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Resonances for transparent obstacles. (English) Zbl 1103.35354
Proceedings of the conference on partial differential equations, Saint-Jean-de-Monts, France, May 31–June 4, 1999. Exp. Nos. I–XIX (1999). Nantes: Université de Nantes (ISBN 2-86939-146-3/pbk). Exp. No. X, 13 p. (1999).
This paper is concerned with the distribution of the resonances near the real axis for the transmission problem for a strictly convex bounded obstacle \(\mathcal O\) in \(\mathbb R^n\), \(n\geq 2\), with a smooth boundary. We consider two distinct cases. If the speed of propagation in the interior of the body is strictly less than that in the exterior, we obtain an infinite sequence of resonances tending rapidly to the real axis. These resonances are associated with a quasimode for the transmission problem the frequency support of which coincides with the corresponding gliding manifold \(\mathcal K\). To construct the quasimode, we first find a global symplectic normal form for pairs of glancing hypersurfaces in a neighborhood of \(\mathcal K\) and then we separate the variables microlocally near the whole glancing manifold \(\mathcal K\). If the speed of propagation inside \(\mathcal O\) is bigger than that outside \(\mathcal O\), then there exists a strip in the upper half plane containing the real axis, which is free of resonances. We also obtain a 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.
For the entire collection see [Zbl 0990.00047].
MSC:
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35A25 Other special methods applied to PDEs
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35L05 Wave equation
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