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浙江大学学报（理学版）
浙江大学学报（理学版）  2019, Vol. 46 Issue (6): 680-685    DOI: 10.3785/j.issn.1008-9497.2019.06.009
数学与计算机科学     
σ2 Hessian方程的Pogorelov型C2 内估计及应用
缪正武
浙江工业大学 理学院，浙江 杭州 310023
The Pogorelov interior C2 estimate of σ2 Hessian equations and its application
MIAO Zhengwu
College of Science, Zhejiang University of Technology, Hangzhou 310023, China
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摘要： 提出利用拉格朗日乘子法重新证明σ2算子的最优凹性,并定义了一个凸锥Γ3?=λ=(λ1,λ2,?,λn)∈Rn:σ1(λ)>0,σ2(λ|i)>0,1≤i≤n。利用σ2算子的最优凹性，给出了σ2Hessian方程Pogorelov型C2内估计，进而证明了σ2(D2u(x))=1,x∈Rn的满足二次多项式增长条件的Γ3?-凸整解为二次多项式。
关键词： σ2Hessian程;  最优凹性;  C2内估计    
Abstract: The Hessian equation is an important class of completely nonlinear partial differential equations. In this paper, the author re-proves the concaveness by using the Lagrange multiplier method and defines a convex cone Γ3?=λ=(λ1,λ2,?,λn)∈Rn:σ1(λ)>0,σ2(λ|i)>0,1≤i≤n. And further uses the optimal concave ofσ2 operator to give the Pogorelov interiorC2 estimate ofσ2 Hessian equations. Then, to prove that the Γ3?- convex entire solution of σ2(D2u(x))=1,x∈Rnis a quadratic polynomial if usatisfies a quadratic growth condition.
Key words: σ2 Hessian equations    optimal concavity    interior C2 estimate
收稿日期: 2018-09-12 出版日期: 2019-11-25
CLC:  O175.29  
基金资助: 浙江省大学生科技创新活动计划——新苗人才计划(2017R403049).
作者简介: 缪正武(1998—)，ORCID:http://orcid.org/0000-0003-3569-1975,男，主要从事椭圆偏微分方程研究,E-mail:mzhengwu@163.com.
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引用本文:

缪正武. σ2 Hessian方程的Pogorelov型C2 内估计及应用[J]. 浙江大学学报（理学版）, 2019, 46(6): 680-685.

MIAO Zhengwu. The Pogorelov interior C2 estimate of σ2 Hessian equations and its application. Journal of ZheJIang University(Science Edition), 2019, 46(6): 680-685.

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https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2019.06.009    或    https://www.zjujournals.com/sci/CN/Y2019/V46/I6/680

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