Geodesic
Existence, nonexistence and regularity theorems in a problem with a free boundary
Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 67-84 
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/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Demidov A. S., “Ob odnoi zadache so svobodnoi granitsei v teorii ravnovesnoi plazmy”, Trudy seminara im. I. G. Petrovskogo, 1978, no. 4, 65–82 | MR | Zbl

[2] Demidov A. S., Configurations du plasma stationnaire équilibré, V. I (Proceedings of a seminar held in Pavia. 1979), Roma, 1980 | MR

[3] Badjadi A., Existence, non-existence et régularité de la solution d'un probléme à frontiére libre, These de Magister, Université d'Alger “Houari Boumediene”, 1982

[4] Natanson I. P., Teoriya funktsii veschestvennoi peremennoi, 3-e izd., Nauka, M., 1974 | MR

[5] Petrovskii I. G., Lektsii ob uravneniyakh s chastnymi proizvodnymi, 3-e izd., Gostekhizdat, M., 1961 | MR

[6] Miranda C., Partial Differential Equations of Elliptic Type, Springer-Verlag, 1970 | MR

[7] Monakhov V. N., Kraevye zadachi so svobodnymi granitsami dlya ellipticheskikh sistem uravnenii, Nauka, Novosibirsk, 1977 | MR

[8] Kartan A., Elementarnaya teoriya analiticheskikh funktsii odnogo i neskolkikh kompleksnykh peremennykh, IL, M., 1963, s. 187