Hessian estimates for the sigma-2 equation in dimension three
Micah Warren, Yu Yuan
我们得到了三维特殊拉格朗日方程$σ_{2}=1$的先验内部Hessian估计。
We derive a priori interior Hessian estimates for the special Lagrangian equation $σ_{2}=1$ in dimension three.
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Interior Hessian estimates for Sigma-2 equations in dimension three
本文在三维空间中证明了 σ_2 = f 的先验内部 C2 估计，推广了 Warren-Yuan 的经典结果。
We derive a priori interior Hessian estimates for semiconvex solutions to the sigma-2 equation. An elusive Jacobi inequality, a transformation rule under the Legendre-Lewy transform, and a mean value inequality for the still nonuniformly elliptic equation without area structure are the key to our arguments. Previously, this result was known for almost convex solutions.
Hessian Estimates for the Sigma-2 Equation with Variable Right-Hand Side Terms in Dimension 4
本研究针对4维带正C^{1,1}右端项的sigma-2 Hessian方程得到先验内部Hessian估计与正则性，高维下需解满足动态半凸条件，推广了已有相关结果。
Hessian estimates for the sigma-2 equation in dimension four
本文研究 $σ_2$ 方程的内部 Hessian 估计与正则性，在四维建立先验估计，并将方法用于三维结果的新证明及高维动态半凸光滑解。
Hessian and gradient estimates for three dimensional special Lagrangian Equations with large phase
本文给出了三维空间中具有至少一个临界值的特殊拉格朗日方程的先验内点Hessian估计和梯度估计。
Hessian estimates for special Lagrangian equations with critical and supercritical phases in general dimensions
在一般高维情形下，我们得到了一类特殊的具有临界和超临界相的拉格朗日方程的先验内点Hessian估计。我们的统一的方法导致更尖锐的估计，即使是以前已知的三维和凸解的情况下。
Singular Solutions of Hessian Elliptic Equations in Five Dimensions
证明了对任意$δ\in [0,1)$，R ^[5]中的一致椭圆型Hessian方程存在齐次$2-δ$解析外零解。
In this paper, we propose a coupled system of complex Hessian equations which generalizes the equation for constant scalar curvature Kähler (cscK) metrics. We show this system can be realized variationally as the Euler-Lagrange equation of a Hessian version of the Mabuchi K-energy in an infinite dimensional space of $k$-Hessian potentials, which can be seen as an infinite dimensional Riemannian manifold with negative sectional curvature. Finally, we prove an a priori $C^0$-estimate for this system which depends on the Entropy, which generalizes a fundamental result of Chen and Cheng for cscK m…
Entire Solutions and Asymptotic Behavior to a Class of Parabolic $k$-Hessian Equations with More General Right-Hand Side Terms
研究一类右端项更一般的抛物型 k-Hessian 方程，利用欧拉折线法获得整解的存在性与不存在性，并确定精确渐近幂指数。
Non-classical Solution to Hessian Equation from Cartan Isoparametric Cubic
给出了如何利用5维Cartan等参立方构造Hessian完全非线性二阶一致椭圆型方程的非光滑解。
Singular Solutions of Hessian Fully Nonlinear Elliptic Equations
我们研究了Hessian完全非线性一致椭圆型方程，证明了这些方程（在12维或更多维）的黏性解的二阶导数可以在区域的内点爆破.我们证明了这类解的最优内正则性不超过C^{1+ε}，从而证明了已知内正则性结果的最优性.同样的道理也适用于Isaacs方程。证明了11维完全非线性Hessian一致椭圆型方程非光滑解的存在性。我们还研究了定义在点邻域内的Hessian方程的解的可能奇异性，证明了穿孔球上的Hessian一致椭圆型方程的齐次0<α<1阶解应该是径向的.
Nonclassical Solutions of Fully Nonlinear Elliptic Equations II. Hessian Equations and Octonions
利用八元数代数构造了21维或更高维Hessian完全非线性一致椭圆型方程的奇异解.这些解的正则性是最不可能的解。同样的道理也适用于Isaacs方程。
Interior Hessian estimates for Hessian quotient equations in dimension three
本文研究了三维中2凸解的内部Hessian估计，以及更高维中半凸解的估计，并提出新方法证明光滑解的倍增不等式，同时在更高维中引入动态半凸性条件。
Interior Hessian estimates for the quadratic Hessian equation
该研究针对任意维度正分支上的二次Hessian方程，建立了其粘性解的内部Hessian估计与光滑性，为相关分析提供关键理论支撑。
Auxiliary Hessian Equations
该研究提出k-Hessian方程作为Kähler类退化为数值维数k的nef类的辅助方程，给出Hessian方程的一致均值不等式与先验估计，为相关退化问题提供概念验证结果。
Remarks on Hessian quotient equations on Riemannian manifolds
针对 Delanoë 和 Urbas 提出的相关问题，本文在二维黎曼流形上利用新测试函数与商算子凹性，证明实 Hessian 商方程具有无障碍二阶先验估计，并揭示其与复几何中 $J$-方程障碍的差异。
Interior $C^2$ estimate for Hessian quotient equation in dimension three
本文针对三维 Hessian 商方程，利用 Jacobi 不等式建立了内部 $C^2$ 估计，是该领域的重要进展。
Energy identity for a class of approximate biharmonic maps into sphere in dimension four
本文证明了四维弱收敛的近似双调和方程列在球面上，其双张力场有界于$L^p$中，对某些$p>1$ .我们证明了一个能量恒等式，它用$\mathbb R^4$上的100个非平凡双调和映射的Hessian能量之和来解释Hessian能量的损失.
