Begoña Barrios, Leandro M. Del Pezzo
We study the existence of positive supersolutions of nonlocal equations $(-Δ)^s u+ |\nabla u|^q=λf(u)$ in exterior domains where the datum $f$ can be comparade with $u^{p}$ near the origin. We prove that the existence or bounded supersolutions its depend of the values of $p$, $q$ and $s$.
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We show the existence of solution for some classes of nonlocal problems. Our proof combines the presence of sub and supersolution with the pseudomonotone operators theory.
In the present paper we study the existence of solutions for some nonlocal problems involving Orlicz-Sobolev spaces. The approach is based on sub-supersolutions.
In the present paper we study the existence of solutions for a class of nonlocal system involving the p(x)-Laplacian operator. The approach is based on a new sub-supersolution result.
We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional $p$-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi's method. Furthermore, by means of a new algebraic inequality, we show that positive weak supersolutions satisfy a reverse Hölder inequality. Finally, we also prove a logarithmic decay estimate for positive supersolutions.
Supersolutions and superharmonic functions for nonlocal operators with Orlicz growth
本文研究具有 Orlicz 增长的非局部算子对应的超解与超调和函数，建立其精细性质和相互关系，并证明非局部障碍问题的适定性及内、边界正则性估计。
We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional $p$-Laplace equations which includes the fractional parabolic $p$-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results …
On weak and viscosity solutions to a nonhomogeneous mixed local-nonlocal equation
研究有界Lipschitz域中非齐次混合局部-非局部p-Laplace方程的弱解与粘性解关系，利用比较原理证明连续弱上解是粘性上解（1<p<∞），并证明有界粘性上解是弱上解（p≥2）。
Sub-supersolutions method for singular quasilinear systems involving gradient terms
本文提出下上解方法，证明含梯度项的奇异拟线性椭圆系统解的存在性、正则性及位置，并应用于对流和吸收系统正解的存在性证明。
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.
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。
In this paper we propose a counterexample to the validity of the Comparison Principle and of the Sub and Supersolution Method for nonlocal problems like the stationary Kirchhoff Equation. This counterexample shows that in general smooth bounded domains in any dimension, these properties cannot hold true if the nonlinear nonlocal term $M(\|u\|^2)$ is somewhere increasing with respect to the $H_0^1$-norm of the solution. Comparing with existing results, this fills a gap between known conditions on $M$ that guarantee or prevent these properties, and leads to a condition which is necessary and suf…
Liouville Theorems for the Lane-Emden Equation Involving a Mixed Local-Nonlocal Operator
研究混合局部-非局部Lane-Emden方程正上解存在的条件，证明当且仅当q>n/(n-2s)时存在分布上解，并建立了基本解的渐近估计、分布意义下的最大值原理和等价积分不等式。
We investigate which elliptic PDEs that have the property that every viscosity supersolution is $W^{1,q}_{loc}(Ω)$, $Ω\subseteq\mathbb{R}^n$. The asymptotic cone of the operator's sublevel set seems to be essential. It turns out that much can be said if we know how this cone compares to the sublevel set of a certain minimal operator associated with the exponent $q$.
Regularity for Mixed Local and Nonlocal Degenerate Elliptic Equations in the Heisenberg Group
本文研究了Heisenberg群中混合局部与非局部退化椭圆方程的正则性，通过推导Caccioppoli型不等式和对数估计，证明了弱解的局部有界性、Hölder连续性及Harnack不等式。
We consider a combination of local and nonlocal $p$-Laplace equations and discuss several regularity properties of weak solutions. More precisely, we establish local boundedness of weak subsolutions, local Hölder continuity of weak solutions, Harnack inequality for weak solutions and weak Harnack inequality for weak supersolutions. We also discuss lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. Our approach is purely analytic and it is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. The…
In the present paper we study the existence of solutions for some nonlocal problems involving the p(x)-Laplacian operator. The approach is based on a new sub-supersolution method
We prove that if the elliptic problem $-Δu+b(x)|\nabla u|=c(x)u$ with $c\ge0$ has a positive supersolution in a domain $Ω$ of $ \IR^{N\ge 3}$, then $c,b$ must satisfy the inequality \[\sqrt{ \int_Ωcϕ^2}\le \sqrt{ \int_Ω| \nablaϕ|^2}+\sqrt{ \int_Ω\frac{b^2}{4}ϕ^2},~~~ϕ\in C_c^\infty(Ω).\] As an application, we obtain Liouville type theorems for positive supersolutions in exterior domains when $c(x)-\frac{b^2(x)}{4}>0$ for large $|x|$, but unlike the known results we allow the case $\liminf_{|x|\rightarrow\infty}c(x)-\frac{b^2(x)}{4}=0$. Also the weights $b$ and $c$ are allowed to be unbounded. …
For a class of quasilinear elliptic equations involving the p-Laplace operator, we develop an abstract critical point theory in the presence of sub-supersolutions. Our approach is based upon the proof of the invariance under the gradient flow of enlarged cones in the $W^{1,p}_0$ topology. With this, we prove abstract existence and multiplicity theorems in the presence of variously ordered pairs of sub-supersolutions. As an application, we provide a four solutions theorem, one of the solutions being sign-changing.
