A geometric problem and the Hopf Lemma. I

On the Höpf Boundary Lemma for quasilinear problems involving singular nonlinearities and applications

In this paper we consider positive solutions to quasilinear elliptic problem with singular nonlinearities. We provide a Höpf type boundary lemma via a suitable scaling argument that allows to deal with the lack of regularity of the solutions up to the boundary.

对数拉普拉斯算子的反对称最大值原理与Hopf引理及其在对称性结果中的应用

Antisymmetric maximum principles and Hopf's lemmas for the Logarithmic Laplacian, with applications to symmetry results

本文为对数拉普拉斯算子建立反对称最大值原理和Hopf型引理，并应用于证明半线性问题解的对称性及平行曲面问题的刚性结果。

Hopf Lemma and regularity results for quasilinear anisotropic elliptic equations

We consider a class of quasi-linear anisotropic elliptic equations, possibly degenerate or singular, which are of interest in several applications such as computer vision and continuum mechanics. We prove a Hopf Lemma as well as local and global regularity estimates for positive solutions, generalizing previous results known in the context of p-Laplacian equations.

A Hopf lemma and regularity for fractional $p-$Laplacians

In this paper, we study qualitative properties of the fractional $p$-Laplacian. Specifically, we establish a Hopf type lemma for positive weak super-solutions of the fractional $p-$Laplacian equation with Dirichlet condition. Moreover, an optimal condition is obtained to ensure $(-\triangle)_p^s u\in C^1(\mathbb{R}^n)$ for smooth functions $u$.

A Hopf-type boundary point lemma for pairs of solutions to quasilinear equations

We show a Hopf boundary point lemma for $u=u_{1}-u_{2},$ given $u_{1},u_{2} \in C^{1,α}$ each weak solutions to a quasilinear equation $\sum_{i=1}^{n} D_{i}(A^{i}(x,u,Du))+B(x,u,Du)=0$ under mild boundedness assumptions on $A^{1},\ldots,A^{n},B.$

Sub-solutions and a point-wise Hopf's Lemma for Fractional p-Laplacian

We prove a Hopf's lemma in the point-wise sense. The essential technique is to prove $(-Δ)^s_p u(x)$ is uniformly bounded in the unit ball $B_1\subset\mathbb{R}^n$, where $u(x)=(1-|x|^2)^s_{+}$. Also we study the global Hölder continuity of bounded positive solutions for $(-Δ)^s_p u(x)=f(x,u).$

A Hopf's Lemma and the Boundary Regularity for the Fractional P-Laplacian

We begin the paper with a Hopf's lemma for a fractional p-Laplacian problem on a half-space. Specifically speaking, we show that the derivative of the solution along the outward normal vector is strictly positive on the boundary of the half-space. Next we show that positive solutions to a fractional p-Laplacian equation possess certain Holder continuity up to the boundary.

A counterexample to the Hopf-Oleinik lemma (elliptic case)

We construct a new counterexample confirming the sharpness of the Dini-type condition for the boundary of $Ω$. In particular, we show that for convex domains the Dini-type assumption is the necessary and sufficient condition which guarantees the Hopf-Oleinik type estimates.

涉及对数拉普拉斯算子的狄利克雷问题的最优边界正则性与Hopf型引理

Optimal boundary regularity and a Hopf-type lemma for Dirichlet problems involving the logarithmic Laplacian

本文研究涉及对数拉普拉斯算子的狄利克雷问题，利用Kelvin变换构造障碍函数证明最优边界正则性，并由此得到非负弱解的Hopf型引理及非线性问题解的唯一性。

Quantitative uniqueness of continuation result related to Hopf's lemma

The classical Hopf's lemma can be reformulated as uniqueness of continuation result. We aim in the present work to quantify this property. We show precisely that if a solution $u$ of a divergence form elliptic equation attains its maximum at a boundary point $x_0$ then both $L^1$-norms of $u-u(x_0)$ on the domain and on the boundary are bounded, up to a multiplicative constant, by the exterior normal derivative at $x_0$.

A Hopf's lemma and a strong minimum principle for the fractional $p$-Laplacian

Our propose here is to provide a Hopf Lemma and a strong minimum principle for week supersolutions of \[ (-Δ_p)^s u= c(x)|u|^{p-2}u \quad \text{ in } Ω\] where $Ω$ is an open set of $\mathbb{R}^N,$ $s\in(0,1),$ $p\in(1,+\infty),$ $c\in C(\overlineΩ)$ and $(-Δ_p)^s$ is the fractional $p$-Laplacian.

