The third boundary value problem in potential theory for domains with a piecewise smooth boundary

Abstract

The paper investigates the third boundary value problem $$\frac{{\partial u}}{{\partial n}}{ } + { \lambda }u{ } = { }\mu $$ for the Laplace equation by the means of the potential theory. The solution is sought in the form of the Newtonian potential (1), (2), where ν is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure $$T\nu . { Denote by }T:{ }\nu { } \to { }T\nu $$ the corresponding operator on the space of signed measures on the boundary of the investigated domain G. If there is α ≠ 0 such that the essential spectral radius of $$\left( {\alpha I - T} \right)$$ is smaller than |α| (for example, if G ⊂ R 3 is a domain “with a piecewise smooth boundary” and the restriction of the Newtonian potential $$U{ \lambda }$$ on ∂G is a finite continuous functions) then the third problem is uniquely solvable in the form of a single layer potential (1) with the only exception which occurs if we study the Neumann problem for a bounded domain. In this case the problem is solvable for the boundary condition $$\mu { } \in { }C'$$ for which μ(∂G) = 0.