Resonances near the real axis for transparent obstacles. (English) Zbl 0951.35036

Let \(O\subset \mathbb{R}^n\), \(n\geq 2\), be a bounded strictly convex domain with \(C^\infty\) boundary \(\Gamma\) and \(\Omega= \mathbb{R}^n/\overline O\), \(\alpha> 0\). This paper deals with the transmission problem \[ \begin{cases} (c^2\Delta+ \lambda^2)u_1= 0\quad &\text{in }O,\\ (\Delta+ \lambda^2) u_2= 0\quad &\text{in }\Omega,\\ u_1- u_2= 0\quad &\text{on }\Gamma,\\ \partial_{-n}u_1+ \partial_n u_2= 0\quad &\text{on }\Gamma,\\ u_2\text{-}\lambda\text{-outgoing},\end{cases}\tag{1} \] \(\lambda\in \mathbb{C}\) will be said to be a resonance for the transmission problem associated to \(O\), if the problem (1) has a nontrivial solution. On this basis the authors show: There exists an infinite sequence \(\{\lambda_j\}\) of different resonances of (1) such that \[ 0< \text{Im }\lambda_j\leq C_N|\lambda_j|^{-N},\quad \forall N\geq 1. \] The basic step concerns the construction of a quasi mode of the frequency support, which is concentrated at the glancing manifold \(\kappa\) of the interior problem.
To do this, they provide a global symplectic normal form for pairs of glancing hypersurfaces in a neighbourhood of \(\kappa\) and then separate the variables microlocally near the whole glancing manifold. The authors conclude their investigation by looking at the geometry of the billiard flow.


35J25 Boundary value problems for second-order elliptic equations
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
58J05 Elliptic equations on manifolds, general theory
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