Geodesic
The Neumann problem for the Laplace equation on general domains
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 4, pp. 1107-1139.

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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.
Classification : 31B10, 35D05, 35J05, 35J25
Keywords: Laplace equation; Neumann problem; potential; boundary integral equation method 
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     author = {Medkov\'a, Dagmar},
     title = {The {Neumann} problem for the {Laplace} equation on general domains},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1107--1139},
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     number = {4},
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     zbl = {1174.31305},
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     url = {http://geodesic.mathdoc.fr/item/CMJ_2007__57_4_a3/}
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Medková, Dagmar. The Neumann problem for the Laplace equation on general domains. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 4, pp. 1107-1139. http://geodesic.mathdoc.fr/item/CMJ_2007__57_4_a3/