Hessian equations with elementary symmetric functions

关于Hessian商方程的一个凹性不等式

A Concavity Inequality for Hessian Quotient Equations

本文利用对称多项式基变换证明Hessian商算子σ_k/σ_{k-1}的凹性不等式，消除了相关方程凸解C²估计的额外假设，还得到初等对称多项式的若干有用不等式。

A fully nonlinear Sobolev trace inequality

The $k$-Hessian operator $σ_k$ is the $k$-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the $k$-Hessian equation $σ_k(D^2u)=f$ with Dirichlet boundary condition $u=0$ is variational; indeed, this problem can be studied by means of the $k$-Hessian energy $-\int uσ_k(D^2u)$. We construct a natural boundary functional which, when added to the $k$-Hessian energy, yields as its critical points solutions of $k$-Hessian equations with general non-vanishing boundary data. As a consequence, we prove a sharp Sobolev trace inequality for $k$-admissible functions $u$…

The Dirichlet problem for mixed Hessian equations on Hermitian manifolds

In this paper we study the Dirichlet problem for a class of Hessian type equation with its structure as a combination of elementary symmetric functions on Hermitian manifolds. Under some conditions with the initial data on manifolds and admissible subsolutions, we derive a priori estimates for this complex mixed Hessian equation and solvability of the corresponding Dirichlet problem.

The Dirichlet problem for Hessian quotient equations on exterior domains

In this paper, we consider the exterior Dirichlet problem for Hessian quotient equations with the right hand side $g$, where $g$ is a positive function and $g=1+O(|x|^{-β})$ near infinity, for some $β>2$. Under a prescribed generalized symmetric asymptotic behavior at infinity, we establish an existence and uniqueness theorem for viscosity solutions, by using comparison principles and Perron's method. This extends the previous results for Monge--Ampère equations and Hessian equations.

有界域中对称增广$k$-Hessian型方程Dirichlet问题可容许下解的存在性

Existence of Admissible Subsolutions to the Dirichlet Problem for Symmetric Augmented $k$-Hessian Type Equations in Bounded Domains

本文证明有界域中对称增广$k$-Hessian型方程Dirichlet问题可容许下解的存在性，以一致$(k-1)$-$A$-凸性为充要条件，推广了Caffarelli-Nirenberg-Spruck的经典结果。

On the two symmetries in the theory of m-Hessian operators

We show that the modern theory of fully nonlinear operators had been started by the skew symmetry of minors in cooperation with the symmetry of symmetric functions. The paper presents some consequences of this interaction for the m-Hessian operators. One of them is a setting of the isoperimetric variational problem for Hessian integral. The m-admissible minimizer is found, what brings out a new simple proof of the well known Poincare - type inequalities for Hessian integrals. Also a new set of inequalities, generated by a special finite set of functions, is found.

On Axially Symmetric Solutions of Fully Nonlinear Elliptic Equations

We show that in dimension 3 axial-symmetric viscosity solutions of uniformly elliptic Hessian equations are in fact the classical ones.

关于超临界k-Hessian型Trudinger-Moser不等式及极值函数

On a supercritical k-Hessian inequality of Trudinger-Moser type and extremal functions

该研究针对ℝ^{N}中单位球上的k-容许径向对称函数空间，建立了k-Hessian算子的超临界Trudinger-Moser型不等式，并证明了对应极值函数的存在性，拓展了相关不等式理论。

A Pohozaev identity and critical exponents of some complex Hessian equations

In this note, we prove some non-existence results for Dirichlet problems of complex Hessian equations. The non-existence results are proved using the Pohozaev method. We also prove existence results for radially symmetric solutions. The main difference of the complex case with the real case is that we don't know if a priori radially symmetric property holds in the complex case.

