Geodesic
Existence, nonexistence and regularity theorems in a~problem with a~free boundary
Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 67-84

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A problem for the Laplace equation in a plane region, part of whose boundary (namely the inside boundary of the region) is unknown, is studied. On this portion of the boundary, denoted by $\gamma$, the solution of the Laplace equation satisfies the zero Dirichlet condition and a given Neumann type condition. On the outside (given) boundary of the region, the solution assumes a constant value. Solvability, as well as insolvability, conditions are studied for the problem with an a priori given topological type of free boundary (the curve $\gamma$ is homeomorphic to a circle or to the union of two circles). The question of regularity of the curve $\gamma$ is considered. Figures: 3. Bibliography: 8 titles. 
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     author = {A. Badzhadi and A. S. Demidov},
     title = {Existence, nonexistence and regularity theorems in a~problem with a~free boundary},
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     url = {http://geodesic.mathdoc.fr/item/SM_1985_50_1_a4/}
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A. Badzhadi; A. S. Demidov. Existence, nonexistence and regularity theorems in a~problem with a~free boundary. Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 67-84. http://geodesic.mathdoc.fr/item/SM_1985_50_1_a4/