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Uspekhi Mat. Nauk, 1970, Volume 25, Issue 3(153), Pages 49–112 (Mi umn5343)  

This article is cited in 3 scientific papers (total in 3 papers)

Geometric problems in quasilinear elliptic equations

I. Ya. Bakelman


Abstract: In their survey reports A. D. Aleksandrov and A. V. Pogorelov [1] and N. V. Efimov [2] give a detailed account of the deep relationships between the theory of surfaces and the theory of partial differential equations; they also highlight the main results and research problems on the boundary of geometry and analysis connected with Gaussian curvature of surfaces and its various generalizations. More recently problems on the boundary of geometry and the theory of quasilinear equations have been intensively investigated in various directions. In this article we shall be concerned with just two closely related problems from among the diverse questions that have been studied in this field: the geometric methods of estimating the solution to the Dirichlet problem for a quasilinear elliptic equation, and the construction of a non-parametric hypersurface with given mean curvature in Riemannian space. Even for the case of zero mean curvature (minimal surfaces) these questions present considerable difficulties. We shall not pay special attention to the minimal surface situation, but we draw attention to the very full presentation of the research in this field in the surveys by R. Osserman [3], [40] and J. C. C. Nitsche [4]. This article is a considerably enlarged account of the lectures given by the author in the Second and Third All-Union symposia on geometry in the large in 1967 and 1969`[15], [35].

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English version:
Russian Mathematical Surveys, 1970, 25:3, 45–109

Bibliographic databases:

UDC: 513.73+517.944
MSC: 35H30, 14J70, 35B45, 35A30, 53A10, 35D05
Received: 27.10.1969

Citation: I. Ya. Bakelman, “Geometric problems in quasilinear elliptic equations”, Uspekhi Mat. Nauk, 25:3(153) (1970), 49–112; Russian Math. Surveys, 25:3 (1970), 45–109

Citation in format AMSBIB
\Bibitem{Bak70}
\by I.~Ya.~Bakelman
\paper Geometric problems in quasilinear elliptic equations
\jour Uspekhi Mat. Nauk
\yr 1970
\vol 25
\issue 3(153)
\pages 49--112
\mathnet{http://mi.mathnet.ru/umn5343}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=340814}
\zmath{https://zbmath.org/?q=an:0217.41401}
\transl
\jour Russian Math. Surveys
\yr 1970
\vol 25
\issue 3
\pages 45--109
\crossref{https://doi.org/10.1070/RM1970v025n03ABEH003802}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Stefan Hildebrandt, Helmut Kaul, “Two-dimensional variational problems with obstructions, and Plateau's problem for h-surfaces in a Riemannian manifold”, Comm Pure Appl Math, 25:2 (1972), 187  crossref  mathscinet  zmath
    2. I. Ya. Bakelman, B. E. Kantor, “Estimates for solutions of quasilinear elliptic equations connected with problems of geometry “in the large””, Math. USSR-Sb., 20:3 (1973), 348–363  mathnet  crossref  mathscinet  zmath
    3. M. Dajczer, J. Ripoll, “Constant mean curvature hypersurfaces with single valued projections on planar domains”, Journal of Differential Equations, 250:3 (2011), 1493  crossref
  • Успехи математических наук Russian Mathematical Surveys
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