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A priori bounds and necessary conditions for solvability of prescribed curvature equations

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Abstract

We prove an estimate for the magnitude of solutions of the prescribed higher order mean curvature equations and examine the necessity of our conditions. Our results include well known sharp estimates for the mean and Gauss curvature and our previous estimate for scalar curvature as special cases.

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References

  1. Bakel’man, I. Ya.,Geometric methods for solving elliptic equations, Moscow, Izdat. Nauka 1965 [Russian]

    Google Scholar 

  2. Burago, Yu. D. and Zalgaller, V.A.,Geometric inequalities, Springer-Verlag, Berlin, (1988)

    MATH  Google Scholar 

  3. Caffarelli, L., Nirenberg, L. and Spruck, J.,Nonlinear second-order elliptic equations V.,The Dirichlet problem for Weingarten surfaces, Comm. Pure Appl. Math.41, 47–70 (1988)

    MATH  MathSciNet  Google Scholar 

  4. Federer, H.,Geometric Measure Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1969

    MATH  Google Scholar 

  5. Gilbarg, D., and Trudinger, N.S.,Elliptic differential equations of second order, Second edition, Springer-Verlag, Berlin (1983)

    MATH  Google Scholar 

  6. Ivochkina, N.M.,Solution of the Dirichlet problem for the equation of curvature of order m, Mat. Sbornik180, 867–887 (1989)

    MATH  Google Scholar 

  7. Ivochkina, N.M.,Solution of the Dirichlet problem for the equation of curvature of order m, J. Algebra and Analysis6 (1989), to appear

  8. Korevaar, N.J.,A priori interior gradient bounds for solutions to elliptic Weingarten equations, Ann. Inst. Henri Poincaré, Analyse Non Lineaire4, 405–421 (1987)

    MATH  MathSciNet  Google Scholar 

  9. Krylov, N.V.,Nonlinear elliptic and parabolic equations of the second order, Reidel, Dordrecht, 1987

    MATH  Google Scholar 

  10. Miranda, M.Una maggiorazione integrale per le curvature delle ipersuperfici minimale, Rend. Sem. Mat. Univ. Padova38, 91–107 (1967)

    MathSciNet  MATH  Google Scholar 

  11. Serrin, J.,The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A264, 413–496 (1969)

    MATH  MathSciNet  Google Scholar 

  12. Trudinger, N.S.,A priori bounds for graphs with prescribed curvature, In: Festschrift for Jürgen Moser, Academic Press, 1989

  13. Trudinger, N.S.,The Dirichlet problem for the prescribed curvature equations, Arch. Rat. Mech. Anal. (to appear)

  14. Trudinger, N.S.,Isoperimetric inequalities for quermassintegralen, (in preparation)

  15. Trudinger, N.S. and Urbas, J.I.E.,The Dirichlet problem for the equation of prescribed Gauss curvature, Bull. Aust. Math. Soc.28, 217–231 (1983)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Centre for Mathematical Analysis, Australien National University, P.O. Box 4, 2607, Canberra, A.C.T., Australia

    Neil S. Trudinger

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  1. Neil S. Trudinger
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Trudinger, N.S. A priori bounds and necessary conditions for solvability of prescribed curvature equations. Manuscripta Math 67, 99–112 (1990). https://doi.org/10.1007/BF02568424

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