Estimates for $k$-Hessian operator and some applications

Wan, Dongrui. "Estimates for $k$-Hessian operator and some applications." Czechoslovak Mathematical Journal 63.2 (2013): 547-564. <http://eudml.org/doc/260655>.

@article{Wan2013,
abstract = {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 $k$, $1\le k\le n$, of the eigenvalues of the Hessian matrix $D^\{2\}u$. For example, $F_\{1\}[u]$ is the Laplacian $\Delta u$ and $F_\{n\}[u]$ is the real Monge-Ampère operator det $D^\{2\}u$, 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.},
author = {Wan, Dongrui},
journal = {Czechoslovak Mathematical Journal},
keywords = {$k$-convex function; $k$-Hessian operator; $k$-Hessian measure; $k$-Green function; -convex function; -Hessian operator; -Hessian measure; -Green function},
language = {eng},
number = {2},
pages = {547-564},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Estimates for $k$-Hessian operator and some applications},
url = {http://eudml.org/doc/260655},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Wan, Dongrui
TI - Estimates for $k$-Hessian operator and some applications
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 2
SP - 547
EP - 564
AB - 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 $k$, $1\le k\le n$, of the eigenvalues of the Hessian matrix $D^{2}u$. For example, $F_{1}[u]$ is the Laplacian $\Delta u$ and $F_{n}[u]$ is the real Monge-Ampère operator det $D^{2}u$, 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.
LA - eng
KW - $k$-convex function; $k$-Hessian operator; $k$-Hessian measure; $k$-Green function; -convex function; -Hessian operator; -Hessian measure; -Green function
UR - http://eudml.org/doc/260655
ER -