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Keywords:
Laplace equation; Neumann problem; potential; boundary integral equation method
Laplace equation; Neumann problem; potential; boundary integral equation method
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.
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