## Necessary and sufficient conditions to Bernstein theorem of a Hessian equation

HTML articles powered by AMS MathViewer

- by Shi-Zhong Du PDF
- Trans. Amer. Math. Soc.
**375**(2022), 4873-4892 Request permission

## Abstract:

The Hessian quotient equations \begin{equation} S_{k,l}(D^2u)\equiv \frac {S_k(D^2u)}{S_l(D^2u)}=1, \ \ \forall x\in {\mathbb {R}}^n \end{equation} were studied for $k-$th symmetric elementary function $S_k(D^2u)$ of eigenvalues $\lambda (D^2u)$ of the Hessian matrix $D^2u$, where $0\leq l<k\leq n$. For $l=0$, (0.1) is reduced to a $k-$Hessian equation \begin{equation} S_k(D^2u)=1, \ \ \forall x\in {\mathbb {R}}^n. \end{equation} Two quadratic growth conditions were found by Bao-Cheng-Guan-Ji [American J. Math. 125 (2013), pp. 301–316] ensuring the Bernstein properties of (0.1) and (0.2) respectively. In this paper, we will drop the point wise quadratic growth condition of Bao-Cheng-Guan-Ji 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 $L^p$-integrability respectively. Our new volume growth or $L^p-$integrable conditions improve largely various previously known point wise conditions provided Bao et al.; Chen and Xiang [J. Differential Equations 267 (2019), pp. 52027–5219]; Cheng and Yau [Comm. Pure Appl. Math. 39 (1986), pp. 8397–866]; Li, Ren, and Wang [J. Funct. Anal. 270 (2016), pp. 26917–2714]; Yuan [Invent. Math. 150 (2002), pp. 1177–125], etc.## References

- Jiguang Bao, Jingyi Chen, Bo Guan, and Min Ji,
*Liouville property and regularity of a Hessian quotient equation*, Amer. J. Math.**125**(2003), no. 2, 301–316. MR**1963687**, DOI 10.1353/ajm.2003.0007 - Eugenio Calabi,
*Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens*, Michigan Math. J.**5**(1958), 105–126. MR**106487** - L. Caffarelli,
*Monge-Ampère equation, div-curl theorems in Lagrangian coordinates, compression and rotation*, Lecture Notes, 1997. - L. Caffarelli, L. Nirenberg, and J. Spruck,
*The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian*, Acta Math.**155**(1985), no. 3-4, 261–301. MR**806416**, DOI 10.1007/BF02392544 - Kai-Seng Chou and Xu-Jia Wang,
*A variational theory of the Hessian equation*, Comm. Pure Appl. Math.**54**(2001), no. 9, 1029–1064. MR**1835381**, DOI 10.1002/cpa.1016 - Li Chen and Ni Xiang,
*Rigidity theorems for the entire solutions of 2-Hessian equation*, J. Differential Equations**267**(2019), no. 9, 5202–5219. MR**3991557**, DOI 10.1016/j.jde.2019.05.028 - Shiu Yuen Cheng and Shing-Tung Yau,
*Complete affine hypersurfaces. I. The completeness of affine metrics*, Comm. Pure Appl. Math.**39**(1986), no. 6, 839–866. MR**859275**, DOI 10.1002/cpa.3160390606 - Sun-Yung Alice Chang and Yu Yuan,
*A Liouville problem for the sigma-2 equation*, Discrete Contin. Dyn. Syst.**28**(2010), no. 2, 659–664. MR**2644763**, DOI 10.3934/dcds.2010.28.659 - Pengfei Guan and Guohuan Qiu,
*Interior $C^2$ regularity of convex solutions to prescribing scalar curvature equations*, Duke Math. J.**168**(2019), no. 9, 1641–1663. MR**3961212**, DOI 10.1215/00127094-2019-0001 - Konrad Jörgens,
*Über die Lösungen der Differentialgleichung $rt-s^2=1$*, Math. Ann.**127**(1954), 130–134 (German). MR**62326**, DOI 10.1007/BF01361114 - F. John, Extremum problems with inequalities as subsidiary conditions, in: Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, NY, 1948, pp 187–204.
- N. V. Krylov,
*Nonlinear elliptic and parabolic equations of the second order*, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. MR**901759**, DOI 10.1007/978-94-010-9557-0 - Ming Li, Changyu Ren, and Zhizhang Wang,
*An interior estimate for convex solutions and a rigidity theorem*, J. Funct. Anal.**270**(2016), no. 7, 2691–2714. MR**3464054**, DOI 10.1016/j.jfa.2016.01.008 - Matt McGonagle, Chong Song, and Yu Yuan,
*Hessian estimates for convex solutions to quadratic Hessian equation*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**36**(2019), no. 2, 451–454. MR**3913193**, DOI 10.1016/j.anihpc.2018.07.001 - A. V. Pogorelov,
*On the improper convex affine hyperspheres*, Geometriae Dedicata**1**(1972), no. 1, 33–46. MR**319126**, DOI 10.1007/BF00147379 - Ravi Shankar and Yu Yuan,
*Hessian estimate for semiconvex solutions to the sigma-2 equation*, Calc. Var. Partial Differential Equations**59**(2020), no. 1, Paper No. 30, 12. MR**4054864**, DOI 10.1007/s00526-019-1690-1 - Neil S. Trudinger and Xu-Jia Wang,
*Hessian measures. II*, Ann. of Math. (2)**150**(1999), no. 2, 579–604. MR**1726702**, DOI 10.2307/121089 - Micah Warren,
*Nonpolynomial entire solutions to $\sigma _k$ equations*, Comm. Partial Differential Equations**41**(2016), no. 5, 848–853. MR**3508325**, DOI 10.1080/03605302.2015.1123277 - Micah Warren and Yu Yuan,
*Hessian estimates for the sigma-2 equation in dimension 3*, Comm. Pure Appl. Math.**62**(2009), no. 3, 305–321. MR**2487850**, DOI 10.1002/cpa.20251 - Yu Yuan,
*A Bernstein problem for special Lagrangian equations*, Invent. Math.**150**(2002), no. 1, 117–125. MR**1930884**, DOI 10.1007/s00222-002-0232-0 - Yu Yuan,
*Global solutions to special Lagrangian equations*, Proc. Amer. Math. Soc.**134**(2006), no. 5, 1355–1358. MR**2199179**, DOI 10.1090/S0002-9939-05-08081-0

## Additional Information

**Shi-Zhong Du**- Affiliation: Department of Mathematics, Shantou University, Shantou 515063, People’s Republic of China
- MR Author ID: 817723
- Email: szdu@stu.edu.cn
- Received by editor(s): February 16, 2021
- Received by editor(s) in revised form: October 22, 2021, and December 17, 2021
- Published electronically: May 4, 2022
- Additional Notes: The author was partially supported by NSFC (12171299), and GDNSF (2019A1515010605)
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 4873-4892 - MSC (2020): Primary 35J60; Secondary 53C23, 53C42
- DOI: https://doi.org/10.1090/tran/8686
- MathSciNet review: 4439494

Dedicated: This paper is dedicated to the memory of Professor Dong-Gao Deng.