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Distribution of the resonances and local energy decay in the transmission problem. (English) Zbl 0931.35115
Summary: We study the resonances associated to the transmission problem for a strictly convex obstacle provided that the speed of propagation of the waves in the interior of the obstacle is strictly greater than the speed in the exterior. We prove that there are no resonances in a region of the form $$\text{Im} z\leq C_1|z|^{-1}$$, $$|\text{Re} z|\geq C_2>0$$. Using this we obtain some uniform estimates on the decay of the local energy.

##### MSC:
 35P15 Estimates of eigenvalues in context of PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation