A general Liouville type theorem for some conformally invariant fully nonlinear equations

Super-Liouville Equations on Closed Riemann Surfaces

Motivated by the supersymmetric extension of Liouville theory in the recent physics literature, we couple the standard Liouville functional with a spinor field term. The resulting functional is conformally invariant. We study geometric and analytic aspects of the resulting Euler-Lagrange equations, culminating in a blow up analysis.

共形不变方程中的一个Liouville型定理

A Liouville-type theorem in conformally invariant equations

本文证明了在满足多项式锥条件及 $x\cdot\nabla K\leq 0$ 的光滑函数 $K$ 下，方程 $-\Delta u=K(x)e^{2u}$ 在 $\mathbb{R}^2$ 上无满足有界性和可积性条件的解，推广了Struwe的结果，并讨论了高阶情形。

Conformally invariant fully nonlinear elliptic equations and isolated singularities

We study solutions to conformally invariant equations with isolated singularties.

Local gradient estimates of solutions to some conformally invariant fully nonlinear equations

This paper concerns local gradient estimates to solutions of general conformally invariant fully nonlinear elliptic equations of second order.

A fully nonlinear version of the Yamabe problem on locally conformally flat manifolds with umbilic boundary

In this paper we establish existence and compactness of solutions to a general fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.

关于具有常边界平均曲率的完全非线性Yamabe问题. I

On the fully nonlinear Yamabe problem with constant boundary mean curvature. I

本文研究了完全非线性Yamabe问题，证明了共形不变二阶椭圆方程在半欧氏空间中的最优Liouville型定理。

Harnack inequalities and Bôcher-type theorems for conformally invariant fully nonlinear degenerate elliptic equations

We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.

On some conformally invariant fully nonlinear equations

We will report some results concerning the Yamabe problem and the Nirenberg problem. Related topics will also be discussed. Such studies have led to new results on some conformally invariant fully nonlinear equations arising from geometry. We will also present these results which include some Liouville type theorems, Harnack type inequalities, existence and compactness of solutions to some nonlinear version of the Yamabe problem.

Towards a Liouville theorem for continuous viscosity solutions to fully nonlinear elliptic equations in conformal geometry

We study entire continuous viscosity solutions to fully nonlinear elliptic equations involving the conformal Hessian. We prove the strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations when one of the competitors is $C^{1,1}$. We obtain as a consequence a Liouville theorem for entire solutions which are approximable by $C^{1,1}$ solutions on larger and larger compact domains, and, in particular, for entire $C^{1,1}_{\rm loc}$ solutions: they are either constants or standard bubbles.

关于半空间中$k$-曲率方程的李雅普诺夫定理

Liouville Theorem for $k-$curvature equation in half space with fully nonlinear boundary condition

本文研究了半空间中$k$-曲率方程的李雅普诺夫定理，提出了基于边界曲率的正常数曲率方程，并通过变分函数发现边界曲率的来源。

Min-oo conjecture for fully nonlinear conformally invariant equations

In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that our results do not assume concavity assumption on the fully nonlinear equations we will work with. This proves rigidity for compact connected locally conformally flat manifolds $(M,g)$ with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere $\partial D(r)$…

Concentration-compactness phenomena in the higher order Liouville's equation

We investigate different concentration-compactness phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in $R^{2m}$, then that of a closed manifold and, finally, the particular case of the sphere $S^{2m}$. In all cases we allow the sign of the Q-curvature to vary, and show that in the case of a closed manifold, contrary to the case of open domains in $R^{2m}$, concentration phenomena can occur only at points of positive Q-curvature. As a consequence, on a locally conformally flat manifold of non-positive Euler characteristic we always have…

Boundary pointwise regularity and Liouville theorems for fully nonlinear equations on cones

In this paper, we prove a boundary pointwise regularity for fully nonlinear elliptic equations on cones. In addition, based on this regularity, we give simple proofs of the Liouville theorems on cones.

半空间上全非线性椭圆方程的Liouville定理

Liouville theorems for fully nonlinear elliptic equations on half spaces

本文通过边界正则性、Hopf和Carleson估计，简洁证明了半空间上全非线性椭圆方程的两个Liouville定理。

Liouville-type theorems for fully nonlinear elliptic equation in half spaces

In \cite{LWZ}, we establish Liouville-type theorems and decay estimates for solutions of a class of high order elliptic equations and systems without the boundedness assumptions on the solutions. In this paper, we continue our work in \cite{LWZ} to investigate the role of boundedness assumption in proving Liouville-type theorems for fully nonlinear equations. We remove the boundedness assumption of solutions which was required in the proof of Liouville-type theorems for fully nonlinear elliptic equations or systems in half spaces. We also prove the Liouville-type theorems for supersolutions of…

Heisenberg群上完全非线性偏微分方程的若干新Liouville型结果

Some new Liouville-type results for fully nonlinear PDEs on the Heisenberg group

该研究针对Heisenberg群上由Heisenberg向量场构造的完全非线性方程，通过退化Hadamard三球面定理证明新的尖锐Liouville型性质，并将欧几里得框架下Pucci极值算子受扰情况的结果推广到Heisenberg群。

A Liouville theorem for fully nonlinear problems with infinite boundary conditions and applications

We prove a Liouville type classification theorem in half-spaces for infinite boundary value problems related to fully nonlinear, uniformly elliptic operators. We then apply the result in order to obtain gradient boundary blow up rates for ergodic functions in bounded domains related to degenerate/singular operators, and, as a further consequence, we deduce the uniqueness of the ergodic functions.

Liouville-type results in exterior domains for radial solutions of fully nonlinear equations

We give necessary and sufficient conditions for the existence of positive radial solutions for a class of fully nonlinear uniformly elliptic equations posed in the complement of a ball in $\mathbb R^N$, and equipped with homogeneous Dirichlet boundary conditions.

Liouville properties and critical value of fully nonlinear elliptic operators

We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully nonlinear parabolic equations. The second is the solvability of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique critical value of the operator.

A fully nonlinear version of the Yamabe problem and a Harnack type inequality

We present some results on a fully nonlinear version of the Yamabe problem and a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations.

On a fully nonlinear Yamabe problem

We solve the $σ_2$-Yamabe problem for a non locally conformally flat manifold of dimension $n>8$.

半空间外部区域中完全非线性椭圆方程的Liouville型定理

Liouville type theorems for fully nonlinear elliptic equations in exterior domains in half spaces

针对半空间外部区域中的完全非线性一致椭圆方程，在二次边界数据和二次增长条件下，证明了任何粘性解在无穷远处趋于二次多项式加上低阶项，收敛速率至少为|x|^{-α-n}，并应用于Monge-Ampère方程、k-Hessian方程和特殊Lagrangian方程。

关于完全非线性椭圆方程Liouville定理的一个注记

A note on the Liouville theorem of fully nonlinear elliptic equations

本文提出一种不依赖$F$的$C^2$正则性和维数$n$的新方法，用于研究完全非线性一致椭圆方程$F(D^2u)=0$在外部域中解的渐近行为。

On the Liouville property for fully nonlinear equations with superlinear first-order terms

We consider in this note one-side Liouville properties for viscosity solutions of various fully nonlinear uniformly elliptic inequalities, whose prototype is $F(x,D^2u)\geq H_i(x,u,Du)$ in $\mathbb{R}^N$, where $H_i$ has superlinear growth in the gradient variable. After a brief survey on the existing literature, we discuss the validity or the failure of the Liouville property in the model cases $H_1(u,Du)=u^q+|Du|^γ$, $H_2(u,Du)=u^q|Du|^γ$ and $H_3(x,u,Du)=\pm u^q|Du|^γ-b(x)\cdot Du$, where $q\geq0$, $γ>1$ and $b$ is a suitable velocity field. Several counterexamples and open problems are tho…

Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians

In this paper, we are concerned with equations \eqref{PDE} involving higher-order fractional Laplacians. By introducing a new approach, we prove the super poly-harmonic properties for nonnegative solutions to \eqref{PDE} (Theorem \ref{Thm0}). Our theorem seems to be the first result on this problem. As a consequence, we derive many important applications of the super poly-harmonic properties. For instance, we establish Liouville theorems, integral representation formula and classification results for nonnegative solutions to fractional higher-order equations \eqref{PDE} with general nonlineari…

Fully nonlinear equations on Riemannian manifolds with negative curvature

We consider the problem of conformally deforming a metric to one with a prescribed symmetric function of the eigenvalues of the Ricci tensor, in the case of negative curvature.