分数阶 $p$-拉普拉斯算子的Hopf引理与边界点结果

Hopf's lemmas and boundary point results for the fractional $p$-Laplacian

本文针对分数阶 $p$-拉普拉斯算子（$p \geq 2$）建立了Hopf边界引理的新证明，并推广到变号势情形，无需全局符号条件，且应用于边界点引理和非局部超定问题。

A Hopf lemma for the regional fractional Laplacian

We provide a Hopf boundary lemma for the regional fractional Laplacian $(-Δ)^s_Ω$, with $Ω\subset\mathbb{R}^N$ a bounded open set. More precisely, given $u$ a pointwise or weak super-solution of the equation $(-Δ)^s_Ω u = c(x)u$ in $Ω$, we show that the ratio $u(x)/(\mathrm{dist}(x,\partialΩ))^{2s-1}$ is strictly positive as $x$ approaches the boundary $\partialΩ$ of $Ω$. We also prove a strong maximum principle for distributional super-solutions.

A Generalization of the Hopf's Lemma for the 1-D Moving-Boundary Problem for the Fractional Diffusion Equation and its Application to a Fractional Free-Boundary Problem

This paper deals with a theoretical mathematical analysis of a one-dimensional-moving-boundary problem for the time-fractional diffusion equation, where the time-fractional derivative of order $\al$ $\in (0,1)$ is taken in the Caputo's sense. A generalization of the Hopf's lemma is proved, and then this result is used to prove a monotonicity property for the free-boundary when a fractional free-boundary Stefan problem is considered.

双重散度形式椭圆方程的Hopf-Oleinik引理

Hopf-Oleinik lemma for elliptic equations in double divergence form

本文首次为双重散度形式的均匀椭圆方程建立了Zaremba-Hopf-Oleinik型边界点引理，并据此在$C^{1,α}$区域中得到了非散度形式二阶椭圆方程Green函数的精确双边估计。

Orlicz-Sobolev空间中分数阶拉普拉斯算子解的Hopf引理与边界行为

Hopf's lemmas and boundary behaviour of solutions to the fractional Laplacian in Orlicz-Sobolev spaces

本文在Orlicz-Sobolev空间框架下，将Hopf边界引理推广到分数阶$a-$拉普拉斯算子，研究满足内部球条件的区域边界附近解的行为，并同时处理常号与变号位势问题。

关于变号解的分数阶 Hopf 引理

A fractional Hopf Lemma for sign-changing solutions

本文研究了分数阶椭圆问题解的边界行为，在不对解的符号作全局假设的条件下建立了 Hopf 引理，并证明非平凡径向解在边界处不能有无穷多个零点累积，同时给出实例证明结果精确性。

非退化稳定算子的边界正则性与Hopf引理

Boundary regularity and Hopf lemma for nondegenerate stable operators

本文针对满足外部C^{1,dini}性质的有界开集，证明了非退化稳定积分微分算子方程解的尖锐边界Hölder正则性与Hopf型边界引理，并展示了估计在s→1-时的稳健性，从而恢复二阶方程经典结果。

完全非线性椭圆方程在Reifenberg区域上的边界Lipschitz正则性与Hopf引理

Boundary Lipschitz Regularity and the Hopf Lemma on Reifenberg Domains for Fully Nonlinear Elliptic Equations

该研究针对完全非线性椭圆方程，提出统一方法证明Reifenberg区域上的边界Lipschitz正则性与Hopf引理，明确内外Reifenberg $C^{1,\mathrm{Dini}}$条件对应的成立情形，拓展了常规条件下的已有结论。

A note on Hopf's lemma and strong minimum principle for nonlocal equations with non-standard growth

Let $Ω\subset \mathbb{R}^n $ be any open set and $u$ be a weak supersolution of $\mathcal{L}u=c(x)g(|u|)\frac{u}{|u|}$ where \[\mathcal{L}u(x)=\text{p.v.} \int_{\mathbb{R}^n} g\left(\frac{|u(x)-u(y)|}{|x-y|^s}\right) \frac{u(x)-u(y)}{|u(x)-u(y)|} K(x,y)\frac{dy}{|x-y|^s}\] and $g=G^{\prime}$ for some Young function $G.$ This note imparts a Hopf's type lemma and strong minimum principle for $u$ when $c(x)$ is continuous in $\barΩ$ that extend the results of Del Pezzo and Quaas (JDE-2017) in fractional Orlicz-Sobolev setting.

