Nonlocal Harnack inequalities

双重非线性混合局部与非局部抛物型方程的Harnack不等式

Harnack inequality for doubly nonlinear mixed local and nonlocal parabolic equations

本文针对双重非线性混合局部与非局部抛物型方程，利用比较原理、局部有界性估计和正性扩张方法，建立了非负弱解的Harnack不等式。

双重非线性非局部p-Laplace方程的弱Harnack估计

Weak Harnack estimates for a doubly nonlinear nonlocal p-Laplace equation

针对双重非线性非局部p-Laplace方程的弱上解，建立了带最优抛物尾的新型弱Harnack估计，利用正性扩张和测度理论，仅需局部正性，凸显非局部特征。

具有非标准增长的非局部问题的Harnack不等式

Harnack inequality for nonlocal problems with non-standard growth

本文基于作者此前相关研究，证明了具有非标准增长的非局部问题的局部极小值与弱解的完整Harnack不等式，核心辅助结果为对应De Giorgi类函数的局部有界性及弱Harnack不等式。

Harnack inequality and its application to nonlocal eigenvalue problems in unbounded domains

We prove the Harnack inequality for general nonlocal elliptic equations with zero order terms. As an application we prove the existence of the principal eigenvalue in general domains. Furthermore, we study the eigenvalue problem associated to the existence of self-similar solutions to the parabolic problem and provide estimates on the decay rate.

不连通区域中的非局部Harnack不等式

Nonlocal Harnack inequality in a disconnected region

本文在不连通区域中建立非局部方程弱解的Harnack不等式，比较解在不同连通分支上的值，并给出基于局部化极大值原理和Poisson核估计的两种证明方法。

标量曲率类型方程的Harnack不等式

Harnack inequalities for equations of type prescribed scalar curvature

本文研究标量曲率类型方程在n≥4维空间中解的Harnack不等式，提出新的不等式形式并分析其应用条件。

非局部内蕴Harnack估计

Nonlocal intrinsic Harnack estimates

本文把DiBenedetto内蕴Harnack估计推广至分数阶p-Laplacian型积分微分方程，建立内蕴表示型非局部估计，并证明p>2时非负局部解满足非局部问题独有的强极大值原理，类似结果亦适用于多孔介质型非局部方程。

海森堡群中的混合局部-非局部抛物方程：具有最优尾部的哈纳克不等式

Mixed Local-Nonlocal Parabolic Equations in Heisenberg Group : Harnack Inequality with an Optimal Tail

本文通过正性扩展方法和聚类引理，证明了海森堡群上线性混合局部-非局部抛物方程具有最优尾部的哈纳克不等式，避免了Moser迭代和对数估计。

非局部漂移-扩散方程的 Harnack 估计

Harnack estimates for nonlocal drift-diffusion equations

本文研究带漂移项的非局部扩散方程，采用 DeGiorgi 类式测度论方法，首次建立其 Harnack 估计，并改进 Hölder 正则性结果，同时得到椭圆情形的非局部弱 Harnack 估计。

非局部热方程 Harnack 不等式的几何方面

Geometric aspects of the Harnack Inequality for a nonlocal heat equation

本文揭示分数热方程正解的精确双边 Harnack 界与高维圆形几何的关联，并推广了前人结果。

Weak Harnack inequality for a mixed local and nonlocal parabolic equation

This article proves a weak Harnack inequality with a tail term for sign changing supersolutions of a mixed local and nonlocal parabolic equation. Our argument is purely analytic. It is based on energy estimates and the Moser iteration technique. Instead of the parabolic John-Nirenberg lemma, we adopt a lemma of Bombieri to the mixed local and nonlocal parabolic case. To this end, we prove an appropriate reverse Hölder inequality and a logarithmic estimate for weak supersolutions.

