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The Neumann problem for the Laplace equation on general domains. (English) Zbl 1174.31305
Summary: The solution of the weak Neumann problem for the Laplace equation with a distribution as a boundary condition is studied on a general open set  \(G\) in the Euclidean space. It is shown that the solution of the problem is the sum of a constant and the Newtonian potential corresponding to a distribution with finite energy supported on  \(\partial G\). If we look for a solution of the problem in this form we get a bounded linear operator. Under mild assumptions on  \(G\) a necessary and sufficient condition for the solvability of the problem is given and the solution is constructed.

31B10 Integral representations, integral operators, integral equations methods in higher dimensions
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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