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 → ∞ . In the high frequency regime the key problem is the dependence of mapping norms on the parameter λ . For single layer potentials, we find that the L 2 ( ∂ Ω ) → L 2 ( Ω ) norms decay in λ . The rate of decay depends on the curvature of ∂Ω: The norm is λ − 3 / 4 in piecewise smooth domains and λ − 5 / 6 if the boundary ∂Ω is positively curved. The double layer potential, however, displays uniform L 2 ( ∂ Ω ) → L 2 ( Ω ) bounds independent of curvature. By various examples, we show that all our estimates on layer potentials are sharp. Appendix A by Galkowski gives bounds L 2 ( ∂ Ω ) → L 2 ( ∂ Ω ) 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.