Modulus of continuity of solutions to complex Hessian equations
type: Item
instance of: scholarly article
title: Modulus of continuity of solutions to complex Hessian equations
zbMATH Open document ID: 1337.32046
DOI: 10.1142/S0129167X16500038
author: Mohamad Charabati
published in: International Journal of Mathematics
publication date: 2016-03-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1401.8254
Mathematics Subject Classification ID: 32U40
Mathematics Subject Classification ID: 35J99
Mathematics Subject Classification ID: 32U05
zbMATH DE Number: 6549422
zbMATH Keywords: complex Hessian equations
zbMATH Keywords: Hölder continuity
zbMATH Keywords: modulus of continuity
MaRDI profile type: Publication
arXiv ID: 1401.8254
cites work: Q5849361
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cites work: The Dirichlet problem for the multidimensional Monge-Ampère equation
cites work: Q5727762
cites work: Q5542428
OpenAlex ID: W2963183807
review text: It is known that the Dirichlet problem for the complex Hessian equation in a strongly \(m\)-pseudoconvex domain in \(\mathbb C ^n\), with \(C^{1,1}\) boundary data and the density in \(L^p\), \(p>n/m\), has a continuous solution. The bound for \(p\) is optimal. In a recent paper, \textit{N. C. Nguyen} [Potential Anal. 41, No. 3, 887--902 (2014; Zbl 1302.32034)] proved Hölder continuity of those solutions with an explicit exponent, but assuming a controlled growth of the right hand side near the boundary. In the present paper the author is able to remove this extra assumption. For \(p>2m/n\) he obtains also a better Hölder exponent. On the way, a sharp result on the modulus of continuity of the solution, when the right hand side is continuous, is shown. The proof uses a rather delicate barrier construction.
reviewed by: Slawomir Kołodziej
description: scientific article; zbMATH DE number 6549422