Bounding acoustic layer potentials via oscillatory integral techniques

Abstract

We consider the Helmholtz single-layer operator (the trace of the single-layer potential) as an operator on $$L^2(\Gamma )$$ where $$\Gamma $$ is the boundary of a 3-d obstacle. We prove that if $$\Gamma $$ is $$C^2$$ and has strictly positive curvature then the norm of the single-layer operator tends to zero as the wavenumber $$k$$ tends to infinity. This result is proved using a combination of (1) techniques for obtaining the asymptotics of oscillatory integrals, and (2) techniques for obtaining the asymptotics of integrals that become singular in the appropriate parameter limit. This paper is the first time such techniques have been applied to bounding norms of layer potentials. The main motivation for proving this result is that it is a component of a proof that the combined-field integral operator for the Helmholtz exterior Dirichlet problem is coercive on such domains in the space $$L^2(\Gamma )$$ .