Liouville results for supersolutions of fractional $p$-Laplacian equations with gradient nonlinearities
本文研究了分数 $p$-拉普拉斯方程中含梯度非线性项的超解的利乌维尔性质，证明在特定参数条件下，非负粘性解必须为常数。
We consider mixed local and nonlocal quasilinear parabolic equations of $p$-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.
We study viscosity solutions to a system of nonlinear degenerate parabolic partial integro-differential equations with interconnected obstacles. This type of problem occurs in the context of optimal switching problems when the dynamics of the underlying state variable is described by an $n$-dimensional Levy process. We first establish a continuous dependence estimate for viscosity sub- and supersolutions to the system under mild regularity, growth and structural assumptions on the partial integro-differential operator and on the obstacles and terminal conditions. Using the continuous dependenc…
We establish existence results for a class of mixed anisotropic and nonlocal $p$-Laplace equation with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, Harnack inequality for solutions and weak Harnack inequality for supersolutions.
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。
We prove global existence, uniqueness and regularity of the mild, Lp and classical solution of a non-linear Fokker-Planck equation arising in an adaptive importance sampling method for molecular dynamics calculations. The non- linear term is related to a conditional expectation, and is thus non-local. The proof uses tools from the theory of semigroups of linear operators for the local existence result, and an a priori estimate based on a supersolution for the global existence result.
Local existence of solutions and comparison principle for initial boundary value problem with nonlocal boundary condition for a nonlinear parabolic equation with memory
本文研究带记忆的非线性抛物方程在非线性非局部边界条件下的初值问题，建立了局部极大解的存在性，证明比较原理并给出解的正性条件，进而推出唯一性定理。
In this paper we establish existence of smooth positive solutions for a singular quasilinear elliptic system involving gradient terms. The approach combines sub-supersolutions method and Schauder's fixed point theorem.
In this article, we perform an asymptotic analysis of a nonlocal reaction-diffusion equation, with a fractional laplacian as the diffusion term and with a nonlocal reaction term. Such equation models the evolutionary dynamics of a phenotypically structured population. We perform a rescaling considering large time and small effect of mutations, but still with algebraic law. We prove that asymptotically the phenotypic distribution density concentrates as a Dirac mass which evolves in time. This work extends an approach based on Hamilton-Jacobi equations with constraint, that has been developed t…
Comparison principle for general nonlocal Hamilton-Jacobi equations with superlinear gradient
该研究在测度满足Wasserstein/全变差联合连续性的弱条件下，证明了带超线性梯度的一般非局部Hamilton-Jacobi方程不连续粘性解的比较原理，并给出多类非局部算子的应用实例。
Holder regularity for nonlocal in time subdiffusion equations with general kernel
本文针对含一般弱奇异 PC 核的时间非局部次扩散方程，证明非负弱超解的弱 Harnack 不等式与弱解的 Hölder 连续性，并将已有分布阶分数阶理论扩展到混合型 Lebesgue 非齐次项情形。
Asymptotic Expansions of the Fundamental Solution for a Mixed Local-Nonlocal Operator
本文研究混合局部-非局部算子 $-\Delta+(-\Delta)^s$ 的基本解，推导其渐近展开并应用于建立相关定理与极大值原理。
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.
Multiple positive solutions to a perturbed Gelfand problem involving mixed local-nonlocal operators and singular nonlinearity
本文研究了带有混合局部-非局部算子和奇异非线性项的扰动Gelfand问题，通过子超解方法和Amann不动点定理，证明了多个正解的存在性。
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。
We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order $s\in (0,1)$ and summability growth $p>1$, whose model is the fractional $p$-Laplacian with measurable coefficients. We state and prove several results for the corresponding weak supersolutions, as comparison principles, a priori bounds, lower semicontinuity, and many others. We then discuss the good definition of $(s,p)$-superharmonic functions, by also proving some related properties. We finally introduce the nonlocal counterpart of the celebrated Perron method…
Semiregular and strongly irregular boundary points for nonlocal Dirichlet problems
本文研究分数阶(s,p)-Laplacian类型非局部非线性方程，利用Kellogg性质和V^{s,p}空间解的可去性结果，将Dirichlet问题的非正则边界点划分为半正则与强非正则两类并予以刻画。
We study a nonlinear, nonlocal Dirichlet problem driven by the degenerate fractional p-Laplacian via a combination of topological methods (degree theory for operators of monotone type) and variational methods (critical point theory). We assume local conditions ensuring the existence of sub- and supersolutions. So we prove existence of two positive solutions, in both the coercive and noncoercive cases.
