The cut-off resolvent can grow arbitrarily fast in obstacle scattering

Abstract

We consider time-harmonic acoustic scattering by a compact sound-soft obstacle $Γ\subset \mathbb{R}^n$ ($n\geq 2$) that has connected complement $Ω:= \mathbb{R}^n\setminus Γ$. This scattering problem is modelled by the inhomogeneous Helmholtz equation $Δu + k^2 u = -f$ in $Ω$, the boundary condition that $u=0$ on $\partial Ω= \partial Γ$, and the standard Sommerfeld radiation condition. It is well-known that, if the boundary $\partial Ω$ is smooth, then the norm of the cut-off resolvent of the Laplacian, that maps the compactly supported inhomogeneous term $f$ to the solution $u$ restricted to some ball, grows at worst exponentially with $k$. In this paper we show that, if no smoothness of $Γ$ is imposed, then the growth can be arbitrarily fast. Precisely, given some modestly increasing unbounded sequence $0<k_1<k_2<\ldots$ and some arbitrarily rapidly increasing sequence $0<a_1<a_2<\ldots$, we construct a compact $Γ$ such that, for each $j\in \mathbb{N}$, the norm of the cut-off resolvent at $k=k_j$ is greater than $a_j$.