关于分数拉普拉斯方程中变号超解的霍普夫引理

On Hopf's Lemma for sign-changing supersolutions to fractional Laplacian equations

本文研究了分数拉普拉斯方程中变号超解的霍普夫引理有效性，通过扩展方法将其与边界点的霍普夫引理联系起来，并改进了最近关于二元性结果的分析。

关于退化全非线性算子的定量Hopf-Oleinik引理及其在自由边界问题中的应用

A quantitative Hopf-Oleinik lemma for degenerate fully nonlinear operators and applications to free boundary problems

本文提出了一种针对退化全非线性算子的定量Hopf-Oleinik引理，并应用于自由边界问题和火焰传播模型的正则性研究。

含广义缓变分数阶$p$-Laplacian的抛物方程的Hopf引理

Hopf's lemma for parabolic equations involving a generalized tempered fractional $p$-Laplacian

本文针对含广义缓变分数阶$p$-Laplacian的非线性抛物系统，建立了相应的Hopf引理，为研究非局部抛物方程解的定性性质提供了有力工具。

完全非线性椭圆方程的边界Lipschitz正则性与Hopf引理

Boundary Lipschitz Regularity and the Hopf Lemma for Fully Nonlinear Elliptic Equations

针对完全非线性椭圆方程粘性解的边界正则性问题，提出将曲边界视为超平面扰动的统一简洁方法，证明内外C^{1,\text{Dini}}条件分别对应边界Lipschitz连续性与Hopf引理成立，并验证该条件的最优性。

混合局部-非局部型拟线性算子的全局梯度正则性与Hopf引理

Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type

本文针对p-Laplacian与分数阶(s, q)-Laplacian之和构成的混合局部-非局部拟线性算子，在边界仅为C^{1, α}且p > s q的条件下，证明了弱解到边界的C^{1, θ}正则性，并建立了正上解的Hopf型引理。

On the Hopf Lemma

The Hopf Lemma for second order elliptic operators is proved to hold in domains with $C^{1,α}$, and even less regular, boundaries. It need not hold for $C^1$ boundaries. Corresponding results are proved for second order parabolic operators.

Symmetry of hypersurfaces and the Hopf Lemma

A classical theorem of A.D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to some results on symmetry properties of hypersurfaces with ordered mean curvature and associated variations of the Hopf Lemma. Some open problems will be discussed.

Partial results on extending the Hopf Lemma

In [1], Theorem 3, the authors proved, in one dimension, a generalization of the Hopf Lemma, and the question arose if it could be extended to higher dimensions. In this paper we present two conjectures as possible extensions, and give a very partial answer. We write this paper to call attention to the problem.

On the maximum principles and the quantitative version of the Hopf lemma for uniformly elliptic integro-differential operators

In the present paper we prove estimates on {subsolutions of the equation $-Av+c(x)v=0$}, $x\in D$, where $D\subset \bbR^d$ is a domain (i.e. an open and connected set) and $A$ is an integro-differential operator of the Waldenfels type, whose differential part satisfies the uniform ellipticity condition on compact sets. In general, the coefficients of the operator need not be continuous but only bounded and Borel measurable. Some of our results may be considered "quantitative" versions of the Hopf lemma, as they provide the lower bound on the outward normal directional derivative at the maximum…

The Hopf Boundary Point Lemma for Vector Bundle Sections

The paper establishes a version of the Hopf boundary point lemma for sections of a vector bundle over a manifold with boundary. This result may be viewed as a counterpart to the tensor maximum principle obtained by R. Hamilton in 1986. Potential applications include the study of various geometric flows and the construction of invariant sets for geometric boundary value problems.

Hopf's lemma for viscosity solutions to a class of non-local equations with applications

We consider a large family of integro-differential equations and establish a non-local counterpart of Hopf's lemma, directly expressed in terms of the symbol of the operator. As closely related problems, we also obtain a variety of maximum principles for viscosity solutions. In our approach we combine direct analysis with functional integration, allowing a robust control around the boundary of the domain, and make use of the related ascending ladder height-processes. We then apply these results to a study of principal eigenvalue problems, the radial symmetry of the positive solutions, and the …

Schrödinger算子的Hopf引理

The Hopf lemma for the Schrödinger operator

本文证明了在势能仅属于 $L_{\mathrm{loc}}^1(Ω)$ 的非负条件下，涉及Schrödinger算子的Dirichlet问题解的Hopf边界点引理，并利用Hopf引理失效的集合刻画了使相应边值问题有解的非负有限Borel测度。

A geometric problem and the Hopf Lemma. II

A classical result of A.D. Alexandrov states that a connected compact smooth $n-$dimensional manifold without boundary, embedded in $\Bbb R^{n+1}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of $M$ in a hyperplane $X_{n+1}=$constant in case $M$ satisfies: for any two points $(X', X_{n+1})$, $(X', \hat X_{n+1})$ on $M$, with $X_{n+1}>\hat X_{n+1}$, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the H…

