Geodesic
On the singularities of the gradient of the solution to the Dirichlet--Neumann problem outside a~plane cut
Matematičeskie zametki, Tome 77 (2005) no. 3, pp. 364-377

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The problem considered in this paper deals with the Laplace equation outside a cut of sufficiently arbitrary form. The Dirichlet condition is given on one side of the cut and the Neumann condition on the other. Using the integral representation for the solution, we obtain explicit asymptotic formulas describing the singularity of the gradient of the solution at the edges of the cut. We discuss the effect of the disappearance of the singularity. 
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     author = {P. A. Krutitskii and A. I. Sgibnev},
     title = {On the singularities of the gradient of the solution to the {Dirichlet--Neumann} problem outside a~plane cut},
     journal = {Matemati\v{c}eskie zametki},
     pages = {364--377},
     publisher = {mathdoc},
     volume = {77},
     number = {3},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_77_3_a4/}
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P. A. Krutitskii; A. I. Sgibnev. On the singularities of the gradient of the solution to the Dirichlet--Neumann problem outside a~plane cut. Matematičeskie zametki, Tome 77 (2005) no. 3, pp. 364-377. http://geodesic.mathdoc.fr/item/MZM_2005_77_3_a4/