YanYan Li, Han Lu, Siyuan Lu
In this paper we classify Möbius invariant differential operators of second order in two dimensional Euclidean space, and establish a Liouville type theorem for general Möbius invariant elliptic equations.
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A general Liouville type theorem for some conformally invariant fully nonlinear equations
对共形不变的完全非线性方程建立了一个普遍的Liouville型定理。
Further results on Liouville type theorems for some conformally invariant fully nonlinear equations
给出了共形不变的完全非线性方程的Liouville型定理的进一步结果。
On some conformally invariant fully nonlinear equations, Part II: Liouville, Harnack and Yamabe
Yamabe问题涉及在给定的闭黎曼流形上寻找一个共形度量，使它具有常数数量曲率。本文主要研究Yamabe问题的一个完全非线性形式和相应的Liouville型问题。
We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single standard bubble in a universal positive distance around each blow-up point, and (iii) the heights of these bubbles are comparable by a universal factor. As an application of this result, we establish a quantitative Liouville theorem.
A Liouville type theorem for some conformally invariant fully nonlinear equations
对一类共形不变的完全非线性方程建立了Liouville型定理
${L^p}$-Liouville Theorems for Invariant Partial Differential Operators in ${\mathbb{R}^n}$
证明了$\mathbb{R}^n$中二阶亚椭圆偏微分算子关于李群合成律左平移不变量的$L^p$ -Liouville定理.两个解决方案和子解决方案的结果。
Super-Liouville Equations on Closed Riemann Surfaces
受近年来物理学文献中刘维尔理论的超对称推广的启发，我们将标准刘维尔泛函与旋量场项耦合起来。所得到的泛函是共形不变的。我们研究几何和分析方面的欧拉-拉格朗日方程，最终在爆破分析。
Liouville theorems for conformally invariant fully nonlinear equations. I
本文研究共形不变完全非线性椭圆方程的Liouville定理，给出其成立及局部梯度估计的充要条件，并应用于闭流形上共形度量的存在性与紧致性。
Degenerate conformally invariant fully nonlinear elliptic equations
建立了退化共形不变方程的Liouville型定理。
Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure
我们提供了一个简单的方法来获得新的Liouville定理的标度不变的超线性抛物问题的梯度结构。为了说明该方法，我们证明了刘维尔定理（保证不存在正古典解）的下列模型问题：标量非线性热方程$$ u_t-Δu=u^p \qquad\hbox{in}\ {\mathbb R}^n\times{\mathbb R}, $$，其向量值的推广与$p$ -齐次非线性和线性热方程在${\mathbb R}^n_+\times{\mathbb R}$补充的非线性边界条件的形式$\partial u/\partialν=u^q$。这里，$ν$表示半空间${\mathbb R}^n_+$边界上的外部单位法线，如果是$n>2$，指数$p,q>1$满足$p<n/(n-2)$和$q<(n-1)/(n-2)$（如果是$n>2$，指数$p,q>1$满足$p<(n+2)/(n-2)$和$q<n/(n-2)$，并且假设解具有某种对称性）。作为不存在性结果的一个典型应用，我们在有界和无界区域上给出了相关问题正解的最优普适估计。
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。
We revisit rescaling methods for nonlinear elliptic and parabolic problems and show that, by suitable modifications, they may be used for nonlinearities that are not scale invariant even asymptotically and whose behavior can be quite far from power like. In this enlarged framework, by adapting the doubling-rescaling method from [37, 38], we show that the equivalence found there between universal estimates and Liouville theorems remains valid. In the parabolic case we also prove a Liouville type theorem for a rather large class of non scale invariant nonlinearities. This leads to a number of ne…
New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces
我们考虑一个奇异的刘维尔方程的紧凑的表面上，所产生的研究的Chern-Simons涡在一个自对偶制度。使用新的改进版本的Moser-Trudinger不等式（其主要特点是尺度不变）和变分方案，我们证明了新的存在性结果。
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。
Liouville theorems for scaling invariant nonlinear parabolic problems in the whole space and/or the halfspace (saying that the problem does not posses positive bounded solutions defined for all times $t\in(-\infty,\infty)$) guarantee optimal estimates of solutions of related initial-boundary value problems in general domains. We prove an optimal Liouville theorem for the linear equation in the halfspace complemented by the nonlinear boundary condition $\partial u/\partialν=u^q$, $q>1$.
