Sharp norm estimates of layer potentials and operators at high frequency

Abstract

In this paper, we investigate single and double layer potentials mapping boundary data to interior functions of a domain at high frequency $λ^2\to\infty$. For single layer potentials, we find that the $L^{2}(\partialΩ)\to{}L^{2}(Ω)$ norms decay in $λ$. The rate of decay depends on the curvature of $\partialΩ$: The norm is $λ^{-3/4}$ in general domains and $λ^{-5/6}$ if the boundary $\partialΩ$ is curved. The double layer potential, however, displays uniform $L^{2}(\partialΩ)\to{}L^{2}(Ω)$ bounds independent of curvature. By various examples, we show that all our estimates on layer potentials are sharp. The appendix by Galkowski gives bounds $L^{2}(\partialΩ)\to{}L^{2}(\partialΩ)$ for the single and double layer operators at high frequency that are sharp modulo $\log λ$. In this case, both the single and double layer operator bounds depend upon the curvature of the boundary.