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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.

##### MSC:
 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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##### References:
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