Interior gradient estimates for mean curvature equations
type: Item
instance of: scholarly article
title: Interior gradient estimates for mean curvature equations
zbMATH Open document ID: 0909.35022
DOI: 10.1007/PL00004604
author: Xu-Jia Wang
published in: Mathematische Zeitschrift
publication date: 1999-03-17T00:00:00Z
review text: The author considers the mean curvature equation \[ (\delta_{ij}-u_iu_j/(1+| \nabla u| ^2))u_{ij}=H(x)(1+| \nabla u| ^2)^{1/2} \] in a ball \(B_r(0)\subset \mathbb{R}^n\). It is proved that if \(u\) is a nonnegative solution of the above equation, of class \(C^3\), if \(| H(x)| \leq C_0\), \(| \nabla H(x)| \leq C_0\), then \(| \nabla u(0)| \leq \exp(C_1+C_2M^2/r^2)\), where \(M=\text{ sup}_{\Omega}u(x)\), \(C_1\) depends on \(n\), \(M\), and \(C_0\); and \(C_2\) depends on \(n\) and \(C_0\). The proof is based on a suitable choice of an auxiliary function. The result is then extended to higher order mean curvature equations. The same method is used to prove Liouville type result.
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 1185893
zbMATH Keywords: gradient estimates
zbMATH Keywords: Liouville type result
reviewed by: J-P Raymond
MaRDI profile type: Publication
full work available at URL: pl00004604
OpenAlex ID: W2066112961
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description: scientific article; zbMATH DE number 1185893