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 Uspekhi Mat. Nauk, 2010, Volume 65, Issue 1(391), Pages 3–96 (Mi umn9341)

Functional geometric method for solving free boundary problems for harmonic functions

A. S. Demidovab

a M. V. Lomonosov Moscow State University
b Moscow Institute of Physics and Technology

Abstract: 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.

DOI: https://doi.org/10.4213/rm9341

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English version:
Russian Mathematical Surveys, 2010, 65:1, 1–94

Bibliographic databases:

UDC: 517.57
MSC: Primary 31A05, 35C20, 35R35, 35Q99, 76D27; Secondary 76W05, 82D10

Citation: A. S. Demidov, “Functional geometric method for solving free boundary problems for harmonic functions”, Uspekhi Mat. Nauk, 65:1(391) (2010), 3–96; Russian Math. Surveys, 65:1 (2010), 1–94

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn9341
• https://doi.org/10.4213/rm9341
• http://mi.mathnet.ru/eng/umn/v65/i1/p3

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Demidov A.S., “Inverse problem for the Grad-Shafranov equation with affine right-hand side”, Russ. J. Math. Phys., 17:2 (2010), 145–153
2. Bezrodnykh S.I., Demidov A.S., “On the uniqueness of solution Cauchy's inverse problem for the equation $\Delta u=au+b$”, Asymptotic Anal., 74:1-2 (2011), 95–121
3. A. S. Demidov, J.-P. Loheac, V. Runge, “Attractors–Repellers in the Space of Contours in the Stokes–Leibenson Problem for Hele–Shaw Flows”, J. Math. Sci., 189:4 (2013), 568–581
4. S. I. Bezrodnykh, V. I. Vlasov, “Application of the multipole method to direct and inverse problems for the Grad–Shafranov equation with a nonlocal condition”, Comput. Math. Math. Phys., 54:4 (2014), 631–695
5. S. I. Bezrodnykh, V. I. Vlasov, “Reshenie obratnoi zadachi dlya uravneniya Greda–Shafranova dlya rascheta magnitnogo polya v tokamake”, Matem. modelirovanie, 26:11 (2014), 57–64
6. S. I. Bezrodnykh, B. V. Somov, “An analysis of magnetic field and magnetosphere of neutron star under effect of a shock wave”, Advances in Space Research, 56:5 (2015), 964–969
7. Demidov A.S., Loheac J.-P., Runge V., “Stokes–Leibenson problem for Hele-Shaw flow: a critical set in the space of contours”, Russ. J. Math. Phys., 23:1 (2016), 35–55
8. S. I. Bezrodnykh, “Finding the Coefficients in the New Representation of the Solution of the Riemann–Hilbert Problem Using the Lauricella Function”, Math. Notes, 101:5 (2017), 759–777
9. S. I. Bezrodnykh, “The Lauricella hypergeometric function $F_D^{(N)}$, the Riemann–Hilbert problem, and some applications”, Russian Math. Surveys, 73:6 (2018), 941–1031
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