In this paper we deal with the well-posedness of Dirichlet problems associated to nonlocal Hamilton-Jacobi parabolic equations in a bounded, smooth domain $Ω$, in the case when the classical boundary condition may be lost. We address the problem for both coercive and noncoercive Hamiltonians: for coercive Hamiltonians, our results rely more on the regularity properties of the solutions, while noncoercive case are related to optimal control problems and the arguments are based on a careful study of the dynamics near the boundary of the domain. Comparison principles for bounded sub and supersolu…
Liouville results for semilinear integral equations with conical diffusion
本文研究锥形区域上半线性非局部方程正上解的不存在性，证明在次临界指数范围内零解是唯一非负上解，并给出临界指数分界及全空间推广。
We show that solutions of time-dependent degenerate parabolic equations with super-quadratic growth in the gradient variable and possibly unbounded right-hand side are locally ${\mathcal C}^{0,α}$. Unlike the existing (and more involved) proofs for equations with bounded right-hand side, our arguments rely on constructions of sub- and supersolutions combined with improvement of oscillation techniques.
On the regularity theory for mixed local and nonlocal weighted quasilinear elliptic equations
该研究针对带Muckenhoupt权的混合局部与非局部退化p-拉普拉斯方程，建立了首个系统的局部正则性理论，扩展了相关现有理论框架。
We obtain existence, uniqueness, and stability results for the modified 1-homogeneous infinity Laplace equation \[ -Δ_\infty u - β|Du| = f, \] subject to Dirichlet or mixed Dirichlet-Neumann boundary conditions. Our arguments rely on comparing solutions of the PDE to subsolutions and supersolutions of a certain finite difference approximation.
In this article, we prove the existence of at least three positive solutions for the following nonlocal singular problem \begin{equation*} (P_\la)\left\{ \begin{split} (-\De)^su &= \la\frac{f(u)}{u^q}, \; \; u>0 \;\; \text{in}\;\; \Om,\\ u &= 0\;\; \text{in}\;\; \mb R^n \setminus \Om \end{split} \right. \end{equation*} where $(-\De)^s$ denotes the fractional Laplace operator for $s\in (0,1)$, $n>2s$, $q \in (0,1)$, $\la>0$ and $\Om$ is smooth bounded domain in $\mb R^n$. Here $f :[0,\infty) \to [0,\infty)$ is a continuous nondecreasing map satisfying $\lim\limits_{u\to \infty}\frac{f(u)}{u^{q+…
This paper mainly focus on the front-like entire solution of a classical nonlocal dispersal equation with ignition nonlinearity. Especially, the dispersal kernel function $J$ may not be symmetric here. The asymmetry of $J$ has a great influence on the profile of the traveling waves and the sign of the wave speeds, which further makes the properties of the entire solution more diverse. We first investigate the asymptotic behavior of the traveling wave solutions since it plays an essential role in obtaining the front-like entire solution. Due to the impact of $f'(0)=0$, we can no longer use the …
Regularity of weak solutions for mixed local and nonlocal double phase parabolic equations
本文研究了混合局部与非局部双相抛物型方程，利用 De Giorgi-Nash-Moser 迭代证明了弱解的局部有界性，并基于能量估计和 De Giorgi 型引理讨论了弱上下解的点态行为与半连续性。
In this paper we study the existence of solution for the following class of nonlocal problems \[ L_0u =f(x,u)+g(x) , \ \mbox{in} \ Ω, \] where $Ω\subset \mathbb{R}^{N}$, $N\geq 1$, is a bounded connected open, $g \in C(\overlineΩ)$, $f:\overlineΩ \times \mathbb{R} \to \mathbb{R}$ are function, and $L_0 : C(\overlineΩ) \to C(\overlineΩ)$ is a nonlocal dispersal operator. Using a sub-supersolution method and the degree theory for $γ$-Condensing maps, we have obtained a result of the Ambrosetti-Prodi type, that is, we obtain a necessary condition on $g$ for the non-existence of solutions, the exi…
We are concerned with the well-posedness of Neumann boundary value problems for nonlocal Hamilton-Jacobi equations related to jump processes in general smooth domains. We consider a nonlocal diffusive term of censored type of order less than 1 and Hamiltonians both in coercive form and in noncoercive Bellman form, whose growth in the gradient make them the leading term in the equation. We prove a comparison principle for bounded sub-and supersolutions in the context of viscosity solutions with generalized boundary conditions, and consequently by Perron's method we get the existence and uniquen…
Nonlocal gradient operators are prototypical nonlocal differential operators thatare very important in the studies of nonlocal models. One of the simplest variational settings for such studies is the nonlocal Dirichlet energies wherein the energy densities are quadratic in the nonlocal gradients. There have been earlier studies to illuminate the link between the coercivity of the Dirichlet energies and the interaction strengths of radially symmetric kernels that constitute nonlocal gradient operators in the form of integral operators. In this work we adopt a different perspective and focus on …
The large time behaviour of nonnegative solutions to a quasilinear degenerate diffusion equation with a source term depending solely on the gradient is investigated. After a suitable rescaling of time, convergence to a unique profile is shown for global solutions. The proof relies on the half-relaxed limits technique within the theory of viscosity solutions and on the construction of suitable supersolutions and barrier functions to obtain optimal temporal decay rates and boundary estimates. Blowup of weak solutions is also studied.
