Jianchun Chu
Guan-Ren-Wang established the curvature estimate of convex hypersurface satisfying the Weingarten curvature equation $σ_{k}(κ(X)) = f(X,ν(X))$. In this note, we give a simple proof of this result.
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In this paper, we establish a curvature estimate for semi-convex solutions of Hessian equations in hyperbolic space. We also obtain a curvature estimate for admissible solutions to prescribed curvature measure type problem in hyperbolic space. A crucial ingredient in both estimates is a concavity inequality for Hessian operator.
In this paper, we establish the curvature estimates for a class of Hessian type equations. Some applications are also discussed.
This paper establishes the global curvature estimate for the $n-2$ curvature equation with the general right hand side which partially solves this longstanding problem.
In this paper, we derive a priori interior Hessian estimates for Lagrangian mean curvature equation if the Lagrangian phase is supercritical and has bounded second derivatives.
Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow
该研究在超临界拉格朗日相位假设下，得到拉格朗日平均曲率流四类特殊解的内部Hessian估计，并将结果推广至更广泛的拉格朗日平均曲率型方程，丰富了相关几何分析理论。
The Dirichlet problem for Hessian quotient type curvature equations in Minkowski space
针对非凸区域，在不假设下解和Serrin型条件下建立先验估计，证明闵可夫斯基空间中一类Hessian商型曲率方程Dirichlet问题的存在性。
The existence of $k$-convex hypersurface for a class of Hessian curvature equations
研究一类 Hessian 曲率方程闭星形超曲面存在性，结合先验估计与连续性方法建立 k - 凸超曲面存在唯一性，通过常秩定理证其严格凸。
We derive Hessian estimates for convex solutions to quadratic Hessian equation by a compactness argument.
Non-convexity of level sets for solutions to $k$-Hessian equations in exterior domains
本文研究了k-Hessian方程在外部领域解的非凸性，展示了在无穷远处具有二次增长条件时，解未必quasiconvex，并提供了谐函数在外部领域衰减到零时quasiconvex性的新证明。
Hessian estimates for Lagrangian mean curvature equation with Lipschitz critical and supercritical phases
本文提出新策略，结合Allard型正则性定理与几何测度论等方法，在锐利的Lipschitz临界和超临界相位条件下，获得了Lagrangian平均曲率方程解的内部Hessian估计。
In this paper, we discuss the more general Hessian inequality $σ_{k}^{\frac{1}{k}}(λ(D_i (A\left(|Du|\right) D_j u)))\geq f(u)$ including the Laplacian, p-Laplacian, mean curvature, Hessian, k-mean curvature operators, and provide a necessary and sufficient condition on the global solvability, which can be regarded as generalized Keller-Osserman conditions.
Non-convexity of level sets for $k$-Hessian equations in convex ring
本文构造显式例子，证明凸环上 $k$-Hessian 方程解的下水平集不必为凸集。
In this work, we prove the existence of local convex solution to the degenerate Hessian equation
By using Minkowski addition of convex functions, we prove convexity and rearrangement properties of solutions to some Hessian equations in $\R^3$ and Brunn-Minkowski and isoperimetric inequalities for related functionals.
A functional inequality between Hessians in spaces with non-zero curvature
该研究将正函数平方根与对数Hessian间的函数不等式推广至非零曲率空间，证明了其成立，扩展了原有结果的适用范围。
We prove that convex viscosity solutions to the quadratic Hessian inequality $σ_2(D^2u) \geq 1$ are strictly $2$-convex. As a consequence we obtain short proofs of smoothness and interior $C^2$ estimates for convex viscosity solutions to $σ_2(D^2u) = 1$, which were proven using different methods in recent works of Guan-Qiu \cite{GQ}, McGonagle-Song-Yuan \cite{MSY} and Shankar-Yuan \cite{SY2}.
We consider variational integrals of the form $\int F(D^2u)$ where $F$ is convex and smooth on the Hessian space. We show that a critical point $u\in W^{2,\infty}$ of such a functional under compactly supported variations is smooth if the Hessian of $u$ has a small oscillation.
