On representation of the solution of the Neumann problem in a domain with peak by the harmonic simple layer potential

Abstract

The problem of finding a solution of the Neumann problem for the Laplacian 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 second kind to be solved for density. The Neumann problem is examined in a bounded n-dimensional domain Ω+ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω− = R n \Ω+. Let Γ be the common boundary of the domains Ω±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over R n , and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω− with a cusp of an inward peak may be represented as Vρ−, where ρ− ∈ Tr(Γ)* is uniquely determined for all Ψ− ∈ Tr(Γ)*. If Ω+ is a domain with an inward peak and if Ψ+ ∈ Tr(Γ)*, Ψ+ ⊥ 1, then the solution of the Neumann problem for Ω+ has the representation u + = Vρ+ for some ρ+ ∈ Tr(Γ)* which is unique up to an additive constant ρ0, ρ0 = V −1(1). These results do not hold for domains with outward peak.