Hessian estimates for convex solutions to quadratic Hessian equation

当2k>n时Hessian商方程k凸解的内部Hessian估计

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。

凸域中Hessian特征值的Brunn-Minkowski不等式

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。

双调和函数的Hessian在凸区域中的有界性

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。

Second order estimates for convex solutions of degenerate $k$-Hessian equations

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.

二次Hessian方程的内部$C^{2,α}$正则性

Interior $C^{2,α}$ Regularity for the Quadratic Hessian Equation

本文针对正分支上右端项为正$C^α$的二次Hessian方程容许解，建立内部$C^{2,α}$正则性，采用定量大迹传播论证与自适应Dirichlet比较法，明确了该方程解的正则性结论。

Hessian 商方程凸解的 Hessian 估计：基于紧致性的证明

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。

Hessian商方程的$k$-凸解的内部Hessian估计

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。

非凸环上Hessian方程的外Dirichlet问题

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。

Hermitian流形上Hessian方程的χ-半凸解的二阶估计

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。

Brunn--Minkowski不等式与凸域中2-Hessian特征值的凸性

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。

关于二次Hessian方程的注记

Remarks on the quadratic Hessian equation

该研究针对二次Hessian方程，证明其粘性解的严格2-凸性相关结论、任意p>2下的先验内部C²估计，并说明这些结果可排除部分构造正则性反例的策略。

Hessian商方程在任意维度下半凸解的内部$C^2$估计

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。

The curvature estimates for convex solutions of some fully nonlinear Hessian type equations

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.

非光滑k-凸域上或存在子解时的黑塞方程

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。

(p,k)-凸超曲面的存在性：一类Hessian商型曲率方程的研究

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。

二次Hessian方程

Quadratic Hessian equation

本文综述二次Hessian方程，涵盖定义、背景、整体解刚性、粘性解正则性、先验估计及开放问题。

三维中Hessian商方程的内部Hessian估计

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。

$k$-Hessian方程凸整体解的刚性

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。

Pogorelov type $C^2$ estimates for Sum Hessian equations and a rigidity theorem

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.

A Bernstein problem for special Lagrangian equations in exterior domains

We establish quadratic asymptotics for solutions to special Lagrangian equations with supercritical phases in exterior domains. The method is based on an exterior Liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded Hessian, and also certain rotation arguments toward Hessian bound. Our unified approach also leads to quadratic asymptotics for convex solutions to Monge-Ampère equations (previously known), quadratic Hessian equations, and inverse harmonic Hessian equations over exterior domains.

On the solutions to weakly coupled system of $\boldsymbol{k_i}$-Hessian equations

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…

Games for eigenvalues of the Hessian and concave/convex envelopes

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…

凸函数的全Hessian矩阵与其行列式的变分竞争

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。

A Liouville type theorem to $2$-Hessian equations

In this paper, we proved that any 2-convex solution $u$ of $σ_2(D^2u)=1$ with a quadratic growth must be a quadratic polynomial in $\mathbb{R}^n\ (n\geq 3 )$ by using a Pogorelov estimate and the global gradient estimate. And we give a positive answer to the unresolved issue in \cite{CX}.

Curvature estimates for semi-convex solutions of Hessian equations in hyperbolic space

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.

Necessary and Sufficient Conditions to Bernstein Theorem of a Hessian Equation

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 res…

A Pogorelov estimate and a Liouville type theorem to parabolic $k$-Hessian equations

We consider Pogorelov type estimates and Liouville type theorems to parabolic $k$-Hessian equations of the form $-u_t σ_k (D^2u) =1$ in $\mathbb{R}^n\times (-\infty, 0]$. We derive that any \textbf{$k+1$-convex-monotone} solution to $-u_t σ_k (D^2u) =1$ when $u(x,0)$ satisfies a quadratic growth and $0<m_1\le -u_t\le m_2$ must be a linear function of $t$ plus a quadratic polynomial of $x$.

