Eigenvalues and resonances of dissipative acoustic operator for strictly convex obstacles

We examine the wave equation in the exterior of a strictly convex bounded domain $K$ with dissipative boundary condition ${\mathrm{\partial }}_{\nu }u-\gamma (x){\mathrm{\partial }}_{t}u=0$ on the boundary $\mathrm{\Gamma }$ and $0<\gamma (x)<1,\mathrm{\forall }x\in \mathrm{\Gamma }\mathrm{.}$ The solutions are described by a contraction semigroup $V(t)=e^{tG},t\ge 0.$ The poles $\lambda$ of the meromorphic incoming resolvent $(G-\lambda {)}^{-1}:{\mathcal{H}}_{comp}\to {\mathcal{D}}_{loc}$ are eigenvalues of G if $\mathrm{R}\mathrm{e}\lambda <0$ and incoming resonances if $\mathrm{R}\mathrm{e}\lambda >0$. We obtain sharper results for the location of the eigenvalues of $G$ and incoming resonances in $\mathrm{\Lambda }=\{\lambda \in \mathbb{C}:\mathrm{\mid }\mathrm{R}\mathrm{e}\lambda \mathrm{\mid }\le C_2(1+\mathrm{\mid }\mathrm{I}\mathrm{m}\lambda \mathrm{\mid }{)}^{-2},\mathrm{\mid }\mathrm{I}\mathrm{m}\lambda \mathrm{\mid }\ge A_2>1\}$ and we prove a Weyl formula for their asymptotic. For $K=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }\le 1\}$ and $\gamma$ constant we show that $G$ has no eigenvalues so the Weyl formula concerns only the incoming resonances.