Boundary value problems for some fully nonlinear elliptic equations

We consider a nonlinear version of the Yamabe problem on locally conformally flat compact manifolds with boundary. The main technique we used is to derive boundary $C^2$ estimates directly from boundary $C^0$ estimates. In particular, the result is a generalization of the work by Escobar.

Explicit subsolutions and a Liouville theorem for fully nonlinear uniformly elliptic inequalities in halfspaces

We prove a Liouville type theorem for arbitrarily growing positive viscosity supersolutions of fully nonlinear uniformly elliptic equations in halfspaces. Precisely, let $M^-_{λ, Λ}$ be the Pucci's inf- operator, defined as the infimum of all linear uniformly elliptic operators with ellipticity constants $Λ\geq λ>0$. Then, we prove that the inequality $M^-_{λ, Λ}(D^2u) +u^p \leq 0$ does not have any positive viscosity solution in a halfspace provided that $-1\leq p \leq (Λ/λn+1)/(Λ/λn-1)$, whereas positive solutions do exist if either $p < -1$ or $p > (Λ/λ(n-1)+2)/(Λ/λ(n-1))$. This will be acc…

Liouville results for fully nonlinear equations modeled on Hörmander vector fields. I. The Heisenberg group

This paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that the operator has a family of generalized subunit vector fields that satisfy the Hörmander condition. A general set of sufficient conditions is given such that all subsolutions bounded above are constant; it includes the existence of a supersolution out of a big ball, that explodes at infinity. Therefore for a large class of operators the problem is reduced to finding such a Lyapunov-like function. This is done here for the vector fields that …

厄米流形上完全非线性椭圆方程的稳定性与梯度估计

Stability and gradient estimates for fully nonlinear elliptic equations on hermitian manifolds

本文针对紧致厄米流形上的凹完全非线性椭圆方程，提出一种不依赖Hou-Ma-Wu估计和Liouville定理的梯度估计直接证明方法，并结合稳定性估计与比较论证，同时给出紧致Kähler流形上复Hessian估计的独立证明。

具有边界退化的完全非线性椭圆与抛物型方程的刘维尔型定理

Liouville-type theorems for fully nonlinear elliptic and parabolic equations with boundary degeneracy

本文研究完全非线性边界退化椭圆与抛物型方程，证明算子退化生成隐式边界条件，且除平凡解外不存在有界解，方法借鉴 Hamilton-Jacobi 方程黏性解唯一性论证。

全非线性椭圆方程的Liouville定理：小振荡与x的周期性和右端项的周期性

Liouville theorem for fully nonlinear elliptic equations with the small oscillation and the periodicity in $x$ and the periodic right hand term

本文研究了具有小振荡和周期性条件的全非线性椭圆方程的二次增长解，建立了Liouville型定理，推广了线性及非线性椭圆方程的已有结果。

Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities

We study fully nonlinear elliptic equations such as \[ F(D^2u) = u^p, \quad p>1, \] in $\R^n$ or in exterior domains, where $F$ is any uniformly elliptic, positively homogeneous operator. We show that there exists a critical exponent, depending on the homogeneity of the fundamental solution of $F$, that sharply characterizes the range of $p>1$ for which there exist positive supersolutions or solutions in any exterior domain. Our result generalizes theorems of Bidaut-Véron \cite{B} as well as Cutri and Leoni \cite{CL}, who found critical exponents for supersolutions in the whole space $\R^n$, i…

Variational status of a class of fully nonlinear curvature prescription problems

Prescribing, by conformal transformation, the kth-elementary symmetric polynomial of the Schouten tensor $P$ to be constant is a generalisation of the Yamabe problem. On compact Riemannian n-manifolds we show that, for k between and including 3 and n, this prescription equation is an Euler-Lagrange equation of some action if and only if the structure is locally conformally flat.