双非线性方程的双倍测度、Poincaré不等式和抛物型Harnack不等式

Doubling measures, Poincaré inequalities and parabolic Harnack inequalities for a doubly nonlinear equation

通过体积双倍和Poincaré不等式刻画满足双非线性方程抛物型Harnack不等式的度量测度空间，采用纯分析方法扩展线性结果。

非局部Trudinger方程的哈拿克估计

Harnack estimates for the nonlocal Trudinger equation

研究非局部Trudinger方程的逐点正则性理论，通过改编精细德乔治 - 莫泽机制，在最优尾条件下建立定量上界及抛物型哈拿克不等式，克服非局部双非线性框架技术难题。

非局部双相位泛函的Harnack型估计

Harnack-type estimates for nonlocal double phase functionals

针对非局部双相位泛函的局部极小化子，建立了Harnack型不等式，通过同心模估计解决了(p,q)-增长与低可积性指数的匹配问题，首次在自然结构条件下给出完整理论。

反对称函数的非局部 Harnack 不等式：一种基于 Bochner 关系与调和分析的方法

The nonlocal Harnack inequality for antisymmetric functions: an approach via Bochner's relation and harmonic analysis

本文利用 Bochner 关系将奇函数的一维 Fourier 变换与径向函数的三维 Fourier 变换相联系，把分数阶 Laplacian 的反对称函数 Harnack 不等式推广至更一般的非局部椭圆算子，将边界型不等式转化为内部不等式。

关于非局部方程的弱 Harnack 估计

On The Weak Harnack Estimate For Nonlocal Equations

本文针对以非局部 Trudinger 方程为模型的双重非线性非局部方程，利用适配非局部框架的正性扩张论证建立弱 Harnack 估计，并避开 Moser 迭代等经典技术。

非线性发展方程的Harnack型不等式

Harnack-type inequalities for nonlinear evolution equations

本文研究非线性发展方程的Harnack不等式，探讨了Li-Yau方法、Auchmuty-Bao扩展以及Moser方法，并应用于孔隙介质方程和p-扩散方程的解。

Weak Harnack inequalities for eigenvalues and constant rank theorems

We consider convex solutions of nonlinear elliptic equations which satisfy the structure condition of Bian-Guan. We prove a weak Harnack inequality for the eigenvalues of the Hessian of these solutions. This can be viewed as a quantitative version of the constant rank theorem.

非合作非局部扩散系统的哈纳克型不等式与行波

Harnack-type inequalities and traveling waves for non-cooperative nonlocal diffusion systems

本文针对实数域上弱耦合非局部扩散系统，建立了不依赖耦合矩阵合作性或不可约性的ℓ¹-范数哈纳克型不等式，结合重标度论证等方法，得到了一类带网络结构的非合作非局部扩散系统行波的最小波速存在性及波剖面负无穷处的精确渐近行为。

The boundary Harnack principle for nonlocal elliptic operators in non-divergence form

We prove a boundary Harnack inequality for nonlocal elliptic operators $L$ in non-divergence form with bounded measurable coefficients. Namely, our main result establishes that if $Lu_1=Lu_2=0$ in $Ω\cap B_1$, $u_1=u_2=0$ in $B_1\setminusΩ$, and $u_1,u_2\geq0$ in $\mathbb R^n$, then $u_1$ and $u_2$ are comparable in $B_{1/2}$. The result applies to arbitrary open sets $Ω$. When $Ω$ is Lipschitz, we show that the quotient $u_1/u_2$ is Hölder continuous up to the boundary in $B_{1/2}$.

黎曼流形上Fisher-KPP型方程的微分Harnack不等式

Differential Harnack inequalities for Fisher-KPP type equations on Riemannian manifolds

该研究在Bakry-Émery里奇曲率有下界的黎曼流形上，针对含Fisher-KPP、Newell-Whitehead方程的一类非线性抛物方程，得到无需正解额外条件的几乎最优微分Harnack不等式，并推导出最优Liouville性质。

Scaling invariant Harnack inequalities in a general setting

In a setting, where only "exit measures" are given, as they are associated with an arbitrary right continuous strong Markov process on a separable metric space, we provide simple criteria for the validity of Harnack inequalities for positive harmonic functions. These inequalities are scaling invariant with respect to a metric on the state space which, having an associated Green function, may be adapted to the special situation. In many cases, this also implies continuity of harmonic functions and Hölder continuity of bounded harmonic functions. The results apply to large classes of Lévy (and s…

Nonlocal Harnack inequalities in the Heisenberg group

We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group $\mathbb{H}^n$, whose prototype is the Dirichlet problem for the $p$-fractional subLaplace equation. These problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase transition problems, in image segmentations models, and so on, when non-Euclidean geometry frameworks and nonlocal long-range interactions do naturally occur. We prove general Harnack inequalities for the related weak solutions. Also, in the case when the growth exponent is $p=2$, we investigat…

