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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)

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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