Boundary asymptotical behavior of large solutions to complex Hessian equations
type: Item
instance of: scholarly article
title: Boundary asymptotical behavior of large solutions to complex Hessian equations
zbMATH Open document ID: 1215.32018
author: Ni Xiang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2010-11-19T00:00:00Z
review text: The author considers the following fully nonlinear singular Dirichlet problem for symmetric functions of the complex Hessian over bounded smooth domains \(\Omega \subset \mathbb C^n\) \[ \sigma_k\left(\frac {\partial^2 u}{\partial z_i \partial \overline z_j}\right)=g(z)f(u) \;\text{in} \;\Omega \subset \mathbb C^n,\quad u=+\infty \;\text{on} \;\partial \Omega, \] for \(k\leq n\), where \(\sigma_k\) is the \(k\)-th symmetric polynomial of the eigenvalue of the complex Hessian, and \(u\) is strictly \(\Gamma_k\)-subharmonic. (A \(C^2\) function \(u\) is strictly \(\Gamma_k\)-subharmonic if \(\sigma_j(\frac {\partial^2 u}{\partial z_i \partial \overline z_j})>0\) for \(j=1,2,\dots ,k\).) Under the assumption that \(f\) satisfies \(\lim_{t\to \infty} \frac {f(\lambda t)}{f(t)} =\lambda^q\) for \(q>n\), and \(g(z)\geq 0 \) and \(\frac {g(z)}{m^{k+1}(dz)}\) are bounded from above and below by some positive constant as \(z\to \partial \Omega\) (where \(m\) is some nondecreasing function on \((0,\mu)\) such that \(\lim_{t\to 0}\frac d{dt}(\int_0^tm(s)ds/m(t))=l\)), the author establishes the asymptotical behavior of any \(C^2\) solution to the above problem. The statement of the main result and the techniques of the proof are quite similar to the work on singular Dirichlet problems for real Monge-Ampère equation by \textit{F. C. Cîrstea} and \textit{C. Trombetti} [Calc. Var. Partial Differ. Equ. 31, No. 2, 167--186 (2008; Zbl 1148.35022)].
Mathematics Subject Classification ID: 32W50
Mathematics Subject Classification ID: 32A05
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 5817665
zbMATH Keywords: complex Hessian equation
zbMATH Keywords: boundary blow-up
zbMATH Keywords: asymptotical behavior
zbMATH Keywords: \(\Gamma\)-subharmonic
reviewed by: Lee Taichung
MaRDI profile type: Publication
full work available at URL: j.na.2010.08.027
OpenAlex ID: W1988747062
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DOI: 10.1016/J.NA.2010.08.027
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description: scientific article; zbMATH DE number 5817665