Power convexity of a class of Hessian equations in the ball

  • Authors

    • Yunhua Ye Jia Ying University
    2015-08-24
    https://doi.org/10.14419/gjma.v3i3.5096
  • Admissible solution, Hessian equations, Power convexity, Strict convexity.

  • Power convexities of a class of Hessian equations are considered in this paper. It is proved that some power functions of the smooth admissible solutions to the Hessian equations are strictly convex in the ball. For a special case of the equation, a lower bound principal curvature and Gaussian curvature estimates are given.

  • References

    1. [1] L.G. Makar-Limanov, Solution of Dirichlet's problem for the equation on a convex region, Math. Notes Acad. Sci. USSR 9 , (1971), 52--53.

      [2] H.J. Brascamp and E.H. Lieb, On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log-concave functions, with an application to the diffusion equation, J. Func.Anal., 22 (1976), 366--389.

      [3] X.N. Ma and X. Lu, The convexity of solution of a class Hessian equation in bounded convex domain in , J. Funct. Anal. 255 (2008), no. 7, 1713--1723.

      [4] P.Liu, X.N.Ma and L.Xu, A Brunn-Minkowski inequatlity for the Hessian eigenvalue

      in three-dimensional convex domain, Adv.Math., 225 (2010), 1616--1633.

      [5] G.Huisken, C.Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183 (1),(1999), 45--70.

      [6] R. C. Reilly, On the Hessian of a function and the curvatures of its graph, Michigan Math.J., 20, (1973), 373--383.

      [7] M. P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1976.

      [8] K.S. Chou and X.J.Wang, A variational theory of the Hessian equations, Comm. Pure Appl.Math., 54 (2001), 1029--1064.

      [9 ]K.S. Tso, Remarks on critical exponents for Hessian operators, Ann. Inst. H.Poincaré Anal. Non Lineairé 7 (1990), No.2, 113--122.

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  • How to Cite

    Ye, Y. (2015). Power convexity of a class of Hessian equations in the ball. Global Journal of Mathematical Analysis, 3(3), 134-139. https://doi.org/10.14419/gjma.v3i3.5096