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The Boundary Value Problems for Non-Linear Elliptic Equations and the Maximum Principle for Euler-Lagrange Equations

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Analysis and Continuum Mechanics

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Abstract

In this paper we investigate the boundary value problems for non-linear elliptic equations connected with infinite convex hypersurfaces. The asymptotic cone of such hypersurfaces is very important in our considerations. We study the problem of existence of solutions for Monge-Ampère equations on the entire space with prescribed asymptotic cone and the maximum principle and estimates for Euler-Lagrange equations. These problems have natural mutual connections.

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References

  1. I. BAKELMAN, Generalized solutions of the Monge-Ampère equations, Dokl. Akad. Nauk USSR, 111 (1957), 1143–1145.

    MathSciNet  Google Scholar 

  2. I. BAKELMAN, Geometric methods of solution of elliptic equations, Monograph, Nauka, Moscow, (1965), 1–340.

    Google Scholar 

  3. I. BAKELMAN, R-curvature, estimates and stability of solutions for elliptic equations, Journal of Diff. Equations, 43, 1 (1982), 106–133.

    Article  MATH  MathSciNet  Google Scholar 

  4. I. BAKELMAN, The boundary value problems for N-dimensional Monge-Ampère equation; N-dimensional plasticity equation, IHES, Bures-sur-Yvette, (1984), preprint, 1–107.

    Google Scholar 

  5. A. D. ALEXANDROV. Existence and uniqueness of convex surface with prescribed integral curvature, Dokl. Akad. Nauk USSR, 35, 8. (1942), 143–147.

    Google Scholar 

  6. A. D. ALEXANDROV. Convex polyhedra, Monograph, Moscow-Leningrad (1950), 1–428.

    Google Scholar 

  7. I. BAKELMAN, On the theory of Monge-Ampère equations, Vestnik LGU, NI, (1958), 25–38.

    Google Scholar 

  8. I. BAKELMAN, A. WERNER, & B. KANTOR, Introduction to global differential geometry, Monograph, Nauka, Moscow, (1973), 1–440.

    Google Scholar 

  9. H. BUSEMANN, Convex surfaces, Monograph, Interscience Publishers, Inc., New York, (1958), 1–238.

    Google Scholar 

  10. A. ALEXANDROV, Intrinsic geometry of convex surfaces, Monograph, Moscow-Leningrad (1948), 1–427.

    Google Scholar 

  11. A. POGORELOV, Exterior geometry of convex surfaces, Monograph, Moscow, Nauka, (1969), 1–759; Engl. transi. Transi. Math. Monographs, v. 35, Amer. Math. Soc. 1973.

    Google Scholar 

  12. S. N. BERNSTEIN, Collection of papers, vol. 3, Akad. Nauk of USSR, (1960), 1–439.

    Google Scholar 

  13. O. A. LADYZHENSKYA & N. N. URALTZEVA, Linear and quasilinear elliptic equations, Nauka, (1973), 2“d ed., 1–576.

    Google Scholar 

  14. D. GILBARG & N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, (monograph), Second Edition, Springer Verlag, (1983), 1–511.

    Google Scholar 

  15. I. BAKELMAN, Geometric problems in quasilinear elliptic equations, Uspekhi Mathem. Nauk, 25: 3, (1970), 42–112; Russian Mathem. Surveys (1970), 25, N° 3, 45–109.

    Google Scholar 

  16. J. SERRIN, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables. Philos. Trans. Roy. Soc., (1969), 413–496.

    Google Scholar 

  17. I. BAKELMAN, Notes concerning the torsion problem of hardening rods, IMA Preprint Series, (1986), 1–38.

    Google Scholar 

  18. CH. MORREY, Multiple integrals, Monograph. Springer Verlag (1966).

    MATH  Google Scholar 

  19. M. PROTTER & H. F. WEINBERGER, Maximum principles in Differential equations. Monograph, Prentice-Hall, Inc, (1967), 1–261.

    Google Scholar 

  20. L. CAFFARELLI, L. NIRENBERG, & J. SPRUCK, The Dirichlet problem for Non-linear Second Order Elliptic Equations I. Monge-Ampère equations, Comm. pure and Appl. Mathematics, (1984).

    Google Scholar 

  21. S. Y. CHENG & S. T. YAU, On the regularity of the solution of the Monge-Ampère equations det (u1) = F(x, u). Comm. pure and Appl. Mathem., 29, (1976), 495–516.

    Article  MATH  MathSciNet  Google Scholar 

  22. A. V. POGORELOV, The Minkowski multidimensional problem, New York, J. Wiley, (1978).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Texas A. & M. University, College Station, USA

    Ilya J. Bakelman

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  1. Ilya J. Bakelman
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Dedicated to James Serrin on his Sixtieth birthday

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© 1989 Springer-Verlag Berlin Heidelberg

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Bakelman, I.J. (1989). The Boundary Value Problems for Non-Linear Elliptic Equations and the Maximum Principle for Euler-Lagrange Equations. In: Analysis and Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83743-2_10

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  • DOI: https://doi.org/10.1007/978-3-642-83743-2_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50917-2

  • Online ISBN: 978-3-642-83743-2

  • eBook Packages: Springer Book Archive

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