研究了黑塞矩阵D^2u的特征值的k阶对称初等函数S_k（D^2u）的Hessian商方程. Bao-Cheng-Guan-Ji（[1]，American J. Math.，2003，125，301-316），分别确保了Hessian商方程或k-Hessian方程的Bernstein性质。本文抛弃了文[1]中的点式二次增长条件，分别利用一个逆等周型不等式、体积增长或Lp-可积证明了（0.1）和（0.2）的Bernstein性质的三个充要条件，我们的体积增长或Lp-可积条件大大改进了文[1，6，7，13，18]等中已知的各种点式条件.
英文摘要：
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.
A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights
本文考虑带权Hessian方程和系统的正径向解的存在性，给出了正径向解大的一个必要条件和一个充分条件.证明定理的方法基本上是基于逐次逼近。我们的结果完善和改进了Zhang和Zhou最近发表的工作（Existence of entire positive k-convex radial solutions to Hessian equations and systems with weights，应用数学快报，第50卷，2015年12月，第48--55页）.
Existence of Admissible Subsolutions to the Dirichlet Problem for Symmetric Augmented $k$-Hessian Type Equations in Bounded Domains
Hanoi Pedagogical University 2（河内师范大學二校）；University of Economics - Technology for Industries（經濟與工業技術大學）；Hanoi University of Science and Technology（河內科技大學）；Quang Binh University（廣平大學）
Quasilinear and Hessian type equations with exponential reaction and measure data
本文证明了在有界区域$Ω$或整个$\mathbb{R}^N$中$-Δ_pu=P(u)+μ$型方程对于$p>1$和$F_k[-u]=P(u)+μ$ with $1\leq k<\frac{N}{2}$的存在性结果，其中$μ$是正Radon测度，而$P(u)\sim e^{au^β}$ with $a>0$和$β\geq 1$ .存在的充分条件表示在$μ$的分数极大潜力。利用$μ$的一些精确Wolff势，得到了解的双边估计.根据Orlicz容量得到了必要条件。我们还在$\mathbb{R}^N$中建立了形式为$u={\bf W}_{α,p}^R[P(u)]+f$的一般Wolff势方程的存在性结果，其中0
A sufficient condition for a finite-time $ L_2 $ singularity of the 3d Euler Equation
通过对压力Hessian特征值的适当假设，得到了三维不可压Euler方程有限时间$L_2$奇异的充分条件.在此条件下，$\lim_{t \to T_*} \sup | \frac{D ø} {Dt} |_{L_2(\vO)} = \infty$，其中$~ \vO \subset \R3$随流体移动。特别地，$|ø|$、$|§_{ij}| , and $| ¶_{ij}| $ all become unbounded at one point $（x_1，T_1）$, $ T_1 $ being the first blow-up time in $ L_2$.
