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Local solvability of the k-Hessian equations

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Abstract

We give a classification of second-order polynomial solutions for the homogeneous k-Hessian equation σ k [u] = 0. There are only two classes of polynomial solutions: One is convex polynomial; another one must not be (k + 1)-convex, and in the second case, the k-Hessian equations are uniformly elliptic with respect to that solution. Base on this classification, we obtain the existence of C ∞ local solution for nonhomogeneous term f without sign assumptions.

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Authors and Affiliations

  1. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, 430071, China

    GuJi Tian

  2. The School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, 430073, China

    Qi Wang

  3. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Chao-Jiang Xu

  4. Laboratoire de Mathématiques, CNRS, UMR 6085, Université de Rouen, Saint-Etienne du Rouvray, 76801, France

    Chao-Jiang Xu

Authors
  1. GuJi Tian
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  2. Qi Wang
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Correspondence to Qi Wang.

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Tian, G., Wang, Q. & Xu, CJ. Local solvability of the k-Hessian equations. Sci. China Math. 59, 1753–1768 (2016). https://doi.org/10.1007/s11425-016-5135-4

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