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The third boundary value problem in potential theory for domains with a piecewise smooth boundary. (English) Zbl 0978.31003
Author’s summary: The paper investigates the third boundary value problem $$\frac{\partial u}{\partial n}+\lambda u=\mu$$ for the Laplace equation by means of potential theory. The solution is sought in the form of the Newtonian potential (1), (2), where $$\nu$$ is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure $${\mathcal T}\nu$$. Denote by $${\mathcal T}:\nu\to{\mathcal T}\nu$$ the corresponding operator on the space of signed measures on the boundary of the investigated domain $$G$$. If there is $$\alpha\neq 0$$ such that the essential spectral radius of $$(\alpha I-{\mathcal T})$$ is smaller than $$|\alpha|$$ (for example, if $$G\subset \mathbb{R}^3$$ is a domain “with a piecewise smooth boundary” and the restriction of the Newtonian potential $${\mathcal U} \lambda$$ on $$\partial G$$ 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 $$\mu\in{\mathcal C}'$$ for which $$\mu(\partial G)=0$$.
Reviewer: A.Kufner (Praha)

##### MSC:
 31B20 Boundary value and inverse problems for harmonic functions in higher dimensions 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35J25 Boundary value problems for second-order elliptic equations
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