Rigidity for general semiconvex entire solutions to the sigma-2 equation
type: Item
instance of: scholarly article
title: Rigidity for general semiconvex entire solutions to the sigma-2 equation
DOI: 10.1215/00127094-2022-0034
author: Ravi Shankar
author: Yu Yuan
published in: Duke Mathematical Journal
publication date: 2022-10-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/2108.00093
review text: The equation \[ \sigma_2(D^2 u)\equiv \sum\limits_{1\le i<j\le n} \lambda_i\lambda_j= 1, \] where \(\lambda_1,\lambda_2,\dots, \lambda_n\) are the eigenvalues of the Hessian matrix \[ D^2u= \left(\frac{\partial^2 u}{\partial x_i\partial x_j}\right)_{m\times n} \] has many applications. In two variables it reduces to the Monge-Ampère equation \[ u_{xx} u_{yy}- u^2_{xy}= 1. \] The contribution of this work is a proof of the theorem below, which was conjectured in 2009 by one of the authors. Theorem. Let \(u= u(x_1,x_2,\dots,x_n)\) be a smooth solution of the equation \(\sigma_2(D^2u)= 1\) in the whole \(\mathbb{R}^n\). If \(D^2u\ge -KI\) for a constant \(K\), then \(u\) is a quadratic polynomial. For \(n=2\) this is Jörgen's classical theorem for the Monge-Ampère equation. The case \(n=3\) has been proved earlier by Y. Yuan. The present proof is technically advanced, based on PDE methods and identities from differential geometry. There are non-polynomial entired solutions that do not satisfy the quasi-convexity assumptions \(D^2u\ge -KI\).
reviewed by: Peter Lindqvist
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B08
zbMATH DE Number: 7608261
zbMATH Keywords: Hessian equation
zbMATH Keywords: Trace Jacobi inequality
zbMATH Keywords: entire solutions
MaRDI profile type: Publication
OpenAlex ID: W4306815854
cites work: Liouville property and regularity of a Hessian quotient equation
cites work: Interior \(C^{2,\alpha}\) regularity theory for a class of nonconvex fully nonlinear elliptic equations.
cites work: A priori estimates for solutions of fully nonlinear equations with convex level set
cites work: A Liouville problem for the sigma-2 equation
cites work: Rigidity theorems for the entire solutions of 2-Hessian equation
cites work: \(C^{2,\alpha}\) estimates for nonlinear elliptic equations of twisted type
cites work: Convergence of the \(J\)-flow on toric manifolds
cites work: Elliptic partial differential equations of second order
cites work: Interior \(C^2\) regularity of convex solutions to prescribing scalar curvature equations
cites work: A Priori Limitations for Solutions of Monge-Ampere Equations II
cites work: Q4354424
cites work: Nonclassical solutions of fully nonlinear elliptic equations
cites work: Elementary proof of Bernstein's theorem on minimal surfaces
cites work: Hessian estimate for semiconvex solutions to the sigma-2 equation
cites work: Nonpolynomial entire solutions to σkequations
cites work: Hessian estimates for the sigma‐2 equationin dimension 3
cites work: A Bernstein problem for special Lagrangian equations
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description: scientific article; zbMATH DE number 7608261