Principal eigenvalues for k-Hessian operators by maximum principle methods

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…

Liouville property and existence of entire solutions of Hessian equations

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.

An interior estimate for convex solutions and a rigidity theorem

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.

Oblique boundary value problems for augmented Hessian equations I

In this paper, we study global regularity for oblique boundary value problems of augmented Hessian equations for a class of general operators. By assuming a natural convexity condition of the domain together with appropriate convexity conditions on the matrix function in the augmented Hessian, we develop a global theory for classical elliptic solutions by establishing global a priori derivative estimates up to second order. Besides the known applications for Monge-Amp`ere type operators in optimal transportation and geometric optics, the general theory here embraces prescribed mean curvature p…

椭圆与抛物型和 Hessian 方程的 Neumann 问题

Neumann Problems for Elliptic and Parabolic Sum Hessian Equations

本文研究与 Hessian 方程的 Neumann 边值问题，在几乎凸且一致 $(k-1)$-凸区域中建立 $(k-1)$-容许解的先验 $C^2$ 估计与存在性，并推广至一致凸区域及抛物型情形。

一类 Hessian 曲率方程的 k - 凸超曲面的存在性

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。

Rigidity of the entire solutions of 2-Hessian equation

In this paper, we prove some rigidity theorems for the entire 2-convex solutions of 2-Hessian equation in Euclidean space. As an application, we obtain a Bernstein type theorem for global special Lagrangian graphs.

松弛凸性假设、斜边界条件下完全非线性椭圆方程的精确 Hessian 估计及应用

Sharp Hessian estimates for fully nonlinear elliptic equations under relaxed convexity assumptions, oblique boundary conditions and applications

本文利用几何切向方法与 Recession 算子，在弱于凸性的结构假设及斜边界条件下建立完全非线性椭圆方程粘性解的全局 Hessian/BMO 估计，并推广至障碍型问题。

Hessian商方程的一个凹性不等式及内部$C^2$估计

A concavity inequality and interior $C^2$ estimate for Hessian quotient equations

该研究建立了特定参数下Hessian商算子的凹性不等式与对应Jacobi不等式，结合已有框架得到Hessian商方程凸解的内部Hessian估计，并证明了具有二次增长的整体凸解必为二次多项式。

Interior estimates of derivatives and a Liouville type theorem for Parabolic $k$-Hessian equations

In this paper, we establish the gradient and Pogorelov estimates for $k$-convex-monotone solutions to parabolic $k$-Hessian equations of the form $-u_tσ_k(λ(D^2u))=ψ(x,t,u)$. We also apply such estimates to obtain a Liouville type result, which states that any $k$-convex-monotone and $C^{4,2}$ solution $u$ to $-u_tσ_k(λ(D^2u))=1$ in $\mathbb{R}^n\times(-\infty,0]$ must be a linear function of $t$ plus a quadratic polynomial of $x$, under some growth assumptions on $u$.

A simple proof of curvature estimate for convex solution of $k$-Hessian equation

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.

k-Hessian方程在外部领域解的非凸性

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。

A Rigidity theorem for parabolic 2-Hessian equations

In this paper, we consider the entire solutions to the parabolic $2$-Hessian equations of the form $-u_tσ_2(D^2 u)=1$ in $\mathbb{R}^n\times (-\infty,0]$. We prove some rigidity theorems for the parabolic $2$-Hessian equations in $\mathbb{R}^n\times (-\infty,0]$ by establishing Pogorelov type estimates for $2$-convex-monotone solutions of the parabolic $2$-Hessian equations.

