Abstract:
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 γ, 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 γ is homeomorphic to a circle or to the union of two circles). The question of regularity of the curve γ is considered.
Figures: 3.
Bibliography: 8 titles.
Citation:
A. Badzhadi, A. S. Demidov, “Existence, nonexistence and regularity theorems in a problem with a free boundary”, Math. USSR-Sb., 50:1 (1985), 67–84
\Bibitem{BadDem83}
\by A.~Badzhadi, A.~S.~Demidov
\paper Existence, nonexistence and regularity theorems in a~problem with a~free boundary
\jour Math. USSR-Sb.
\yr 1985
\vol 50
\issue 1
\pages 67--84
\mathnet{http://mi.mathnet.ru/eng/sm2276}
\crossref{https://doi.org/10.1070/SM1985v050n01ABEH002733}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=715835}
\zmath{https://zbmath.org/?q=an:0599.35133|0553.35088}
Linking options:
https://www.mathnet.ru/eng/sm2276
https://doi.org/10.1070/SM1985v050n01ABEH002733
https://www.mathnet.ru/eng/sm/v164/i1/p64
This publication is cited in the following 6 articles:
A. S. Demidov, “Functional geometric method for solving free boundary problems for harmonic functions”, Russian Math. Surveys, 65:1 (2010), 1–94
A. S. Demidov, “Reduction of a Nonstationary Euler Hydrodynamic Equation to a System of Ordinary Differential Equations in the Plane Problem of Cavitational Flow with Boundary Control”, Math. Notes, 86:1 (2009), 41–52
A. S. Demidov, “Helmholtz–Kirchhoff method and boundary control of a plane flow”, J. Math. Sci., 150:5 (2008), 2335–2343
Demidov A., “Complete Asymptotics of Dirichlet Problem Solutions to 2D-Laplace Equation with Rapidly Oscillation Boundary Value Data”, Dokl. Akad. Nauk, 346:6 (1996), 732–734
Yong Liu, “The Equilibrium Plasma Subject to Skin Effect”, SIAM J Math Anal, 26:5 (1995), 1157
Demidov A., Petrova V., “Inverse Problem with Free-Boundary in the Theory of Equilibrium Plasma”, Differ. Equ., 30:6 (1994), 958–962
Citation in format AMSBIB
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