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Izv. RAN. Ser. Mat., 1996, Volume 60, Issue 4, Pages 111–158 (Mi izv80)  

This article is cited in 6 scientific papers (total in 6 papers)

Some criteria for parabolicity and hyperbolicity of the boundary sets of surfaces

V. M. Miklyukov

Volgograd State University

Abstract: We give criteria for the parabolicity and hyperbolicity of the boundary sets of surfaces $F=(D,ds^2_F)$, where $D$ is a domain in $\mathbb R^n$ and $ds^2_F$ is the square of the length element on $F$. We prove the parabolicity of certain boundary sets located on the graphs of the solutions of equations of minimal surface type. As an example we present a generalized maximum principle for the derivatives of solution of equations of minimal surface type where domains of $\mathbb R^n$ become “narrow” at infinity. We formulate criteria for the parabolicity and hyperbolicity of boundary sets on the graphs of spacelike surfaces in Minkowski space $\mathbb R_1^{n+1}$, and in particular, we obtain an essential strengthening of the theorem of Choi and Treibergs on the hyperbolicity of the graphs of entire solutions of the constant mean curvature equation in $\mathbb R_1^3$.

DOI: https://doi.org/10.4213/im80

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English version:
Izvestiya: Mathematics, 1996, 60:4, 763–809

Bibliographic databases:

MSC: Primary 53A10, 35J60, 53B30; Secondary 30C65, 30C80, 31B15, 49Q05, 53B25, 58E20
Received: 21.12.1994

Citation: V. M. Miklyukov, “Some criteria for parabolicity and hyperbolicity of the boundary sets of surfaces”, Izv. RAN. Ser. Mat., 60:4 (1996), 111–158; Izv. Math., 60:4 (1996), 763–809

Citation in format AMSBIB
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\vol 60
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\pages 111--158
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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. A. P. Karmazin, “Parabolicity and Hyperbolicity Conditions for Boundary Elements of Surfaces”, Math. Notes, 70:6 (2001), 866–869  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. A. A. Klyachin, “Description of the set of singular entire solutions of the maximal surface equation”, Sb. Math., 194:7 (2003), 1035–1054  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. S. A. Korol'kov, “Harmonic functions on Riemannian manifolds with ends”, Siberian Math. J., 49:6 (2008), 1051–1061  mathnet  crossref  mathscinet  isi  elib
    4. Poluboyarova N.M., “Nekotorye otsenki \it{g}-emkosti ekstremalnykh poverkhnostei i sledstviya iz nikh1”, Vestnik volgogradskogo gosudarstvennogo universiteta. seriya 1: matematika. fizika, 2011, no. 2, 63–74  mathscinet  elib
    5. Yu. V. Goncharov, A. G. Losev, A. V. Svetlov, “Garmonicheskie funktsii na konusakh modelnykh mnogoobrazii”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2014, no. 3(22), 13–22  mathnet
    6. A. N. Kondrashov, “On the asymptotics of solutions of elliptic equations at the ends of non-compact Riemannian manifolds with metrics of a special form”, Izv. Math., 83:2 (2019), 287–314  mathnet  crossref  crossref  adsnasa  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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