 973.  doi: 10.1512/iumj.1976.25.25077.  Google Scholar [9] A. Majda, High frequency asymptotics for the scattering matrix and the inverse problem of acoustical scattering,, Comm. Pure Appl. Math., 29 (1976), 261.  doi: 10.1002/cpa.3160290303.  Google Scholar [10] R. Melrose, Polynomial bound on the distribution of poles in scattering by an obstacle,, Journées Equations aux Dérivées partielles, (1984), 1.  doi: 10.5802/jedp.285.  Google Scholar [11] R. Melrose and M. Taylor, Boundary Problems for Wave Equations with Glancing and Gliding Rays,, Available from , ().   Google Scholar [12] F. Olver, Asymptotics and Special Functions,, Academic Press, (1974).   Google Scholar [13] V. Petkov, Scattering problems for symmetric systems with dissipative boundary conditions,, in Studies in Phase Space Analysis and Applications to PDEs, 84 (2013), 337.  doi: 10.1007/978-1-4614-6348-1_15.  Google Scholar [14] V. Petkov and G. Vodev, Asymptotics of the number of the interior transmission eigenvalues,, Journal of Spectral Theory, ().   Google Scholar [15] G. Popov and G. Vodev, Resonances near the real axis for transparent obstacles,, Commun. Math. Phys., 207 (1999), 411.  doi: 10.1007/s002200050731.  Google Scholar [16] J. Sjöstrand and G. Vodev, Asymptotics of the number of Rayleigh resonances,, Math. Ann., 309 (1997), 287.  doi: 10.1007/s002080050113.  Google Scholar [17] J. Sjöstrand, Weyl law for semi-classical resonances with randomly perturbed potentials,, Mémoire de SMF, 136 (2014).   Google Scholar [18] B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics,, Gordon & Breach Science Publishers, (1989).   Google Scholar [19] G. Vodev, Transmission eigenvalue-free regions},, Commun. Math. Phys., 336 (2015), 1141.  doi: 10.1007/s00220-015-2311-2.  Google Scholar [20] G. Vodev, Transmission eigenvalues for strictly concave domains,, Math. Ann., 366 (2016), 301.  doi: 10.1007/s00208-015-1329-2.  Google Scholar
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