## Differential and Integral Equations

### Convexity and asymptotic estimates for large solutions of Hessian equations

#### Abstract

We consider the smallest viscosity solution of the Hessian equation $S_k(D^2u)=f(u)$ in $\omega$, that becomes infinite at the boundary of $\omega\subset\mathbb R^n$; here $S_k(D^2u)$ denotes the $k$-th elementary symmetric function of the eigenvalues of $D^2u$, for $k\in\{1,\dots, n\}$. We prove that if $\omega$ is strictly convex and $f$ satisfies suitable assumptions, then the smallest solution is convex. We also establish asymptotic estimates for the behaviour of such a solution near the boundary of $\omega$.

#### Article information

Source
Differential Integral Equations Volume 13, Number 10-12 (2000), 1459-1472.

Dates
First available in Project Euclid: 21 December 2012

https://projecteuclid.org/euclid.die/1356061135

Mathematical Reviews number (MathSciNet)
MR1787077

Zentralblatt MATH identifier
0977.35046

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B40: Asymptotic behavior of solutions

#### Citation

Colesanti, A.; Salani, P.; Francini, E. Convexity and asymptotic estimates for large solutions of Hessian equations. Differential Integral Equations 13 (2000), no. 10-12, 1459--1472.https://projecteuclid.org/euclid.die/1356061135