A New proof of Liouville type theorems for a class of semilinear elliptic equations
本文研究欧氏空间或具非负Ricci曲率的闭流形上的典型半线性椭圆方程，利用不变张量法构造的关键积分恒等式结合积分估计，重新证明了若干经典Liouville定理。
Convergence of bi-spatial pullback random attractors and stochastic Liouville type equations for nonautonomous stochastic p-Laplacian lattice system
本文研究带乘性噪声的非自治随机p-Laplacian格点方程长期行为关于随机项系数的收敛性，证明了双空间拉回随机吸引子上半连续性、不变样本测度收敛性，验证了随机Liouville型方程及样本统计解的收敛结论。
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。
Weighted ${L^p}$-Liouville Theorems for Hypoelliptic Partial Differential Operators on Lie Groups
本文证明了一类二阶亚椭圆型偏微分算子$\mathcal{L}$在李群$\mathbb{G}$上的加权$L^p$-Liouville定理，其底流形是$n$维空间.我们证明了一个自然权是$\mathbb{G}$的右不变测度$\check{H}$.证明了$L^p(\mathbb{G},\check{H})$中$C^2$子解的Liouville型定理。我们提供的运营商的例子，我们的结果适用，共同的应用程序的唯一性的柯西问题的演化运营商$\mathcal{L}-\partial_t$。
A Liouville Property for Isotropic Diffusions in Random Environment
我们得到了随机环境中平稳扩散的Liouville性质，这些随机环境是大于2的空间维上布朗运动的小的、各向同性扰动。准确地说，我们证明了，在一个子集的全概率，常数函数是唯一的严格次线性映射是不变的扩散的发展。证明了常数函数是唯一在演化下不变的有界古映射。这些结果依赖于Fehrman [3]和Sznitman和Zeitouni [7]以前的工作，在第一种情况下，我们的方法是由Benjamini，Duminil-Copin，Kozma和Yadin [1]在离散环境中的工作所激发的。
On Liouville Type of Theorems to the 3-D Incompressible Axisymmetric Navier-Stokes Equations
Liouville型定理在研究三维Navier-斯托克斯方程整体正则性的Blow-up方法中起着关键作用。本文在对涡旋分量速度$u_θ$作适当的尺度不变性假设下，证明了三维轴对称带涡旋斯托克斯方程的Liouville型定理。已知$ru_θ$满足最大值原理。$u_θ$中的假设对于进一步研究三维不可压轴对称Navier-斯托克斯方程的整体正则性是自然和有用的。
Liouville theorems and universal estimates for superlinear parabolic problems without scale invariance
本文针对无尺度不变性的超线性抛物问题，利用椭圆刘维尔定理、能量估计和分量比例化约化方法建立全空间与半空间中的刘维尔型定理，并将其应用于普适奇异性与衰减估计。
A result by Courrège says that linear translation invariant operators satisfy the maximum principle if and only if they are of the form $\mathcal{L}=\mathcal{L}^{σ,b}+\mathcal{L}^μ$ where $$ \mathcal{L}^{σ,b}[u](x)=\text{tr}(σσ^{\texttt{T}} D^2u(x))+b\cdot Du(x) $$ and $$ \mathcal{L}^μ[u](x)=\int \big(u(x+z)-u-z\cdot Du(x) \mathbf{1}_{|z| \leq 1}\big) \,\mathrm{d} μ(z). $$ This class of operators coincides with the infinitesimal generators of Lévy processes in probability theory. In this paper we give a complete characterization of the translation invariant operators of this form that satisfy …
Liouville theorem of the subcritical biharmonic equation on complete manifolds
本文针对具非负Ricci曲率的完备非紧黎曼流形，用不变张量法推导微分恒等式并结合Bernstein技巧与连续性方法得关键估计，建立次临界双调和方程在特定条件下无正$C^4$解的刘维尔定理。
Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess positive entire solutions) guarantee optimal universal estimates of solutions of related initial and initial-boundary value problems. In the case of the nonlinear heat equation $$u_t-Δu=u^p\quad\hbox{in}\quad {R}^n\times{R}, \qquad p>1, $$ the nonexistence of positive classical solutions in the subcritical range $p(n-2)<n+2$ has been conjectured for a long time, but all known results require either a more restrictive assumption on $p$ or deal with a special cl…
