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Functional geometric method for solving free boundary problems for harmonic functions. (English. Russian original) Zbl 1205.35343
Russ. Math. Surv. 65, No. 1, 1-94 (2010); translation from Usp. Mat. Nauk. 65, No. 1, 3-96 (2010).
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 functional geometric methods. The core of these methods is an interrelated analysis of the functional and geometric characteristics of the problems under consideration and of the corresponding nonlinear Riemann-Hilbert problems. An extensive list of open questions is presented. The contents of this long paper is given below. Chapter 1. Direct and inverse problems of plasma equilibrium in a tokamak: 1. Solvability theorems for the problem of equilibrium of a plasma discharge having configuration of given topological type. 2. An explicit formula expressing a harmonic function in terms of its Cauchy data on an analytic curve and the inverse problem of plasma equilibrium. Chapter 2. The Stokes-Leibenzon problem (for the Hele-Shaw flow): 3. Theorems on short and infinitely long evolution of a contour. 4. Kochina-Saffman-Taylor “finger instability”. Chapter 3. Extremal free boundary problems: 5. Planar flow with minimal ratio of the external values of the pressure on the free boundary. 6. Selecting an optimal velocity of a flow around an obstacle and optimizing the dynamics of the shape of the obstacle in the Föppl-Lavrent’ev scheme. 7. Kirchhoff scheme of a steady flow around a curvilinear obstacle partially absorbing the energy of the flow, and an estimate of the maximum efficiency of a turbine in an open stream. Chapter 4. Exponentially exact high-frequency asymptotics. 8. Vector fields defining exponentially exact high-frequency asymptotics for harmonic functions. 9. The Oleinik-Temam averaging problem. 10. Exponentially exact asymptotic approximations for a harmonic function in a domain with strongly wrinkled boundary. Appendix 1: Tokamaks and interaction of solutions. Appendix 2: Comparison of the functional geometric and hodograph methods. Appendix 3: “Hele-Shaw flow”, the “multigrid method”, etc.

MSC:
35R35 Free boundary problems for PDEs
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
35C20 Asymptotic expansions of solutions to PDEs
76D27 Other free boundary flows; Hele-Shaw flows
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
30E25 Boundary value problems in the complex plane
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