非局部 Serrin 型问题的稳定性结果、反对称 Harnack 不等式及几何估计

Stability results for nonlocal Serrin-type problems, antisymmetric Harnack inequalities, and geometric estimates

本文研究了非局部偏微分方程的对称性与几何性质，包括由分数阶拉普拉斯算子驱动的超定问题、反对称解的 Harnack 不等式及分数阶平均曲率的几何估计。

A certain critical density property for invariant Harnack inequalities in H-type groups

We consider second order linear degenerate-elliptic operators which are elliptic with respect to horizontal directions generating a stratified algebra of H-type. Extending a result by Gutiérrez and Tournier for the Heisenberg group, we prove a critical density estimate by assuming a condition of Cordes-Landis type. We then deduce an invariant Harnack inequality for the non-negative solutions from a result by Di Fazio, Gutiérrez, and Lanconelli.

黎曼流形上半线性抛物型方程的微分Harnack不等式 I：Bakry-Émery曲率下界

Differential Harnack inequalities for semilinear parabolic equations on Riemannian manifolds I: Bakry-Émery curvature bounded below

本文提出一种统一方法，将Bakry-Émery曲率有下界的黎曼流形上半线性抛物型方程的微分Harnack不等式推导转化为求解常微分方程组，并由此得到对数型方程和Yamabe型方程的精确估计及Liouville型定理。

Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes

We study energy functionals obtained by adding a possibly discontinuous potential to an interaction term modeled upon a Gagliardo-type fractional seminorm. We prove that minimizers of such non-differentiable functionals are locally bounded, Hölder continuous, and that they satisfy a suitable Harnack inequality. Hence, we provide an extension of celebrated results of M. Giaquinta and E. Giusti to the nonlocal setting. To do this, we introduce a particular class of fractional Sobolev functions, reminiscent of that considered by E. De Giorgi in his seminal paper of 1957. The flexibility of these …

时间不敏感的非局部抛物 Harnack 估计

Time-insensitive nonlocal parabolic Harnack estimates

本文针对非局部抛物方程建立不受等待时间现象影响的新型全局解 Harnack 估计，并证明局部解情形下等待时间不可避免。

Harnack inequalities and Bounds for Densities of Stochastic Processes

We consider possibly degenerate parabolic operators in the form $$ \sum_{k=1}^{m}X_{k}^{2}+X_{0}-\partial_{t}, $$ that are naturally associated to a suitable family of stochastic differential equations, and satisfying the Hörmander condition. Note that, under this assumption, the operators in the form $Ł$ has a smooth fundamental solution that agrees with the density of the corresponding stochastic process. We describe a method based on Harnack inequalities and on the construction of Harnack chains to prove lower bounds for the fundamental solution. We also briefly discuss PDE and SDE methods …

不平等与集中现象

Inégalités de Harnack et Phénomène de Concentration

证明了2阶和4阶椭圆算子的乘积sup * inf的不等式。利用这些不等式和集中现象，我们可以描述这些偏微分方程解的渐近行为。

Harnack and pointwise estimates for degenerate or singular parabolic equations

In this paper we give both an historical and technical overview of the theory of Harnack inequalities for nonlinear parabolic equations in divergence form. We start reviewing the elliptic case with some of its variants and geometrical consequences. The linear parabolic Harnack inequality of Moser is discussed extensively, together with its link to two-sided kernel estimates and to the Li-Yau differential Harnack inequality. Then we overview the more recent developements of the theory for nonlinear degenerate/singular equations, highlighting the differences with the quadratic case and introduci…

Nonlocal Harnack inequalities for nonlocal heat equations

In this paper, applying the De Giorgi method, we obtain nonlocal Harnack inequalities for weak solutions of nonlocal parabolic equations given by an integro-differential operator $\rL_K$ as follows; \begin{equation*}\begin{cases} \rL_K u+\pa_t u=0 &\text{ in $\Om\times(-T,0]$ } u=g &\text{ in $\bigl((\BR^n\s\Om)\times (-T,0]\bigr)\cup\bigl(\Om\times\{t=-T\}\bigr)$ } \end{cases}\end{equation*} where $g\in C(\BR^n\times [-T,0])\cap L^{\iy}(\BR^n\times(-T,0])$ and $\,\Om\,$ is a bounded domain in $\BR^n$ with Lipschitz boundary. Moreover, we get nonlocal parabolic weak Harnack inequalities of the…

Elliptic Harnack inequalities for symmetric non-local Dirichlet forms

We study relations and characterizations of various elliptic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces. We allow the scaling function be state-dependent and the state space possibly disconnected. Stability of elliptic Harnack inequalities is established under certain regularity conditions and implication for a priori Hölder regularity of harmonic functions is explored. New equivalent statements for parabolic Harnack inequalities of non-local Dirichlet forms are obtained in terms of elliptic Harnack inequalities.

