Sharp norm estimates of layer potentials and operators at high frequency

TL;DR

This paper derives sharp high-frequency norm estimates for layer potentials and operators, revealing how boundary curvature influences decay rates and providing new bounds that are optimal up to logarithmic factors.

Contribution

It establishes precise decay rates for layer potential norms at high frequency, highlighting the role of boundary curvature and proving the sharpness of these estimates.

Findings

Single layer potential norms decay as λ^{-3/4} generally and λ^{-5/6} for curved boundaries.

Double layer potential bounds are uniform and independent of curvature.

Appendix provides sharp boundary operator bounds, dependent on curvature, up to logarithmic factors.

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.