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Distribution of resonances and local energy decay in the transmission problem. II. (English) Zbl 0968.35035
For part I see G. Popov and G. Vodev [Asymptotic Anal. 19, No. 3-4, 253-265 (1999; Zbl 0931.35115)].
From the introduction: This paper is concerned with the resonances of the transmission problem for a transparent bounded strictly convex obstacle \({\mathcal O}\) with a smooth boundary (which may contain an impenetrable body). If the speed of propagation inside \({\mathcal O}\) is bigger than that outside \({\mathcal O}\), we prove under some natural conditions, that there exists a strip in the upper half plane containing the real axis, which is free of resonances. We also obtain an uniform decay of the local energy for the corresponding mixed problem with an exponential rate of decay when the dimension is odd, and polynomial otherwise. It is well known that such a decay of the local energy holds for the wave equation with Dirichlet (Neumann) boundary conditions for any nontrapping obstacle. In our case, however, \({\mathcal O}\) is a trapping obstacle for the corresponding classical system.

MSC:
35J15 Second-order elliptic equations
35B34 Resonance in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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