Liouville theorems for scaling invariant nonlinear elliptic systems (saying that the system does not possess nontrivial entire solutions) guarantee a priori estimates of solutions of related, more general systems. Assume that $p=2q+3>1$ is Sobolev subritical, $n\le3$ and $β\in{\mathbb R}$. We first prove a Liouville theorem for the system $$\left.\begin{aligned} -Δu &=|u|^{2q+2}u+β|v|^{q+2}|u|^q u, \\ -Δv &=|v|^{2q+2}v+β|u|^{q+2}|v|^q v, \end{aligned}\ \right\} \quad\hbox{in}\quad {\mathbb R}^n,$$ in the class of radial functions $(u,v)$ such that the number of nodal domains of $u,v,u-v,u+v$ i…
The Liouville-type equation and an Onofri-type inequality on closed 4-manifolds
本文针对满足$\textrm{Ric}\geqslant 3κg$且$κ>0$的闭4-流形，研究Liouville型方程，用不变张量法推导微分恒等式分类参数范围内的解，通过连续性方法建立二阶导数估计，还推导了4-球面上的Onofri型不等式并证明其刚性。
On some conformally invariant fully nonlinear equations
我们将报告有关Yamabe问题和Nirenberg问题的一些结果。还将讨论相关议题。这类研究对一些几何上的共形不变的完全非线性方程有了新的结果。我们还将给出这些结果，其中包括一些Liouville型定理，Harnack型不等式，一些非线性版本的Yamabe问题的解的存在性和紧性。
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…
Liouville Theorem for Dunkl Polyharmonic Functions
假设$f$是$\mathbb{R}^n$中的Dunkl多调和（即，对于某个整数$p$，是$(Δ_h)^p f=0$，其中$Δ_h$是与根系$R$和多重性函数$κ$相关联的Dunkl拉普拉斯算子，定义在$R$上并且相对于有限Coxeter群不变）。证明了$f$是$s\ge 2p-2$的$\le s$次多项式的必要条件和成功条件。作为一个直接推论，一个上下有界的Dunkl调和函数是常数。
Metrisability of two-dimensional projective structures
我们实现了R. Liouville \cite{Liouville}构造了一个显式局部障碍，以证明曲面上给定的射影结构$[Γ]$内的Levi-Civita联络的存在。在投射类中的联络的分支中，阻塞是5阶的。它可以表示为积分曲线是$[Γ]$的测地线的二阶常微分方程的点不变量或表示为投影类的加权标量射影不变量。如果障碍为零，我们找到了实解析情形下度量存在的充分条件。在一般的情况下，它们表示为两个6阶不变量在连接中的消失。在退化的情况下，足够的阻塞最多为8级。
The modified Camassa-Holm equation (also called FORQ) is one of numerous $cousins$ of the Camassa-Holm equation possessing non-smoth solitons ($peakons$) as special solutions. The peakon sector of solutions is not uniquely defined: in one peakon sector (dissapative) the Sobolev $H^1$ norm is not preserved, in the other sector (conservative), introduced in [2], the time evolution of peakons leaves the $H^1$ norm invariant. In this Letter, it is shown that the conservative peakon equations of the modified Camassa-Holm can be given an appropriate Poisson structure relative to which the equations …
Liouville-type theorems for the stationary fractional Navier-Stokes equations in $\mathbb{R}^n$
本文在速度场$u$满足可积性条件和分数阶能量的大尺度Morrey型界条件下，证明了稳态分数阶Navier-Stokes方程的Liouville型定理，并推导出在有限分数阶能量条件下$\frac{n}{3} \le α< \frac{n+2}{3}$范围内的Liouville结果。
Liouville theorems, universal estimates and periodic solutions for cooperative parabolic Lotka-Volterra systems
