Local solvability of the k-Hessian equations

GuJi Tian; Qi Wang; Chao-Jiang Xu

doi:10.1007/s11425-016-5135-4

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Local solvability of the k-Hessian equations

Authors

Tian, GuJi¹
Wang, Qi²
Xu, Chao-Jiang^{3, 4}

¹ Chinese Academy of Sciences, Wuhan Institute of Physics and Mathematics, Wuhan, 430071, China , Wuhan (China)
² Zhongnan University of Economics and Law, The School of Statistics and Mathematics, Wuhan, 430073, China , Wuhan (China)
³ Wuhan University, School of Mathematics and Statistics, Wuhan, 430072, China , Wuhan (China)
⁴ Université de Rouen, Laboratoire de Mathématiques, CNRS, UMR 6085, Saint-Etienne du Rouvray, 76801, France , Saint-Etienne du Rouvray (France)

Type

Published Article

Journal

Science China Mathematics

Publisher

Science China Press

Publication Date

Feb 26, 2016

Volume

59

Issue

9

Pages

1753–1768

Identifiers

DOI: 10.1007/s11425-016-5135-4

Source

Springer Nature

Keywords

35J60
35J70

License

Yellow

Abstract

We give a classification of second-order polynomial solutions for the homogeneous k-Hessian equation σk[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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