# Estimates for $k$-Hessian operator and some applications

Czechoslovak Mathematical Journal (2013)

- Volume: 63, Issue: 2, page 547-564
- ISSN: 0011-4642

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topWan, 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 -

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