Single and Double Layer Potentials on Domains with Conical Points I: Straight Cones

Abstract

Let ${\Omega = \mathbb{R}^+ \omega}$ be an open straight cone in ${\mathbb{R}^n, n\geq3}$ , where ${\omega \subset S^{n-1}}$ is a smooth subdomain of the unit sphere. Denote by K and S the double and single layer potential operators associated to Ω and the Laplace operator Δ. Let r be the distance to the origin. We consider a natural class of dilation invariant operators on ∂Ω, called Mellin convolution operators and show that ${K_a :=r^{a}Kr^{-a}}$ and ${S_b := r^{b-\frac{1}{2}}Sr^{-b-\frac{1}{2}}}$ are Mellin convolution operators for ${a \in (-1, n-1)}$ and ${b \in (\frac{1}{2}, n-\frac{3}{2})}$ . It is known that a Mellin convolution operator T is invertible if, and only if, its Mellin transform ${\hat T( \lambda)}$ is invertible for any real λ. We establish a reduction procedure that relates the Mellin transforms of K _a and S _b to the single and, respectively, double layer potential operators associated to some other elliptic operators on ω, which can be shown to be invertible using the classical theory of layer potential operators on smooth domains. This reduction procedure thus allows us to prove that ${\frac{1}{2}\pm K}$ and S are invertible between suitable weighted Sobolev spaces. A classical consequence of the invertibility of these operators is a solvability result in weighted Sobolev spaces for the Dirichlet problem on Ω.