TY - CHAP

T1 - The k-hessian equation

AU - Wang, Xu Jia

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=70350123876&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-01674-5_5

DO - 10.1007/978-3-642-01674-5_5

M3 - Chapter

AN - SCOPUS:70350123876

SN - 9783642016738

T3 - Lecture Notes in Mathematics

SP - 177

EP - 252

BT - Geometric Analysis and PDEs

PB - Springer Verlag

ER -