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 Mat. Sb. (N.S.), 1981, Volume 116(158), Number 1(9), Pages 72–86 (Mi msb2432)

Some singularities in the behavior of solutions of equations of minimal-surface type in unbounded domains

V. M. Miklyukov

Abstract: In this paper the behavior of the solutions of equations of minimal-surface type is studied in unbounded domains. It is established that if the domain is sufficiently narrow in the neighborhood of the point at infinity of $\mathbf R^2$, then any solution having zero Dirichlet or Neumann data on the boundary must be identically constant. A condition on the narrowness of the domain is found under which the solution cannot change sign in the domain. An estimate of the form $\sum_ki(a_k)\leqslant c$ is proved, where $i(a_k)$ is the topological index of the solution at the point $a_k$, $c$ is a constant depending only on the equation, the domain and the number of points of local extremum of the boundary function, and the summation is taken over all critical points of the solution.
Bibliography: 11 titles.

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English version:
Mathematics of the USSR-Sbornik, 1983, 44:1, 61–73

Bibliographic databases:

UDC: 517.54+517.947
MSC: Primary 35J20; Secondary 49F10, 53A10

Citation: V. M. Miklyukov, “Some singularities in the behavior of solutions of equations of minimal-surface type in unbounded domains”, Mat. Sb. (N.S.), 116(158):1(9) (1981), 72–86; Math. USSR-Sb., 44:1 (1983), 61–73

Citation in format AMSBIB
\Bibitem{Mik81} \by V.~M.~Miklyukov \paper Some singularities in the behavior of solutions of equations of minimal-surface type in unbounded domains \jour Mat. Sb. (N.S.) \yr 1981 \vol 116(158) \issue 1(9) \pages 72--86 \mathnet{http://mi.mathnet.ru/msb2432} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=632489} \zmath{https://zbmath.org/?q=an:0556.49022|0495.49028} \transl \jour Math. USSR-Sb. \yr 1983 \vol 44 \issue 1 \pages 61--73 \crossref{https://doi.org/10.1070/SM1983v044n01ABEH000951} 

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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. Vuorinen M., “Conformal Geometry and Quasiregular-Mappings”, Lect. Notes Math., 1319 (1988), 1–&
2. V. M. Miklyukov, “Some criteria for parabolicity and hyperbolicity of the boundary sets of surfaces”, Izv. Math., 60:4 (1996), 763–809
3. Hwang J., “How Many Theorems Can Be Derived From a Vector Function - on Uniqueness Theorems for the Minimal Surface Equation”, Taiwan. J. Math., 7:4 (2003), 513–539
4. Meeks Iii W.H., Perez J., “The Classical Theory of Minimal Surfaces”, Bull. Amer. Math. Soc., 48:3 (2011), 325–407
5. Allen Weitsman, “Spiraling Minimal Graphs”, Comput. Methods Funct. Theory, 2014
6. Erik Lundberg, Allen Weitsman, “On the growth of solutions to the minimal surface equation over domains containing a halfplane”, Calc. Var, 2015
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