Logarithmic convexity of the eigenfunction of the Hessian operator in the ball

  • Authors

    • Yunhua Ye Jia Ying University
    2015-07-25
    https://doi.org/10.14419/ijams.v3i2.4979
  • Eigenfunction, Hessian Operator, Logarithmic Convexity, Strict Convexity.

  • Abstract

    This paper is concerned with the logarithmic convexity of the eigenfunction of the Hessian operator. It is proved that the logarithm of the eigenfunction of the Hessian operator is strictly convex in the ball.

  • References

    1. [1] L.G. Makar-Limanov, Solution of Dirichlet's problem for the equation on a convex region, Math. Notes Acad. Sci. USSR, No.9, (1971), pp.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., No.22, (1976), pp.366--389.

      [3] P.Liu, X.N.Ma and L.Xu, A Brunn-Minkowski inequatlity for the Hessian eigenvalue in three-dimensional convex domain, Adv.Math., No.225, (2010), pp.1616--1633.

      [4] X.J.Wang, A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J., No.43, (1994), pp.25--54.

      [5] G.Huisken, C.Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math.,183 (1999), No. 1, pp.45--70.

      [6] R. C. Reilly, On the Hessian of a function and the curvatures of its graph, Michigan Math.J., No.20,(1973), pp.373--383.

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  • How to Cite

    Ye, Y. (2015). Logarithmic convexity of the eigenfunction of the Hessian operator in the ball. International Journal of Advanced Mathematical Sciences, 3(2), 98-102. https://doi.org/10.14419/ijams.v3i2.4979