非局部动力学方程的Harnack不等式不成立

The Harnack inequality fails for nonlocal kinetic equations

本研究证明非局部动力学方程的Harnack不等式不成立，以分数阶Kolmogorov方程为反例，指出这是纯粹非局部现象，补充了非局部亚椭圆方程理论的相关结论。

李-尤和哈纳克估计非局部扩散问题

Li-Yau and Harnack estimates for nonlocal diffusion problems

本文介绍微分哈纳克不等式，总结了关于非局部扩散问题的李-尤和哈纳克估计的课程主要结果。

非常快扩散方程的正性、局部光滑和Harnack不等式

Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations

研究了具有$m<1$的快扩散方程$\partial_t u =Δ(u^m)/m$在一般非负初值下局部解$u(t,x)\ge 0$的定性性质。我们的主要结果是定量的正性和有界性估计的局部定义的解决方案的形式$[0,T]\times\RR^d$域。它们结合成新的Harnack不等式的形式，是典型的快速扩散方程。这样的结果是新的低$m$在所谓的非常快的扩散范围内，精确地为所有的$m\le m_c=(d-2)/d.$的有界性的声明是真实的，甚至为$m\le 0$，而积极的不能是真实的，在该范围内。

Weak Harnack estimates for supersolutions to doubly degenerate parabolic equations

We establish weak Harnack inequalities for positive, weak supersolutions to certain doubly degenerate parabolic equations. The prototype of this kind of equations is $$\partial_tu-\operatorname{div}|u|^{m-1}|Du|^{p-2}Du=0,\quad p>2,\quad m+p>3.$$ Our proof is based on Caccioppoli inequalities, De Giorgi's estimates and Moser's iterative method.

熵单调性与Harnack不等式的关系

A relation between entropy monotonicity and Harnack inequalities

我们将Perelman的熵计算推广到Ricciflow解内部域的情况。在欧几里得空间作为环境流形的情况下，出现了与Harnack不等式的有趣关系。

混合局部和非局部 $p$-拉普拉斯方程弱赫尔曼不等式两种替代证明

Two alternative proofs of weak Harnack inequality for mixed local and nonlocal $p$-Laplace equations with a nonhomogeneity

本文研究了一类混合局部和非局部 $p$-拉普拉斯方程，给出了保证变号超解弱赫尔曼不等式尾项的 $f $ 的充分条件，采用两种不同方法避免了克里洛夫-萨夫诺夫覆盖引理和正性展开，一种通过约翰-尼伦伯格引理，另一种通过Bombieri-Giusti引理，并为超解的反向霍尔德不等式提供了新证明，同时通过局部有界性结果、尾估计和初始弱赫尔曼不等式建立了解的赫尔曼不等式。

光滑度量测度空间上一类非线性抛物方程的曲率条件、Liouville型定理与Harnack不等式

Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces

本文在光滑度量测度空间上，针对含Witten Laplace算子的非线性抛物方程，建立Souplet-Zhang、Hamilton与Li-Yau型梯度估计，并由此导出Harnack不等式及Liouville型常值性结果。

Allen-Cahn型方程的Liouville定理与Harnack不等式

Liouville theorems and Harnack inequalities for Allen-Cahn type equation

本文研究Allen-Cahn型方程，在Ricci及积分曲率条件下建立对数梯度估计、Harnack不等式与Liouville定理，并推广至含Lane-Emden等方程的非线性类及Bakry-Émery几何框架。

关于非局部双相问题的弱 Harnack 不等式

On the weak Harnack inequalities for nonlocal double phase problems

本文利用正性扩张方法研究非局部双相泛函的弱 Harnack 不等式，通过基于非局部 Caccioppoli 型不等式建立测度理论估计来克服调制系数与增长指数相互作用的挑战，并探讨了极小子的定量有界性。

The Harnack inequality for a class of nonlocal parabolic equations

In this paper we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators $(\partial_t - \mathscr{L})^s$, $0<s<1$, where $\mathscr{L}$ is the infinitesimal generator of a class of symmetric semigroups. As a by-product we also obtain a similar result for the nonlocal operators $(-\mathscr{L})^s$. Our focus is on non-Euclidean situations.

