Czechoslovak Mathematical Journal

, Volume 63, Issue 2, pp 547–564 | Cite as

Estimates for k-Hessian operator and some applications



The k-convex functions are the viscosity subsolutions to the fully nonlinear elliptic equations F k [u] = 0, where F k [u] is the elementary symmetric function of order k, 1 ⩽ ⩽ 6 n, of the eigenvalues of the Hessian matrix D 2 u. For example, F 1[u] is the Laplacian Δu and F n [u] is the real Monge-Ampère operator detD 2 u, while 1-convex functions and n-convex functions are subharmonic and convex in the classical sense, respectively. In this paper, we establish an approximation theorem for negative k-convex functions, and give several estimates for the mixed k-Hessian operator. Applications of these estimates to the k-Green functions are also established.


k-convex function k-Hessian operator k-Hessian measure k-Green function 

MSC 2010

47J20 58C35 31A15 31A05 


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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2013

Authors and Affiliations

  1. 1.College of Mathematics and Computational ScienceShenzhen UniversityShenzhenChina

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