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.

The paper authored by Xiaolong Han and Melissa Tacy now includes an appendix by Jeffrey Galkowski on double layer operators

References in corpus (2)

• Restriction Bounds for the Free Resolvent and Resonances in Lossy Scattering
• The quantisation of normal velocity does not concentrate on hypersurfaces

Cited by in corpus (1)

• Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations