### Abstract

The k-Hessian is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k 2, the k-Hessian equation is a fully nonlinear partial differential equations. It is elliptic when restricted to k-admissible functions. In this paper we establish the existence and regularity of k-admissible solutions to the Dirichlet problem of the k-Hessian equation. By a gradient flow method we prove a Sobolev type inequality for k-admissible functions vanishing on the boundary, and study the corresponding variational problems. We also extend the definition of k-admissibility to non-smooth functions and prove a weak continuity of the k-Hessian operator. The weak continuity enables us to deduce a Wolff potential estimate. As an application we prove the Hölder continuity of weak solutions to the k-Hessian equation. These results are mainly from the papers [CNS2, W2, CW1, TW2, Ld] in the references of the paper.

Original language | English (US) |
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Title of host publication | Geometric Analysis and PDEs |

Publisher | Springer Verlag |

Pages | 253-273 |

Number of pages | 21 |

ISBN (Print) | 9783642016738 |

DOIs | |

State | Published - Jan 1 2009 |

### Publication series

Name | Lecture Notes in Mathematics |
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Volume | 1977 |

ISSN (Print) | 0075-8434 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

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## Cite this

*Geometric Analysis and PDEs*(pp. 253-273). (Lecture Notes in Mathematics; Vol. 1977). Springer Verlag. https://doi.org/10.1007/978-3-642-01674-5_6