Minimizers of functionals of mixed local and nonlocal type are locally $C^{1,β}$-regular.
A general theory of nonlocal elasticity based on nonlocal gradients and connections with Eringen's model
本文提出基于非局部梯度的非局部弹性理论，证明其包含Eringen模型，并在两种情况下展示非局部解向经典局部解收敛。
We consider a Hilbertian and a charges approach to fractional gradient constraint problems of the type $|D^σu|\leq g$, involving the distributional fractional Riesz gradient $D^σ$, $0<σ<1$, extending previous results on the existence of solutions and Lagrange multipliers of these nonlocal problems. We also prove their convergence as $σ\nearrow1$ towards their local counterparts with the gradient constraint $|D u|\leq g$.
We obtain Grönwall type estimates for the gradient of the harmonic functions for a Lévy operator with order strictly larger than 1 and minimal assumptions of its Lévy measure.
$Γ$-convergence involving nonlocal gradients with varying horizon: Recovery of local and fractional models
本文研究非局部超弹性模型的渐近分析，通过对核函数进行各向同性缩放，证明了在范围消失与发散两种极限状态下，非局部梯度分别收敛于经典梯度与Riesz分数阶梯度，并推导出拟凸积分泛函向局部与分数阶模型的$Γ$-收敛。
Gradient estimates for parabolic nonlinear nonlocal equations
本文建立了一类抛物型非线性非局部测度数据问题解的逐点梯度估计，并证明其解的一阶正则性及齐次方程解的Hölder连续空间梯度。
Non-constant functions with zero nonlocal gradient and their role in nonlocal Neumann-type problems
本文研究了有限视域分数阶梯度消失的函数，证明其构成无限维向量空间，并利用伪微分算子理论刻画其特征，进而提出非局部Poincaré不等式等工具，用于研究非局部Neumann型问题的适定性与局部极限。
Nonlinear nonlocal potential theory at the gradient level
本文研究非线性非局部方程的梯度正则性，证明低阶齐次方程弱解梯度 Hölder 连续，并在一般测度数据下建立阶数大于一方程的精确高阶可微性与逐点梯度位势估计。
This paper deals with the periodic homogenization of nonlocal parabolic Hamilton-Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different limit problems. To prove the locally uniform convergence to the unique solution of the Cauchy problem for the effective equation we need a new comparison principle among viscosity semi-solutions of integro-differential equations that can be of independent interest.
We study families of porous medium equation with nonlocal pressure. We construct their weak solutions via JKO schemes for modified Wasserstein distances. We also establish the regularization effect and decay estimates for the $L^p$ norms.
We consider the class of measurable functions defined in all of $\mathbb{R}^n$ that give rise to a nonlocal minimal graph over a ball of $\mathbb{R}^n$. We establish that the gradient of any such function is bounded in the interior of the ball by a power of its oscillation. This estimate, together with previously known results, leads to the $C^\infty$ regularity of the function in the ball. While the smoothness of nonlocal minimal graphs was known for $n = 1, 2$ (but without a quantitative bound), in higher dimensions only their continuity had been established. To prove the gradient bound, we …
Compactness results for a Dirichlet energy of nonlocal gradient with applications
本文证明了半空间非局部梯度有限Dirichlet能量函数空间的两个紧性结果，给出了核函数序列的充分条件保证渐近紧嵌入，并应用于证明一致Poincaré型不等式及非局部扩散与最优控制问题的收敛性。
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