Existence, nonexistence and regularity theorems in a problem with a free boundary

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.
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Bibliography: 8 titles.