Experimental Mathematics

The Spectrum of the Damped Wave Operator for a Bounded Domain in { $\boldsymbol{R^2}$}

Mark Asch and Gilles Lebeau

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Abstract

The spectrum of the damped wave operator for a bounded domain in {$R^2$} is shown to be related to the asymptotic average of the damping function by the geodesic flow. This allows the calculation of an asymptotic expression for the distribution of the imaginary parts of the eigenvalues for a radially symmetric geometry. Numerical simulations confirm the theoretical model. In addition, we are able to exhibit the beautiful structure of the spectrum and the close links between the eigenfunctions, the rays of geometrical optics, and the geometry of the damping region. The MATLAB code used in this paper is provided.

Article information

Source
Experiment. Math., Volume 12, Issue 2 (2003), 227-241.

Dates
First available in Project Euclid: 31 October 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1067634733

Mathematical Reviews number (MathSciNet)
MR2016708

Zentralblatt MATH identifier
1061.35064

Subjects
Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Secondary: 35B37 49J20: Optimal control problems involving partial differential equations 49K20: Problems involving partial differential equations 93C20: Systems governed by partial differential equations

Keywords
Spectrum non-self-adjoint operator damped wave equation

Citation

Asch, Mark; Lebeau, Gilles. The Spectrum of the Damped Wave Operator for a Bounded Domain in { $\boldsymbol{R^2}$}. Experiment. Math. 12 (2003), no. 2, 227--241. https://projecteuclid.org/euclid.em/1067634733


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