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On the equation of minimal surface in R3: various representations, properties of exact solutions, conservation laws

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Various representations of the equation of minimal surface in \( {\mathbb{R}^3} \) are considered. Properties of exact solutions are studied, and a procedure of construction the corresponding conservation laws is suggested. Links between the solutions of this equation and those of the elliptic version of the Monge–Ampere equation are found. Bibliography: 19 titles.

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References

  1. A. V. Pogorelov, Differential Geometry [in Russian], Moscow (1969).

  2. B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry [in Russian], Moscow (1986).

  3. Dao Chong Thi and A. T. Fomenko, Minimal Surfaces and Plateaux Problem [in Russian], Moscow (1991).

  4. S. N. Bernstein, Collected Works [in Russian], Vol. 3, Moscow (1960).

  5. R. Ossermann, Usp. Mat. Nauk, 22, 55 (1967).

    Google Scholar 

  6. R. Osserman (ed.), Minimal Surfaces, Springer (1997).

  7. A. B. Borisov, Dokl. Ros. Akad. Nauk, 389, 1 (2003).

    Google Scholar 

  8. A. Moro, Physics, Optics, 3, 1253 (2009).

    Google Scholar 

  9. V. Blashke, Introduction to Differential Geometry [in Russian], Moscow-Izhevsk (2007).

  10. A. D. Polianin, V. F. Zaitsev, and A. I. Jurov, Methods of Solution of Nonlinear Equations of Mathematical Physics and Mechanics [in Russian], Moscow (2005).

  11. E. V. Ferapontov and Y. Nutku, solv.int/94090004v1 (1994).

  12. A. V. Kiselev, Fund. Appl. Math., 12, 93 (2006).

    Google Scholar 

  13. I. A. Taimanov, Lectures on Differential Geometry [in Russian], Moscow-Izhevsk (2002).

  14. M. Ablowitz and P. A. Clarcson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering, Cambridge Univ. Press (1991).

  15. M. Gourses, solv.-int/19712018v1(1997).

  16. J. C. Brunelli, M. Gürses, and K. Jeltukhin, hep.th/9906233v1 (1999).

  17. P. J. Olver, Application of the Lie Groups to Differential Equations, Springer-Verlag, New York (1986).

    Google Scholar 

  18. A. B. Borisov and V. V. Kisielev, Inverse Problems, 5, 959 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  19. E. Sh. Gutshabash and V. D. Lipovskii, Zap. Nauchn. Semin. LOMI, 180, 53 (1990).

    Google Scholar 

  20. B. Pelloni, Private Communication, Gallipoli (2008).

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Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    E. Sh. Gutshabash

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  1. E. Sh. Gutshabash
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Correspondence to E. Sh. Gutshabash.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 374, 2010, pp. 121–135.

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Gutshabash, E.S. On the equation of minimal surface in R3: various representations, properties of exact solutions, conservation laws. J Math Sci 168, 829–836 (2010). https://doi.org/10.1007/s10958-010-0031-x

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