Exponential Vorticity Hessian Growth in Capillary Liquid Drop in Two Dimensions
针对二维仅受表面张力影响的非平凡体涡度理想流体液滴演化问题，构造初始域为圆盘、初始速度任意小的初始数据，证明涡度海森矩阵随时间无限指数增长。
Interior $C^{2}$ estimate for semi-convex solutions to a class of Hessian quotient equations in arbitrary dimensions
研究Hessian商方程在任意维度下的内部$C^2$估计，探讨半凸条件下的椭圆性，并推导相应求和Hessian方程的结果，同时建立若干刚性结论。
Let $Ω$ be a bounded domain (with smooth boundary) on the hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$. In this paper, by using a priori estimates, we can establish Pogorelov type estimates of $k$-convex solutions to a class of Hessian quotient equations defined over $Ω\subset\mathscr{H}^{n}(1)$ and with the vanishing Dirichlet boundary condition.
Variational integrals on Hessian spaces: partial regularity for critical points
本研究建立Hessian空间中变分积分临界点的正则性理论，证明满足小BMO条件的有界Hessian临界点的光滑性，给出奇异集Hausdorff维数上界，并应用于拉格朗日几何变分问题。
Degenerate complex Hessian equations on compact Kähler manifolds
设$(X,ω)$是维数为$n$的紧致Kähler流形，固定$m\in \mathbb{N}$使得$1\leq m \leq n$ .我们证明了任何$(ω,m)$ -sh函数都可以用光滑的$(ω,m)$ -sh函数来逼近.复Hessian方程的一个位势论也得到了发展，它推广了紧致Kähler流形上的经典多元位势论。然后，我们使用新的变分工具，由于伯曼，Boucksom，Guedj和Zeriahi研究退化复杂的海森方程。
Doubling and the two-dimensional critical valued Lagrangian phase
本文针对二维拉格朗日平均曲率方程，在相变号且梯度沿零集消失时，通过改进的加倍技巧处理退化的Jacobi不等式，建立了内Hessian和梯度估计。
Large dimension behavior of the Hessian eigenvalues of the unit balls
本文研究高维空间中单位球Hessian特征值的渐近行为，证明当n/k有界时k-Hessian特征值有界，并给出其增长阶下界；在k=n时证明Monge-Ampère特征值趋于4。
Boundedness of the Hessian of a biharmonic function in a convex domain
考虑任意凸区域上双调和方程的狄利克雷问题，证明了变分解的二阶导数在所有维上都有界。
Interior estimates for Hessian quotient equations
本文证明了Hessian商方程在次数差为一或二时的内部Hessian估计，并证明该范围最优，方法结合非线性比较函数、加倍极大值原理和Pogorelov估计。
Solutions to degenerate complex Hessian equations
设$(X,ω)$是$n$维紧Kähler流形.研究了$(ω+dd^cφ)^m\wedge ω^{n-m}=F(x,φ)ω^n.$形式的退化复Hessian方程在$F$上的某些自然条件下，该方程存在唯一的连续解.当$(X,ω)$有理齐次时，我们进一步证明了解是Hölder连续的.
Smooth solutions to the complex Hessian equation
设$(X,ω)$是一个维数为$n$的紧致Kähler流形，且固定$1\leq m\leq n.$我们证明了复Hessian方程$(ω+dd^cφ)^m\wedge ω^{n-m}=fω^n$，具有$0<f\in \mathcal{C}^{\infty}(X)$，有光滑的可采纳解$ φ\in \mathcal{C}^{\infty}(X)$ .以前已知当$(X,ω)$具有非负的全纯对分曲率时，这是成立的。
We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope $α'$. Actually we obtain at the same time a solution of the open case $α'>0$, an improved solution of the known case $α'<0$, and solutions for a family of Hessian equations which includes the Fu-Yau equation as a special case. The method is based on the introduction of a more stringent ellipticity condition than the usual $Γ_k$ admissible cone condition, and which can be shown to be preserved by precise estimates with scale.