Liouville results for fully nonlinear equations modeled on Hörmander vector fields: II. Carnot groups and Grushin geometries

The paper treats second order fully nonlinear degenerate elliptic equations having a family of subunit vector fields satisfying a full-rank bracket condition. It studies Liouville properties for viscosity sub- and supersolutions in the whole space, namely, that under a suitable bound at infinity from above and, respectively, from below, they must be constants. In a previous paper we proved an abstract result and discussed operators on the Heisenberg group. Here we consider various families of vector fields: the generators of a Carnot group, with more precise results for those of step 2, in par…

A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound

We prove compactness of solutions of a fully nonlinear Yamabe problem satisfying a lower Ricci curvature bound, when the manifold is not conformally diffeomorphic to the standard sphere. This allows us to prove the existence of solutions when the associated cone $Γ$ satisfies $μ^+_Γ\le 1$, which includes the $σ_k-$Yamabe problem for $k$ not smaller than half of the dimension of the manifold.

实扭曲蒙日-安培方程的刘维尔定理

A Liouville theorem for the real twisted Monge-Ampère equation

本文通过建立无一致椭圆性的Pogorelov型内部$C^2$估计，证明了实扭曲蒙日-安培方程整分裂凸解的刘维尔定理，推广了Streets和Warren的刚性定理。

A fully nonlinear equation on 4-manifolds with positive scalar curvature

We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. We also give a new conformally invariant condition for positivity of the Paneitz operator, which allows us to give many new examples of manifolds admitting metrics with constant $Q$-curvature.

On a fully nonlinear sharp Sobolev trace inequality

We classify local minimizers of $\intσ_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$-ball, $4\leq n\leq 5$, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension $n+1=4$. If minimizers exist, this implies a fully nonlinear sharp Sobolev trace inequality. Our proof is an adaptation of the Frank--Lieb proof of the sharp Sobolev inequality, and in particular does not rely on symmetrization or Obata-type arguments.

Asymptotic and optimal Liouville properties for Wolff type integral systems

This article examines the properties of positive solutions to fully nonlinear systems of integral equations involving Hardy and Wolff potentials. The first part of the paper establishes an optimal existence result and a Liouville type theorem for the integral systems. Then, the second part examines the decay rates of positive bound states at infinity. In particular, a complete characterization of the asymptotic properties of bounded and decaying solutions is given by showing that such solutions vanish at infinity with two principle rates: the slow decay rates and the fast decay rates. In fact,…

On a Conformal Gauss-Bonnet-Chern inequality for LCF manifolds and related topics

In this paper, we prove the following two results: First, we study a class of conformally invariant operators $P$ and their related conformally invariant curvatures $Q$ on even-dimensional Riemannian manifolds. When the manifold is locally conformally flat(LCF) and compact without boundary, $Q$-curvature is naturally related to the integrand in the classical Gauss-Bonnet-Chern formula, i.e., the Pfaffian curvature. For a class of even-dimensional complete LCF manifolds with integrable $Q$% -curvature, we establish a Gauss-Bonnet-Chern inequality. Second, a finiteness theorem for certain classe…

Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equations

In recent years, the study of the interplay between (fully) non-linear potential theory and geometry received important new impulse. The purpose of this work is to move a step further in this direction by investigating appropriate versions of parabolicity and maximum principles at infinity for large classes of non-linear (sub)equations $F$ on manifolds. The main goal is to show a unifying duality between such properties and the existence of suitable $F$-subharmonic exhaustions, called Khas'minskii potentials, which is new even for most of the "standard" operators arising from geometry, and imp…

Liouville theorems for a family of very degenerate elliptic non linear operators

We prove nonexistence results of Liouville type for nonnegative viscosity solutions of some equations involving the fully nonlinear degenerate elliptic operators ${\cal P}^\pm_k$, defined respectively as the sum of the largest and the smallest $k$ eigenvalues of the Hessian matrix. For the operator ${\cal P}^+_k$ we obtain results analogous to those which hold for the Laplace operator in space dimension $k$. Whereas, owing to the stronger degeneracy of the operator ${\cal P}^-_k$, we get totally different results.

