On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian
type: Item
instance of: scholarly article
title: On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian
zbMATH Open document ID: 1068.32024
author: Song-Ying Li
published in: The Asian Journal of Mathematics
publication date: 2005-03-21T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ajm/1087840910
review text: Let \( D\) be a smoothly bounded domain in \( {\mathbb C}^n \). The author investigates the existence and uniqueness of solutions \( u\) for Dirichlet problems \( f(\lambda(H(u)))=\psi \) in \( \overline{D} \), \( u=\phi \) on \( \partial D \), where \( \lambda(H(u)) \) denotes the eigenvalues of the complex Hessian of \( u \) and \( f \) is a symmetric function satisfying suitable concavity, positivity and growth assumptions. In the real case, such problems were treated by \textit{L. Caffarelli, L. Nirenberg} and \textit{J. Spruck} [Acta Math. 155, 261--301 (1985; Zbl 0654.35031)]. In the complex case, the author obtains, among other results, the following statement in the particular situation of elementary symmetric functions \( \sigma^{(k)} \) for \( k=2,\dots,n \): assume that \( \partial D \) is connected and that the eigenvalues \( \lambda({\mathcal L}) \) of the Levi form \( \mathcal L \) satisfy \( \sigma^{(k-1)}(\lambda({\mathcal L}))>0 \) on \( \partial D \). Then, for smooth data \( \psi \) and \( \phi \) with \( \psi>0 \), the problem \( \sigma^{(k)}(\lambda(H(u)))=\psi \) in \( \overline{D} \), \( u=\phi \) on \( \partial D \) has a unique solution \( u\in C^\infty(\overline{D}) \).
reviewed by: Vincent Thilliez
Mathematics Subject Classification ID: 32W20
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 2148243
zbMATH Keywords: Dirichlet problems
zbMATH Keywords: complex differential operators
MaRDI profile type: Publication
DOI: 10.4310/AJM.2004.V8.N1.A8
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description: scientific article; zbMATH DE number 2148243