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…
The curvature estimates of $k$ curvature equations for general right hand side is a longstanding problem. In this paper, we totally solve the $n-1$ case and we also discuss some applications for our estimate.
The main result of this paper gives a plenary proof on the curvature estimates for $k$ curvature equations with general right hand sides with $n<2k$ based on a concavity inequality. We further give a explicit lower bound of the inequality.
Interior Hessian estimates for k-convex solutions of Hessian quotient equations with 2k>n
针对满足2k>n的Hessian商方程，利用凹性不等式和逐点加倍论证，建立了k凸解的内部Hessian估计。
The curvature estimates of quotient curvature equation do not always exist even for convex setting \cite{GRW}. Thus it is natural question to find other type of elliptic equations possessing curvature estimates. In this paper, we discuss the existence of curvature estimate for fully nonlinear elliptic equations defined by symmetric polynomials, mainlly, the linear combination of elementary symmetric polynomials.
Translating solutions and the entire Hessian curvature flow in Minkowski space
该文研究Minkowski空间非紧类空超曲面的$k$-Hessian曲率流，证明了给定渐近行为的平移解存在性，以及满足条件的严格凸初始超曲面的曲率流全局存在且归一化后收敛到平移解。
A Brunn--Minkowski inequality for the Hessian eigenvalue in convex domain
通过形变方法证明有界凸域上一类Hessian方程解的严格对数凹性，进而得到Hessian特征值的Brunn-Minkowski不等式并刻画等号成立条件。
We consider the Dirichlet problem for the biharmonic equation on an arbitrary convex domain and prove that the second derivatives of the variational solution are bounded in all dimensions.
A simple proof of curvature estimates for the n-1 Hessian equation
针对Ren和Wang已证明的n-1型曲率方程曲率估计，本研究提供了该定理的一个更简单的证明方法。
The $C^{1,1}$ estimate of the Dirichlet problem for degenerate $k$-Hessian equations with non-homogenous boundary conditions is an open problem, if the right hand side function $f$ is only assumed to satisfy $f^{1/(k-1)} \in C^{1,1}$. In this paper, we solve this problem for convex solutions defined in the strictly convex bounded domain.
In this paper, complex Hessian equation over Kähler manifold was studied. Under the condition that the underline Kähler manifold has non-negative holomorphic bisectional curvature, the existence and regularity of the solution was proved.
$C^2$ estimates for $k$-Hessian equations and a rigidity theorem
本文通过证明半凸条件下 $k$-Hessian 算子的凹性不等式，建立相关方程解的内部估计与 Liouville 型结果，并简化 $k$-曲率方程全局曲率估计的证明。
Recently, the first named author together with Xinan Ma \cite{ma2015neumann}, have proved the existence of the Neumann problems for Hessian equations. In this paper, we proceed further to study classical Neumann problems for Hessian equations. We prove here the existence of classical Neumann problems under the uniformly convex domain in R^n. As an application, we use the solution of the classical Neumann problem to give a new proof of a family of Alexandrov-Fenchel inequalities arising from convex geometry. This geometric application is motivated from Reilly \cite{Reilly1980}.
