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 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) 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 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) for the single and double layer operators at high frequency that are sharp modulo logλ. In this case, both the single and double layer operator bounds depend upon the curvature of the boundary.