A constant rank theorem for Hermitian \(k\)-convex solutions of complex Laplace equations
type: Item
instance of: scholarly article
title: A constant rank theorem for Hermitian \(k\)-convex solutions of complex Laplace equations
zbMATH Open document ID: 1192.58011
author: Fei Han
author: Xinan Ma
author: Damin Wu
published in: Methods and Applications of Analysis
publication date: 2009-12-22T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.maa/1257170937
review text: The main result of the paper establishes a constant rank theorem for the Hermitian \(k\)-convex solutions of the complex Laplace equation. The proof is based on the strong maximum principle. In some geometry and analysis problems, it is always important to get the existence of convex solutions for partial differential equations. Up to now, we have in general two methods to produce convex solutions for elliptic partial differential equations. They are the macroscopic and microscopic convexity principle. The macroscopic convexity principle was developed by Korevaar, Kennington, Kawohl, and by Alvarez-Lasry-Lions for generally nonlinear partial differential equations. The microscopic convexity principle concentrates on the constant rank theorem for convex solutions of partial differential equations. In differential geometry, there is much interest on the \(k\)-convexity of the solutions of the geometric partial differential equations. Here, the \(k\)-convexity means that the sum of any \(k\) eigenvalues of the Hessian matrix of the solution is nonnegative. Similarly, we can formulate the notions of \(k\)-convexity for the curvature operator and second fundamental forms of hypersurfaces. For the potential application and independently interesting in complex analysis and complex geometry, in this paper, we establish the complex counterpart of the constant rank theorem on the Hermitian \(k\)-convex solutions for the complex Laplace equation. We consider the following equation \[ \Delta u=\sum^n_{\alpha=1}\frac{\partial^2u}{\partial z^\alpha\partial\overline z^\alpha}=f(z)\tag{1.1} \] in a domain \(\Omega\) of \(\mathbb C^n\). Here, we assume that \[ f\in C^{2,\lambda}(\Omega),\quad f>0,\text{ for some }0<\lambda <1.\tag{1.2} \] We are interested in the solution \(u\) of Hermitian \(k\)-convex type. Namely, the sum of any \(k\) eigenvalues of the Hermitian matrix \((u_{\alpha\overline\beta})_{1\leq \alpha,\beta\leq n}\) is nonnegative. We would like to know, under what con…
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 5651166
zbMATH Keywords: Hermitian \(k\)-convex
zbMATH Keywords: constant rank
zbMATH Keywords: complex Laplace equation
zbMATH Keywords: maximum principle
reviewed by: Dumitru Motreanu
MaRDI profile type: Publication
OpenAlex ID: W2042488690
DOI: 10.4310/MAA.2009.V16.N2.A5
description: scientific article; zbMATH DE number 5651166