Necessary and Sufficient Conditions to Bernstein Theorem of a Hessian Equation
Analysis of PDEs
2021-12-30 v2
Abstract
The Hessian quotient equatio were studied for k-th symmetric elementary function S_k(D^2u) of eigenvalues of the Hessian matrix D^2u. Two pointwise quadratic growth conditions were found by Bao-Cheng-Guan-Ji ([1], American J. Math., 2003, 125, 301-316) ensuring Bernstein properties of Hessian quotient equation or k-Hessian equation respectively. In this paper, we will drop the point wise quadratic growth condition of [1] and prove three necessary and sufficient conditions to Bernstein property of (0.1) and (0.2), using a reverse isoperimetric type inequality, volume growth or Lp-integrable respectively.Our volume growth or Lp-integrable conditions improve largely various known point wise conditions in [1,6,7,13,18] etc.
Keywords
Cite
@article{arxiv.2106.06211,
title = {Necessary and Sufficient Conditions to Bernstein Theorem of a Hessian Equation},
author = {Shi-Zhong Du},
journal= {arXiv preprint arXiv:2106.06211},
year = {2021}
}