非局部方程的抛物 Harnack 不等式

The parabolic Harnack inequality for nonlocal equations

本文在变分框架下证明具有界可测系数的线性抛物非局部方程弱解的抛物 Harnack 不等式与 Hölder 估计，核心在于精细分析非局部尾项。

Harnack inequalities for evolving hypersurfaces on the sphere

We prove Harnack inequalities for hypersurfaces flowing on the unit sphere by $p$-powers of a strictly monotone, 1-homogeneous, convex, curvature function $f$, $0<p\leq 1.$ If $f$ is the mean curvature, we obtain stronger Harnack inequalities.

纯分析视角下非局部Dirichlet空间上的抛物Harnack不等式

The parabolic Harnack inequality on non-local Dirichlet spaces in the view of pure analysis

该研究针对无消亡项的正则Dirichlet型，用纯分析的Nash与Moser方法证明抛物Harnack不等式（PHI），推导其重要性质，并结合前期弱Harnack不等式结果扩充了PHI的等价刻画范围。

Harnack inequality for the fractional nonlocal linearized Monge--Ampère equation

The fractional nonlocal linearized Monge--Ampère equation is introduced. A Harnack inequality for nonnegative solutions to the Poisson problem on Monge--Ampère sections is proved.

Differential Harnack inequalities for a family of sub-elliptic diffusion equations on Sasakian manifolds

We prove a version of differential Harnack inequality for a family of sub-elliptic diffusions on Sasakian manifolds under certain curvature conditions.

Differential Harnack inequalities and Perelman type entropy formulae for subelliptic operators

In this paper, under the generalized curvature-dimension inequality recently introduced by F. Baudoin and N. Garofalo, we obtain differential Harnack inequalities for the positive solutions to the Schödinger equation associated to subelliptic operator with potential. As applications of the differential Harnack inequality, we derive the corresponding parabolic Harnack inequality. Also we define the Perelman type entropy associated to subelliptic operators and derive its monotonicity.

A stability theorem for elliptic Harnack inequalities

We prove a stability theorem for the elliptic Harnack inequality: if two weighted graphs are equivalent, then the elliptic Harnack inequality holds for harmonic functions with respect to one of the graphs if and only if it holds for harmonic functions with respect to the other graph. As part of the proof, we give a characterization of the elliptic Harnack inequality.

带右端项的新边界 Harnack 不等式

New boundary Harnack inequalities with right hand side

本文在 Lipschitz 区域中针对带 $L^q$（$q>n$）右端项的散度型与非散度型椭圆方程建立了新的边界 Harnack 不等式，并由此得到完全非线性（薄）障碍问题自由边界的 $\mathcal{C}^{1,α}$ 正则性。

New differential Harnack inequalities for nonlinear heat equations

We prove constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation $ω_t=Δω+aω\ln ω$ on closed manifolds. We also derive a new interpolated Harnack inequality for the equation $ω_t=Δω-ω\lnω+\varepsilon Rω$ on closed surfaces under the $\varepsilon$-Ricci flow. Finally we prove a new differential Harnack inequality for the equation $ω_t=Δω-ω\lnω$ under the Ricci flow without any curvature condition. Among these Harnack inequalities, the correction terms are all time-exponential functions, which are superior to time-polynomial functions.

散度型椭圆偏微分方程的微分Harnack不等式与Morel-Oswald型最大值原理

Differential Harnack inequalities and maximum principles of Morel-Oswald type for elliptic PDE in divergence form

本文为散度型一致椭圆方程正解建立Morel-Oswald型Hopf-Oleinik引理推广及新的微分Harnack估计，并给出最优全局结果。

非局部Harnack不等式用于非局部双重相方程I；具有正有界的调制系数但无Hölder条件

Nonlocal Harnack inequalities for nonlocal double phase equations I ; with positive bounded modulating coefficient with no Hölder condition

本文通过De Giorgi-Nash-Moser理论，证明了非局部双重相方程弱解的非局部Harnack不等式，并研究了其弱超解的局部有界性和弱Harnack不等式。