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

This article is cited in 9 scientific papers (total in 9 papers)

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

Full text: PDF file (2067 kB)
References: PDF file   HTML file

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
Received: 11.12.2009

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

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    Citing articles on Google Scholar: Russian citations, English citations
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    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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    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  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  elib  scopus
    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  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    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  mathnet  mathscinet  elib
    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  crossref  isi  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  crossref  mathscinet  isi  elib
    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  mathnet  crossref  crossref  adsnasa  isi  elib
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