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:
[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

show all references

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

[1]

T. J. Christiansen. Resonances and balls in obstacle scattering with Neumann boundary conditions. Inverse Problems & Imaging, 2008, 2 (3) : 335-340. doi: 10.3934/ipi.2008.2.335

[2]

Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Problems & Imaging, 2009, 3 (1) : 139-149. doi: 10.3934/ipi.2009.3.139

[3]

Laurent Bourgeois, Houssem Haddar. Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Problems & Imaging, 2010, 4 (1) : 19-38. doi: 10.3934/ipi.2010.4.19

[4]

Victor Isakov. On uniqueness of obstacles and boundary conditions from restricted dynamical and scattering data. Inverse Problems & Imaging, 2008, 2 (1) : 151-165. doi: 10.3934/ipi.2008.2.151

[5]

Long Hu, Tatsien Li, Bopeng Rao. Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type. Communications on Pure & Applied Analysis, 2014, 13 (2) : 881-901. doi: 10.3934/cpaa.2014.13.881

[6]

Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113

[7]

Marc Briant. Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions. Kinetic & Related Models, 2017, 10 (2) : 329-371. doi: 10.3934/krm.2017014

[8]

Bruno Fornet, O. Guès. Penalization approach to semi-linear symmetric hyperbolic problems with dissipative boundary conditions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 827-845. doi: 10.3934/dcds.2009.23.827

[9]

Gennaro Infante. Positive solutions of differential equations with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 432-438. doi: 10.3934/proc.2003.2003.432

[10]

Thierry Daudé, Damien Gobin, François Nicoleau. Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds. Inverse Problems & Imaging, 2016, 10 (3) : 659-688. doi: 10.3934/ipi.2016016

[11]

Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, David Isaacson, Jari P. Kaipio, Debra McGivney, Erkki Somersalo, Joseph Volzer. Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results. Inverse Problems & Imaging, 2015, 9 (3) : 749-766. doi: 10.3934/ipi.2015.9.749

[12]

Valery Imaikin, Alexander Komech, Herbert Spohn. Scattering theory for a particle coupled to a scalar field. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 387-396. doi: 10.3934/dcds.2004.10.387

[13]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[14]

Gennaro Infante. Eigenvalues and positive solutions of odes involving integral boundary conditions. Conference Publications, 2005, 2005 (Special) : 436-442. doi: 10.3934/proc.2005.2005.436

[15]

Qiong Meng, X. H. Tang. Solutions of a second-order Hamiltonian system with periodic boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1053-1067. doi: 10.3934/cpaa.2010.9.1053

[16]

Daniel Franco, Donal O'Regan. Existence of solutions to second order problems with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 273-280. doi: 10.3934/proc.2003.2003.273

[17]

Ammar Khemmoudj, Taklit Hamadouche. General decay of solutions of a Bresse system with viscoelastic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4857-4876. doi: 10.3934/dcds.2017209

[18]

Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713

[19]

Zhenhua Zhang. Asymptotic behavior of solutions to the phase-field equations with neumann boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 683-693. doi: 10.3934/cpaa.2005.4.683

[20]

Davide Guidetti. Classical solutions to quasilinear parabolic problems with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 717-736. doi: 10.3934/dcdss.2016024

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]