Unique solvability of the integral equation for harmonic simple layer potential on the boundary of a domain with a peak

Abstract

The problem of finding a solution of the Dirichlet problem for the Laplace equation in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the kind Vρ = f to solve for the density, where f are boundary Dirichlet data. It is shown that if S is the boundary of an n-dimensional domain (n > 2) with an outward peak on S, then the operator V−1, which acts on the smooth functions on S, admits a unique extension to an isomorphism between the spaces of traces on S of functions with finite Dirichlet integral over Rn and the dual space. Thereby the equation Vρ = f is uniquely solvable for the density ρ for every trace f = u|S of function u with finite Dirichlet integral over Rn. Using an explicit description of the space of the traces specified, we can enunciate the theorem on solvability of a boundary integral equation Vρ = f in terms of the function describing the peak cusp.