Interior $C^2$ estimate for Hessian quotient equation in general dimension
针对一般维数下 Hessian 商方程的内部 $C^2$ 正则性问题，论文证明该估计在 $k=n-1,n-2$ 时成立，并在 $k\leq n-3$ 时以奇异解证明其失效。
Some shape functionals for the $k$-Hessian equation
本文研究$k$-Hessian算子的扭转刚度，证明任意维度的Pólya型下界，并通过两个定量估计探究最优集合。
Interior Hessian estimates for Hessian quotient equations
研究针对$l=k-1$和$l=k-2$情形的一般Hessian商方程，借助半凸条件下Hessian商算子的定量凹性不等式，建立了容许半凸解的内部$C^2$估计，为任意维数下对应范围的该类方程提供了统一论证。
Sharp Hessian integrability for fully nonlinear elliptic supersolutions in low dimensions
本文通过谱分辨接触几何机制，解决了平面及三维全非线性椭圆上解的尖锐Hessian可积性猜想，给出了最优指数并识别了剩余差距的谱配置。
We devote this paper to study semi-stable nonconstant radial solutions of $S_k(D^2u) = w(|x|)g(u)$ on the Euclidean space $R^n$. We establish pointwise estimates and necessary conditions for the existence of such solutions (not necessarily bounded) for this equation. For bounded solutions we estimate their asymptotic behavior at infinity. All the estimates are given in terms of the spatial dimension $n$, the values of $k$ and the behavior at infinity of the growth rate function of $w$.
We study a class of elliptic problems, involving a $k$-Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the origin. Furthermore, we study the multiplicity of regular solutions and bifurcation diagrams. An essential ingredient of this study is analyzing the number of intersection points between the singular and regular solutions for rescaled problems. In the particular case of the exponential nonlinearity, we obtain the convergence of regular solutions to the singular …
A Brunn--Minkowski inequality and Convexity for the 2-Hessian eigenvalue in convex domains
本文证明了凸域上2-Hessian特征函数的严格对数凹性与扭转问题解的1/2-凸性，由此建立Brunn--Minkowski不等式，并构造反例说明该性质对3-Hessian方程失效。
We study the long-time behavior of solutions of the $k$-Hessian evolution equation $u_t=S_{k}(D^2 u)$, posed on a bounded domain of the $n$-dimensional space with homogeneous boundary conditions. To this end, we construct a separable solution and we show that the long-time behavior of $u$ is precisely described by this special solution. Further, we initiate the study of that dynamic phenomenon on the entire space, providing a new class of explicit and radially symmetric self-similar solutions that we call $k$-Barenblatt solutions. These solutions present some common properties as those of well…
Trudinger-Moser type inequalities for the Hessian equation with logarithmic weights
该研究建立带对数权重的k-Hessian方程的精确Trudinger-Moser不等式并研究最大化子存在性，将经典结果推广至带对数权重的k-容许函数空间，通过转化为一维加权Sobolev空间结合多种技术求解，补充了相关研究工作。
It is well-known that a celebrated Jörgens-Calabi-Pogorelov theorem for Monge-Ampère equations states that any classical (viscosity) convex solution of $\det(D^2u)=1$ in $\mathbb{R}^n$ must be a quadratic polynomial. Therefore, it is an interesting topic to study the existence and uniqueness theorem of such fully nonlinear partial differential equations' Dirichlet problems on exterior domains with suitable asymptotic conditions at infinity. As a continuation of the works of Caffarelli-Li for Monge-Ampère equation and of Bao-Li-Li for $k$-Hessian equations, this paper is devoted to the solvabil…
Minimum principles and a priori estimates for 2-Hessian problems
本文研究了二阶Hessian方程类，建立了一个P函数的最小原理，利用精确的矩阵不等式推导出关于P函数的微分不等式，并在凸性假设下获得高维最小原理，进而推导出几种经典二阶Hessian边界值问题解的先验界。
We prove a vanishing result for critical points of the supersymmetric nonlinear sigma model on complete non-compact Riemannian manifolds of positive Ricci curvature that admit an Euclidean type Sobolev inequality, assuming that the dimension of the domain is bigger than two and that a certain energy is sufficiently small.
We consider the problem \begin{equation}(1)\;\;\; \begin{cases} S_k(D^2u)= λ|x|^σ (1-u)^q &\mbox{in }\;\; B,\\ u <0 & \mbox{in }\;\; B,\\ u=0 &\mbox{on }\partial B, \end{cases} \end{equation} where $B$ denotes the unit ball in $\mathbb{R}^n$, $n>2k$ ($k\in \mathbb{N}$), $λ>0$, $q > k$ and $σ\geq 0$. We study the existence, uniqueness and multiplicity of negative bounded radially symmetric solutions of (1). The methodology to obtain our results is based on a dynamical system approach. For this, we introduce a new transformation which reduces problem (1) to an autonomous two dimensional generali…
Hessian estimates for Dirichlet and Neumann eigenfunctions of Laplacian
本文借助 Riemannian 流形上的随机分析，建立带边界紧流形中 Laplacian 的 Dirichlet 特征函数 Hessian 上下界及尖锐增长估计，并推广至 Neumann 情形。
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