Sharp norm estimates of layer potentials and operators at high frequency

In this paper, we investigate single and double layer potentials mapping boundary data to interior functions of a domain at high frequency λ2→∞λ2→∞. 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 L2(∂Ω)→L2(Ω)L2(∂Ω)→L2(Ω) norms decay in λ  . The rate of decay depends on the curvature of ∂Ω: The norm is λ−3/4λ−3/4 in piecewise smooth domains and λ−5/6λ−5/6 if the boundary ∂Ω is positively curved. The double layer potential, however, displays uniform L2(∂Ω)→L2(Ω)L2(∂Ω)→L2(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 L2(∂Ω)→L2(∂Ω)L2(∂Ω)→L2(∂Ω) for the single and double layer operators at high frequency that are sharp modulo log⁡λlog⁡λ. In this case, both the single and double layer operator bounds depend upon the curvature of the boundary.