Hessian Estimates for Convex Solutions of Hessian Quotient Equations via Compactness
本文通过紧致性方法，为两类 Hessian 商方程的凸解建立了内部 Hessian 估计，并改进了 Lu 和 Fung 的证明。
Interior Hessian estimates for $k$-convex solutions of Hessian quotient equations
针对Hessian商方程的$k$-凸解，采用Li-Wu的定量分离传播法推导加倍不等式，结合Shankar-Yuan的紧性论证得到内部Hessian估计，该方法也适用于一类一般Hessian方程。
Exterior Dirichlet problem for Hessian equations on a non-convex ring
本文证明了非凸环上Hessian方程外Dirichlet问题光滑解的存在性，拓展了Bao-Li-Li在该问题上的已有研究结果。
Convex functions with symplectic Hessian
该研究针对R²ᵐ上满足det(Hess(u))=1且黑塞矩阵属于Sp(2m,ℝ)的实Monge-Ampère方程凸解，证明三阶导数估计，推广Nitsche的经典证明并改进相关常数与估计，还构造了具特殊几何行为的解的例子。
Second order estimates for $χ$-semi convex solutions of Hessian equations on Hermitian manifolds
本文在Hermitian流形上，针对方程两边含梯度项的复Hessian方程，利用半凸性假设建立修正凹性不等式，从而获得可容许解的二阶估计。
Second boundary value problem for the Hessian curvature flow
本文研究了严格凸超曲面在第二边界条件下沿$k$-Hessian曲率流的演化，通过正交不变性技术克服边界$C^2$估计困难，证明了流的长时间存在性及收敛至平移解，推广了已有Gauss曲率流结果。
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。
The paper introduces a new differential-geometric system which originates from the theory of $m$-Hessian operators. The core of this system is a new notion of invariant differentiation on multidimensional surfaces. This novelty gives rise to the following absolute geometric invariants: invariant derivatives of the surface position vector, an invariant connection on a surface via subsurface, curvature matrices of a hypersurface and its normal sections, $p$-curvatures and $m$-convexity of a hypersurface, etc. Our system also produces a new interpretation of the classic geometric invariants and o…
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。
Dirichlet problem for degenerate Hessian quotient type curvature equations
本文研究退化Hessian商型曲率方程的Dirichlet问题，在特定正则性条件下证明 $C^{1,1}$ 图形超曲面的存在唯一性，并给出退化Hessian型方程在最优条件下的二阶导数估计。
We derive a priori interior Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. Our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases.
Sharp Hessian inequalities and monotonicity for Green's functions of positively curved Einstein manifolds
该研究针对正曲率爱因斯坦流形的格林函数，证明了其尖锐黑塞不等式及相关单调性公式，扩展了Park的结果，并给出几何应用。
The existence of $(\mathbf{p}, k)$-convex hypersurfaces for a class of Hessian quotient type curvature equations
本文研究了一类Hessian商型曲率方程中闭合星形超曲面的存在性，结合先验估计与连续方法，证明了非齐次和齐次方程的(p,k)-凸超曲面存在性和唯一性，并利用新发现的『逆凸性』性质，证明了常秩定理，从而获得严格凸解的存在性和唯一性。
We prove the symmetry of solutions to overdetermined problems for a class of fully nonlinear equations, namely Hessian quotient equations and Hessian quotient curvature equations. Our approach is based on establishing a Rellich-Pohozaev type identity for Hessian quotient equations and using a P function. Our result generalizes the overdetermined problems for k-Hessian equations and k-curvature equations.
The Hessian equation on nonsmooth k-convex domains or in the presence of subsolutions
研究非光滑k-凸域上k-黑塞测度狄利克雷问题，证k>n/2时对特定边界数据有唯一解，严格k-凸域仅需连续边界数据，核心方法未提及，主要贡献是给出狄氏问题解的存在唯一性结论。
Liouville type theorems and Hessian estimates for special Lagrangian equations
本文针对特殊Lagrange方程在特定“凸性”条件下证明了Liouville型定理，核心方法是通过Neumann-Poincaré不等式和均值不等式建立解的整体Hessian估计，并进一步推导了该条件及超临界相位下的内部Hessian估计。
Hessian estimates for Lagrangian mean curvature equation with sharp Lipschitz phase
本文研究了具有尖锐Lipschitz相位的Lagrangian平均曲率方程，建立了凸解和超临界相位解的先验内部$C^{1,1}$估计，并给出了非Lipschitz情况下的反例，进而推导出$C^0$粘性解的内部$C^{2,\alpha}$正则性。
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。
Rigidity for Convex Entire Solutions to the $k$-Hessian Equation
本文证明了在无增长假设下，$k$-Hessian方程（$3\le k\le n-1$）的凸整体解必为二次多项式，揭示了其刚性特征。
We mainly study Pogorelov type $C^2$ estimates for solutions to the Dirichlet problem of Sum Hessian equations. We establish respectively Pogorelov type $C^2$ estimates for $k$-convex solutions and admissible solutions under some conditions. Furthermore, we apply such estimates to obtain a rigidity theorem for $k$-convex solutions of Sum Hessian equations in Euclidean space.