研究了具有等扩散系数的抛物型Lotka-Volterra合作系统在有界和无界区域上的正解.该系统的补充Dirichlet或Neumann边界条件。在对反应项系数的适当假设下，这些问题既有整体解，也有在有限时间内爆破的解。我们发现，任何解决方案$(u,v)$定义的时间间隔$(0,T)$满足普遍估计的形式$$u(x,t)+v(x,t)\leq C(1+t^{-1}+(T-t)^{-1}),$$ $C$不依赖于$x,t,u,v,T$。特别是，这个界保证了整体存在性和有界性的阈值解决方案之间的边界爆破和整体存在。此外，这个界产生最佳的爆破率估计的解决方案，在有限时间内爆破。我们的估计是基于相应的标度不变抛物系统的新的刘维尔型定理，并需要对空间维数$n$：$n\leq5$的最佳限制。作为应用，我们还证明了当系数是时间周期的时，时间周期正解的存在性。我们的方法也可以用于更一般的抛物系统。
A Liouville Theorem for the Axially-symmetric Navier-Stokes Equations
令$v(x, t)= v^r e_r + v^θe_θ+ v^z e_z$为三维不可压轴对称Navier-Stokes方程斯托克斯方程的解。用$b = v^r e_r + v^z e_z$表示径向-轴向矢量场。在$b$上的一般标度不变条件下，证明了量$Γ= r v^θ$在$r = 0$，$t = 0$上是Hölder连续的.作为应用，我们给出了Koch-Nadirashvili-Seregin-Sverak在\cite{KNSS}和Seregin-Sverak在\cite{SS}中关于Liouville性质的一个猜想的部分证明.作为另一个应用，我们证明了如果$b \in L^\infty([0, T], BMO^{-1})$，则$v$是正则的.这为Koch和Tataru在\cite{KochTataru}中提出的关于轴对称情形下Navier-斯托克斯方程的唯一性和正则性的一个开放性问题提供了答案。
We are concerned with solutions to the nonlinear heat equation $u_t=Δu+|u|^{p-1}u$, $x\in \mathbb{R}^N$, that are defined for all positive and negative time. If the exponent $p$ is greater or equal to the Joseph-Lundgren exponent $p_c$ and $|u|$ stays below some positive radially symmetric steady state, under a mild condition on the behaviour of $u$ as $|x|\to \infty$, we show that $u$ is independent of time. Our method of proof uses Serrin's sweeping principle, based on the strong maximum principle, applied to the linearized equation for $u_t$. Our result covers that of Poláčik and Yanagida […
Liouville theorems for nonlocal Lane--Emden inequalities with degenerate kernels
本文通过测试函数方法和二进制分解，证明了非局部算子下兰恩-埃门德不等式的勒维叶型定理，仅在特定条件下存在平凡解。
Liouville theorems and universal estimates for superlinear elliptic problems without scale invariance
本文将刘维尔型定理应用于无尺度不变性的椭圆系统以获得普适奇性和衰减估计，提出了基于 Pohozaev 恒等式等方法证明新定理，并综述改进了现有证明方法。
Liouville theorems for anisotropic $p$-Laplace equations with a semilinear term
针对带半线性项的各向异性 $p$-Laplace 方程，结合 Serrin 技巧与不变张量方法，建立正解、超/子解及稳定变号解的 Liouville 不存在性与分类结果。
In this paper, we investigate blow-up of solutions to semilinear wave equations with scale-invariant damping and nonlinear memory term in $\mathbb{R}^n$, which can be represented by the Riemann-Liouville fractional integral of order $1-γ$ with $γ\in(0,1)$. Our main interest is to study mixed influence from damping term and the memory kernel on blow-up conditions for the power of nonlinearity, by using test function method or generalized Kato's type lemma. We find a new competition, particularly for the small value of $γ$, on the blow-up range between the effective case and the non-effective ca…
A Fulling-Kuchment theorem for the 1D harmonic oscillator
证明了存在一对“非等谱”一维半经典薛定谔算子，其谱模h^ \infty一致.特别地，它们的所有半经典迹不变量都是相同的。我们的证明是基于一个想法的Fulling-Kuchment和阿达玛的变分公式适用于适当的扰动的谐振子。保留字：逆谱问题，半经典薛定谔算子，迹不变量，阿达玛变分公式，谐振子，彭罗斯蘑菇，斯特姆-刘维理论。
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