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