Geodesic
Functional geometric method for solving free boundary problems for harmonic functions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 65 (2010) no. 1, pp. 1-94

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A survey is given of results and approaches for a broad spectrum of free boundary problems for harmonic functions of two variables. The main results are obtained by the functional geometric method. The core of these methods is an interrelated analysis of the functional and geometric characteristics of the problems under consideration and of the corresponding non-linear Riemann–Hilbert problems. An extensive list of open questions is presented. Bibliography: 124 titles.
Keywords: free boundaries, harmonic functions. 
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A. S. Demidov. Functional geometric method for solving free boundary problems for harmonic functions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 65 (2010) no. 1, pp. 1-94. http://geodesic.mathdoc.fr/item/RM_2010_65_1_a0/