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

Popov, Georgi; Vodev, Georgi

Commun. Math. Phys. 207, No. 2, 411-438 (1999).

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.

Reviewer: M.L.Mehra (Bornheim)

MSC:

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

Keywords:

transmission problem; quasi-modes and resonances; glancing hypersurface