关于Hessian方程Dirichlet问题的连续性方法

On a continuity method for Dirichlet problem of Hessian equations

本文发展了一种用于黎曼流形上Hessian方程Dirichlet问题的连续性方法，结合Morse理论与对(f,Γ)对的刻画，在类型2情形下构造了允许函数并求解了Dirichlet问题。

完全非线性椭圆型方程内部估计的新方法

A new approach to interior estimates for fully nonlinear elliptic equations

该研究基于两点最大值原理，通过仅对方程求一次导数的初等方法，证明了Monge-Ampère方程和二次Hessian方程在对应条件下的内部二阶导数估计，拓展了相关方程的估计适用维度。

Radial fractional Laplace operators and Hessian inequalities

In this paper we deduce a formula for the fractional Laplace operator $(-Δ)^{s}$ on radially symmetric functions useful for some applications. We give a criterion of subharmonicity associated with $(-Δ)^{s}$, and apply it to a problem related to the Hessian inequality of Sobolev type: $$\int_{\mathbb{R}^n}|(-Δ)^{\frac{k}{k+1}} u|^{k+1} dx \le C \int_{\mathbb{R}^n} - u \, F_k[u] \, dx, $$ where $F_k$ is the $k$-Hessian operator on $\mathbb{R}^n$, $1\le k < \frac{n}{2}$, under some restrictions on a $k$-convex function $u$. In particular, we show that the class of $u$ for which the above inequal…

On the exterior Dirichlet problem for a class of fully nonlinear elliptic equations

In this paper, we mainly establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for a class of fully nonlinear second-order elliptic equations related to the eigenvalues of the Hessian, with prescribed generalized symmetric asymptotic behavior at infinity. Moreover, we give some new results for the Hessian equations, Hessian quotient equations and the special Lagrangian equations, which have been studied previously.

Admissible solutions to augmented nonsymmetric $k$-Hessian type equations II. A priori estimates and the Dirichlet problem

Using the established $d$-concavity of the $k$-Hessian type functions $F_k(R)=\log(S_k(R)),$ whose variables are nonsymmetric matrices, we prove $ C^{2, α}(\overlineΩ) $ estimates for strictly $(δ, \widetildeγ_k) $-admissible solutions to the Dirichlet problem without the well-known regularity condition. A necessary condition for the existence of strictly $δ$-admissible solutions to the equations is given. By the method of continuity, we provide some sufficient conditions for the unique solvability in the class of strictly $(δ,\widetildeγ_k)$-admissible solutions to the Dirichlet problem, prov…

Asymptotic behavior of solutions of a k-Hessian evolution equation

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…

On polyharmonic regularizations of $k-$Hessian equations: Variational methods

This work is devoted to the study of the boundary value problem \begin{eqnarray}\nonumber (-1)^αΔ^αu = (-1)^k S_k[u] + λf, \qquad x &\in& Ω\subset \mathbb{R}^N, \\ \nonumber u = \partial_n u = \partial_n^2 u = \cdots = \partial_n^{α-1} u = 0, \qquad x &\in& \partial Ω, \end{eqnarray} where the $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum $f$ obeys suitable summability properties. We prove the existence of at least two solutions, of which at least one is isolated, strictly by means of variational methods. We loo…

关于多项式复Hessian方程I：黎曼流形上的非线性非谱椭圆Hessian方程

On Polynomial Complex Hessian Equations I: Non-Linear Non-Spectral Elliptic Hessian Equations on Riemannian Manifolds

本文研究紧致黎曼流形上的一般完全非线性椭圆Hessian方程，通过弱化标准对称性假设，在推导先验估计后利用连续性方法求解。

An elementary proof of the Vigdergauz equations for a class of square symmetric structures

For a periodically perforated structure, for which homogenization takes place in the linear theory of elasticity, the components of the effective elasticity tensor depend in general on the geometry of the holes as well as on the local elastic properties. These dependencies were shown by Vigdergauz in [7] to be separated in an elementary way for one particular class of structures. The original proof of this relation made use of the lattice approach to describe periodic functions using complex variables. In this paper we present a proof of the so-called Vigdergauz equations for a related class o…

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…

Radially bounded solutions of a $k$-Hessian equation involving a weighted nonlinear source