Inhomogeneous Hopf-Oleĭnik Lemma and Applications. Part IV: Sharp Krylov Boundary Gradient Type Estimates for Solutions to Fully Nonlinear Differential Inequalities with unbounded coefficients and $C^{1,Dini}$ boundary data

In this paper we provide another application of the Inhomogeneous Hopf-Ole\uınik Lemma (IHOL) proved in \cite{BM-IHOL-PartI} or \cite{Boyan-2}. As a matter of fact, we also provide a new and simpler proof of a slightly weaker version IHOL for the uniformly elliptic fully nonlinear case which is sufficient for most purposes. The paper has essentially two parts. In the first part, we use IHOL for unbounded RHS to develop a Caffarelli's "Lipschitz implies $C^{1,α}$" approach to prove Ladyzhenskaya-Uraltseva boundary gradient type estimates for functions in $S^{*}(γ, f)$ that vanishes on the bound…

具有非负散度奇异对流线性椭圆问题中 Hopf-Oleinik 引理的失效

Failure of the Hopf-Oleinik lemma for a linear elliptic problem with singular convection of non-negative divergence

本文研究了具有非负散度奇异对流的线性椭圆 Dirichlet 问题，将解的存在性理论推广至 $|E| \in L^2$ 并证明了比较原理，同时揭示此类奇异漂移项会导致 Hopf-Oleinik 引理失效。

Boundary weak Harnack estimates and regularity for elliptic PDE in divergence form

We obtain a global extension of the classical weak Harnack inequality which extends and quantifies the Hopf-Oleinik boundary-point lemma, for uniformly elliptic equations in divergence form. Among the consequences is a boundary gradient estimate, due to Krylov and well-studied for non-divergence form equations, but completely novel in the divergence framework. Another consequence is a new more general version of the Hopf-Oleinik lemma.

A miscellany

We present several results, including some remarks on the Hopf Lemma.

Some remarks on singular solutions of nonlinear elliptic equations. III: viscosity solutions, including parabolic operators

The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.

非局部超定Serrin问题的定量稳定性

Quantitative stability for the nonlocal overdetermined Serrin problem

该研究通过移动平面法构造新反对称障碍，建立非局部Serrin超定问题的定量稳定性，得到优于经典拉普拉斯情形的稳定性估计及相关非局部引理，工具具独立研究价值。

A boundary point lemma for Black-Scholes type operators

We prove a sharp version of the Hopf boundary point lemma for Black-Scholes type equations. We also investigate the existence and the regularity of the spatial derivative of the solutions at the spatial boundary.

On the Boundary Point Principle for divergence-type equations

We provide some versions of the Zaremba-Hopf-Oleinik boundary point lemma for general elliptic and parabolic equations in divergence form under the sharp requirements on the coefficients of equations and on the boundaries of domains.

Two geometric lemmas for $S^{N-1}$-valued maps and an application to the homogenization of spin systems

We prove two geometric lemmas for $S^{N-1}$-valued functions that allow to modify sequences of lattice spin functions on a small percentage of nodes during a discrete-to-continuum process so as to have a fixed average. This is used to simplify known formulas for the homogenization of spin systems.

A Schwarz-type lemma for noncompact manifolds with boundary and geometric applications

We prove a Schwarz-type lemma for noncompact manifolds with possibly noncompact boundary. The result is a consequence of a suitable form of the weak maximum principle of independent interest. The paper is enriched with applications to conformal deformations of noncompact manifolds with boundary, among them a generalization of a classical result by Escobar.

反对称函数在非局部方程中的作用

The role of antisymmetric functions in nonlocal equations

本文建立带零阶项分数阶 Laplace 算子反对称超解的 Hopf 型引理，并结合移动平面法证明半线性分数阶平行曲面问题的径向对称性与区域为球，同时讨论相关最大值原理、Harnack 不等式及局部假设下的反例。

Strong Comparison Principles for Some Nonlinear Degenerate Elliptic Equations

In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form $\nabla^2 ψ+ L(x,\nabla ψ)$, including the conformal hessian operator.

正则性结构的几何嵌入

Geometric embedding for regularity structures

该研究通过建立变形Butcher-Connes-Kreimer霍普夫代数与shuffle霍普夫代数商的对应，将正则性结构中的特定模型视为几何粗糙路径，拓展了两类霍普夫代数的同构关系，依赖正则性结构中的形变形式与post-Lie结构。

The splitting lemmas for nonsmooth functionals on Hilbert spaces I

The Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least $C^2$-smooth functionals. It obstructs one using Morse theory to study most of variational problems of form $F(u)=\int_Ωf(x, u,..., D^mu)dx$ as in (\ref{e:1.1}). In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower than $C^1$) on a Hilbert space $H$ which have higher smoothness (but lower…