半空间中分数阶截断拉普拉斯算子的Liouville型结果

Liouville type results for the fractional truncated Laplacians in a half-space

本文针对半空间中含分数阶截断拉普拉斯算子和幂型非线性项的完全非线性积分方程，研究其粘性上解的存在性，以推导区分存在与不存在情形的阈值指数估计。

Regularity and a Liouville theorem for a class of boundary-degenerate second order equations

We study a class of second-order boundary-degenerate elliptic equations in two dimensions with minimal regularity assumptions. We prove a maximum principle and a Harnack inequality at the degenerate boundary, and assuming local boundedness, we prove continuity. On globally defined non-negative solutions we provide strong constraints on behavior at infinity, and prove a Liouville-type theorem for entire solutions on the closed half-plane. The class of PDE in question includes many from mathematical finance, Keldysh- and Tricomi-type PDE, and the 2nd order reduction of the fully non-linear 4th o…

具有梯度超线性增长的完全非线性椭圆方程的刘维尔型定理

Liouville type theorems for fully nonlinear elliptic equations with superlinear growth in gradient

研究完全非线性椭圆系统正上解，引入有效维数确定临界指数\(q_{c}\)，通过梯度指数、有效维数和非线性耦合相互作用建立刘维尔型不存在性定理，揭示有效维数在确定临界不存在阈值中的作用。

Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing $σ_k$-curvature in the interior and constant $H_k$-curvature on the boundary. When restricting to the closure of the positive $k$-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem which allows us to construct examples of compact Riemannian manifolds $(X,g)$ for which this problem admits multiple non-homoth…

On the $σ_2$-Nirenberg problem on $\mathbb{S}^2$

We establish theorems on the existence and compactness of solutions to the $σ_2$-Nirenberg problem on the standard sphere $\mathbb S^2$. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Möbius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Möbius invariant degenerate elliptic equations, a priori estimates of first and second order der…

全分数阶主方程的 Liouville 定理及其应用

Liouville theorem for fully fractional master equations and its applications

本文研究了全分数阶主方程，证明了在特定缓增函数空间与渐近假设下齐次方程解必为常数的 Liouville 型定理，建立了非齐次伪微分方程与积分方程的等价性，并获得了衰减率的最优估计。

完全非线性 Loewner-Nirenberg 问题：Liouville 定理与局部边界估计的反例

The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates

本文完整分类了上半空间中完全非线性 Loewner-Nirenberg 问题的正粘性解，证明解的唯一性或单参数族结构取决于锥参数 μ_Γ^+ 是否大于 1，并据此构造了局部边界 C^0 估计的反例。

具单调系数的临界p-拉普拉斯方程：刘维尔分类与Schoen型哈纳克不等式

Critical $p$-Laplace equations with monotone coefficients: Liouville classification and a Schoen-type Harnack inequality

本文研究具单调系数的临界p-拉普拉斯方程，完成其正整体解的刘维尔分类，建立Schoen型哈纳克不等式，为经典半线性方法提供拟线性对应及替代证明。

Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations

We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison principle, which we refer to as the principle of propagation of touching points, for operators of the form $\nabla^2 ψ+ L(x,ψ,\nabla ψ)$ which are non-decreasing in $ψ$.

无温度假设的三维定常热带气候模型的两个改进型Liouville型定理

Two improved Liouville-type theorems for the 3D stationary tropical climate model without temperature assumptions

该研究针对三维定常热带气候模型，在无温度假设下，通过L²估计等方法建立改进型Liouville型定理，获特定可积性条件下解平凡性，引入对数修正优化定理。

On a class of fully nonlinear elliptic equation in dimension two

We study existence and asymptotic behavior of radial positive solutions of some fully nonlinear equations involving Pucci's extremal operators in dimension two. In particular we prove the existence of a positive solution of a fully nonlinear version of the Liouville equation in the plane. Moreover for the $\mathcal{M}^-_{λ,Λ}$ operator, we show the existence of a critical exponent and give bounds for it.