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…

Existence of radial solutions to biharmonic $k-$Hessian equations

This work presents the construction of the existence theory of radial solutions to the elliptic equation \begin{equation}\nonumber Δ^2 u = (-1)^k S_k[u] + λf(x), \qquad x \in B_1(0) \subset \mathbb{R}^N, \end{equation} provided either with Dirichlet boundary conditions \begin{eqnarray}\nonumber u = \partial_n u = 0, \qquad x \in \partial B_1(0), \end{eqnarray} or Navier boundary conditions \begin{equation}\nonumber u = Δu = 0, \qquad x \in \partial B_1(0), \end{equation} where the $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix…

超临界k-Hessian Sobolev型不等式的极值函数

Extremal functions for a supercritical k-Hessian inequality of Sobolev-type

本文研究单位球上径向对称k-容许函数空间中k-Hessian算子的超临界Sobolev型不等式，证明了相应变分问题极值函数的存在性及相关超临界增长k-Hessian方程的可解性。

一个k-Hessian方程解的存在性及其与自相似解的联系

Existence of solutions for a $k$-Hessian equation and its connection with self-similar solutions

本文研究一类k-Hessian方程的径向对称解，在不使用相平面方法的情况下证明了解的存在唯一性，并利用这些结果构造了热方程的一族新的有限时间爆裂II型自相似解。

Polyharmonic $k-$Hessian equations in $\mathbb{R}^N$

This work is focused on the study of the nonlinear elliptic higher order equation \begin{equation}\nonumber \left( -Δ\right)^m u = S_k[-u] + λf, \qquad x \in \mathbb{R}^N, \end{equation} where the $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix of the solution and the datum $f$ belongs to a suitable functional space. This problem is posed in $\mathbb{R}^N$ and we prove the existence of at least one solution by means of topological fixed point methods for suitable values of $m \in \mathbb{N}$. Questions related to the regularity…

实与复退化黑塞方程的刘维尔刚性

Liouville Rigidity for Real and Complex Degenerate Hessian Equations

研究实与复退化黑塞方程的刘维尔刚性，通过刘维尔容许性这一递归几何条件，证明当且仅当容许集满足该条件时，相关方程的有界全局\(C^{0,\alpha}\)整体粘性解为常数，还给出了相关例子及各向异性构造。

Admissible solutions to augmented nonsymmetric $k-$Hessian type equations I. The $d-$concavity of the $k-$Hessian type functions

We establish for $2 \le k \le n-1$ the strict concavity of the function $f_k(λ)=\log(σ_k(λ))$ on a subset of the positive cone $Γ_n=\{λ=(λ_{1}, λ_{2}, \cdots,λ_{n})\in \mathbb{R}^n; λ_j>0,j=1,\cdots, n\}$ where $σ_{k}(λ)$ is the basic symmetric polynomial of degree $k,$ $2 \leq k \leq n.$ Then we apply the result to study the so-called $d-$concavity of the $k-$Hessian type function $F_{k}(R)=\log \left(S_{k}(R)\right),$ where $S_{k}(R)=σ_{k}(λ(R)), λ(R)= \left(λ_{1}, λ_{2}, \cdots, λ_{n}\right) \in \mathbb{C}^{n}$ is eigenvalue-vector of $R \in \mathbb{R}^{n \times n},$ $R=ω+β, ω^{T}=ω, ω>0, \…

埃尔米特流形上复赫斯方程的正则化与赫尔德连续性

Regularization and Hölder continuity for complex Hessian equations on Hermitian manifolds

针对紧埃尔米特流形上的复赫斯方程，提出$\u0393$-容许函数的定量正则化方法，证明右端项$L^p$（$p>\frac{n}{m}$）的复$m$-赫斯方程解的赫尔德连续性，并将结果推广至满足特定结构条件的一般类方程。

空间形式中具有常数Dirichlet边界条件的完全非线性方程的过定问题

Overdetermined problems for fully nonlinear equations with constant Dirichlet boundary conditions in space forms

本文研究空间形式中两类完全非线性方程带常数Dirichlet边界条件的过定问题，证明星形域上Hessian商问题解的径向对称性，并利用Rellich-Pohožaev型恒等式在无星形假设下证明k-Hessian问题的径向对称性。