带刚性定理与抛物型版本的和Hessian方程的Pogorelov估计

The Pogorelov estimates for the sum Hessian equation with rigidity theorem and parabolic versions

该研究在半凸假设下推导和Hessian方程(k-1)-凸解的Pogorelov型C²估计，据此得到整体解的刚性定理，并将相关估计与刚性定理拓展至抛物型版本的和Hessian方程，推进了已有相关研究。

凸k-Hessian解内部奇异集的Sharp Hausdorff界

Sharp Hausdorff Bounds for the Interior Singular Set of Convex $k$-Hessian Solutions

针对凸k-Hessian方程σ_k(D²u)=1的凸粘性解，证明其局部C²正则性缺失集的(n-1)维Hausdorff测度为0，得到3≤k<n时的余维1精细正则性结果，指数最优，还推导了相关正则性结论并刻画了平坦方向数。

2-Hessian方程凸解的内部$C^{2,\alpha}$正则性

Interior $C^{2,α}$ Regularity for Convex Solutions of the 2-Hessian Equation

该研究针对$\mathbb{R}^n$开子集上的2-Hessian方程凸容许粘性解，证明其具有局部$C^{2,\alpha}$正则性，并在特定条件下得到$B_{1/4}$内的一致正则估计。

外区域中Hessian方程的一类Serrin型超定问题

A Serrin-type over-determined problem for Hessian equations in the exterior domain

研究外区域Hessian方程的超定问题，证明存在唯一有界区域使问题有唯一严格凸解。

Hessian商方程的内部导数估计与Bernstein定理

Interior derivative estimates and Bernstein theorem for Hessian quotient equations

本文针对椭圆型和抛物型Hessian商方程建立了内部导数估计，并在此基础上证明了抛物型Hessian商方程的Bernstein定理，刻画了全局抛物凸解必为线性时间项与凸二次多项式之和的形式。

黑森测度II

Hessian measures II

在我们以前关于这个主题的文章中，我们引入了与欧几里德n-空间中的区域\Om中的连续k-凸函数相关联的k-Hessian测度的概念，k=1，.，n，并证明了一个关于局部一致收敛性的弱连续性结果.本文考虑k-凸函数，不一定连续，证明了相应的k-Hessian测度关于依测度依测度收敛的弱连续性。证明依赖于k-凸函数梯度的局部积分估计。

带有Lipschitz右端项的σ₂/σ₁方程的半凸解的内部Hessian估计

Interior Hessian Estimates for Semi-convex Solutions of the $σ_2/σ_1$ Equation with Lipschitz Right-Hand Sides

针对带有Lipschitz右端项的σ₂/σ₁方程的半凸解，证明了依赖于右端项Lipschitz范数的内部Hessian估计，应用于二维凸粘性解的正则性且确定右端项Lipschitz正则性要求最优。

Hessian estimate for semiconvex solutions to the sigma-2 equation

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 quotient equations on exterior domains

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…

$L_p$对偶Christoffel-Minkowski类型问题对于一类Hessian商方程的类

The $L_p$ dual Christoffel-Minkowski type problem for a class of Hessian quotient equations

本文研究了Hessian商算子σ_k(Λ)/σ_l(Λ)的L_p对偶Christoffel-Minkowski类型问题，利用新发现的'逆凸性'性质，在适当结构假设下建立了满秩定理，并结合先验估计证明了严格球面凸解的存在性和唯一性。

2-Hessian方程Dirichlet问题的很弱解

Very weak solutions of the Dirichlet problem for 2-Hessian equation

本文针对2-Hessian方程Dirichlet问题，利用凸积分与截断技术，构造了任意小$\alpha$下无穷多个$C^{1,\alpha}$很弱解。

四维空间中 σ₂ 方程的 Hessian 估计

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。

k-偏导数方程的均匀估计与Brezis-Merle型不等式

Uniform estimates and Brezis-Merle type inequalities for the $k$-Hessian equation

本文研究k-凸函数在边界上消失的Brezis-Merle型不等式，建立中间Hessian方程的Alexandrov-Bakelman-Pucci型估计，并提出k-Hessian方程解的集中-紧致原理。

二次 Hessian 方程的非常弱解

Very weak solutions of quadratic Hessian equations

研究\(n\geq 2\)时\(\mathbb{C}^n\)中区域内复\(\sigma_2\)方程的非常弱解，通过扩展相关方法，提高了实情形下反例的正则性阈值。