Distribution of resonances and local energy decay in the transmission problem. II

Abstract

This paper is concerned with the resonances of the transmission problem for a transparent bounded strictly convex obstacle O with a smooth boundary (which may contain an impenetrable body). If the speed of propagation inside O is bigger than that outside 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, O is a trapping obstacle for the corresponding classical system. Let O1 ⊂ R n ,n ≥ 2, be a bounded domain with a connected C ∞ boundary Γ1. Let also O2 ⊂ R n be a bounded domain with a connected C ∞ boundary Γ and such that O1 ⊂O 2 and Γ1 ∩ Γ= ∅. Set Ω = R n O2 and Ω1 = R n O1. Consider in O = O2 O1 the operator �