In this paper, the existence and multiplicity of nontrivial radial convex solutions to general coupled system of $k_i$-Hessian equations in a unit ball are studied via a fixed-point theorem. In particular, we obtain the uniqueness of nontrivial radial convex solution and nonexistence of nontrivial radial $\boldsymbol{k}$-admissible solution to a power-type system coupled by $k_i$-Hessian equations in a unit ball. Moreover, using a generalized Krein-Rutman theorem, the existence of $\boldsymbol{k}$-admissible solutions to an eigenvalue problem in a general strictly $(k-1)$-convex domain is also…
We study the PDE $λ_j(D^2 u) = 0$, in $Ω$, with $u=g$, on $\partial Ω$. Here $λ_1(D^2 u) \leq ... \leq λ_N (D^2 u)$ are the ordered eigenvalues of the Hessian $D^2 u$. First, we show a geometric interpretation of the viscosity solutions to the problem in terms of convex/concave envelopes over affine spaces of dimension $j$. In one of our main results, we give necessary and sufficient conditions on the domain so that the problem has a continuous solution for every continuous datum $g$. Next, we introduce a two-player zero-sum game whose values approximate solutions to this PDE problem. In addit…
Variational competition between full Hessian and its determinant for convex functions
该研究针对二维单位球扇形区域上满足特定边界条件的凸函数，证明了含全Hessian矩阵与其行列式的变分泛函的上下界，该设定受非线性弹性启发，还得到了Föppl-von-Kármán能量相关变分问题的推论。
Starshaped compact hypersurfaces in warped product manifolds II: a class of Hessian type equations
本文通过导出全局曲率估计，证明了一类星形紧致超曲面的存在性，将既有结果从欧几里得空间推广至 Riemannian 扭曲积，并利用类似论证建立了相关完全非线性椭圆偏微分方程容许解的内部二阶先验估计。
The Neumann Problem for Hessian Equations
本文证明了一致凸区域中Hessian方程Neumann边界问题经典解的存在性，通过建立二阶先验导数估计，肯定回答了Trudinger 1986年的猜想。
For fully nonlinear $k$-Hessian operators on bounded strictly $(k-1)$-convex domains $Ω$ in ${\mathbb R}^N$, a characterization of the principal eigenvalue associated to a $k$-convex and negative principal eigenfunction will be given as the supremum over values of a spectral parameter for which admissible viscosity supersolutions obey a minimum principle. The admissibility condition is phrased in terms of the natural closed convex cone $Σ_k$ in the space of symmetric N by N matrices, which is an elliptic set in the sense of Krylov [Trans. AMS, 1995] and which corresponds to using $k$-convex fu…
In this paper, we establish the existence and uniqueness theorem for entire solutions of Hessian equations with prescribed asymptotic behavior at infinity. This extends the previous results on Monge-Ampère equations. Our approach also makes the prescribed asymptotic order optimal within the range preset in exterior Dirichlet problems. In addition, we show a Liouville type result for $k$-convex solutions. This partly removes the $(k+1)$- or $n$-convexity restriction imposed in existing work.
In our previous paper on this topic, we introduced the notion of k-Hessian measure associated with a continuous k-convex function in a domain \Om in Euclidean n-space, k=1,...,n, and proved a weak continuity result with respect to local uniform convergence. In this paper, we consider k-convex functions, not necessarily continuous, and prove the weak continuity of the associated k-Hessian measure with respect to convergence in measure. The proof depends upon local integral estimates for the gradients of k-convex functions.
We establish an interior $C^2$ estimate for $k+1$ convex solutions to Dirichlet problems of $k$-Hessian equations. We also use such estimate to obtain a rigidity theorem for $k+1$ convex entire solutions of $k$-Hessian equations in Euclidean space.
Hessian operators, overdetermined problems, and higher order mean curvatures: symmetry and stability results
本文将Serrin对称性结果与Alexandrov肥皂泡定理的联系扩展至Hessian算子超定问题与常高阶平均曲率边界，提供新对称性结果、定量稳定性估计及多种替代证明。
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