Eigenvalues and resonances of dissipative acoustic operator for strictly convex obstacles

Abstract

We examine the wave equation in the exterior of a strictly convex bounded domain $K$ with dissipative boundary condition $\partial_ν u - γ(x) \partial_t u = 0$ on the boundary $Γ$ and $0 < γ(x) <1, \:\forall x \in Γ.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \: t \geq 0.$ The poles $λ$ of the meromorphic incoming resolvent $(G - λ)^{-1}: \:{ \mathcal H}_{comp} \rightarrow {\mathcal D}_{loc}$ are eigenvalues of G if ${\rm Re}\: λ< 0$ and incoming resonances if ${\rm Re}\: λ> 0$. We obtain sharper results for the location of the eigenvalues of $G$ and incoming resonances in $Λ= \{λ\in \mathbb C:\: |{\rm Re}\: λ| \leq C_2(1 + |{\rm Im}\: λ|)^{-2},\: |{\rm Im}\: λ| \geq A_2 > 1\}$ and we prove a Weyl formula for their asymptotic. For $K = \{x \in {\mathbb R}^3:\:|x| \leq 1\}$ and $γ$ constant we show that $G$ has no eigenvalues so the Weyl formula concerns only the incoming resonances.