Chenchen Mou
This paper is concerned with existence of viscosity solutions of non-translation invariant nonlocal fully nonlinear equations. We construct a discontinuous viscosity solution of such nonlocal equation by Perron's method. If the equation is uniformly elliptic in the sense of \cite{SS}, we prove the discontinuous viscosity solution is Hölder continuous and thus it is a viscosity solution.
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We study the Dirichlet problem for non-homogeneous equations involving the fractional $p$-Laplacian. We apply Perron's method and prove Wiener's resolutivity theorem.
Perron solutions and boundary regularity for nonlocal nonlinear Dirichlet problems
研究分数\(p -\)拉普拉斯型非线性算子非局部狄利克雷问题，考虑索伯列夫解与佩龙解，证明二者对应的边界正则性概念等价，给出佩龙解新定义及相关可解性、唯一性结果，揭示其在特定扰动下的不变性。
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…
Stability and Perron's method for Monge solutions of eikonal equations in metric spaces
本文研究度量空间中 eikonal 方程 Monge 解的稳定性，引入球面子斜率条件克服子斜率的不稳定性，建立无需测试函数的稳定性，还发展了适用于一般完备长度空间的 Perron 方法并得到新的存在性结果。
On an evolutionary equation involving the nonlocal infinity Laplacian
研究非局部无穷拉普拉斯算子的演化方程，在柱体区域上利用 Perron 方法证明 Dirichlet 问题解的存在性和唯一性。
We use Perron's method to construct viscosity solutions of fully nonlinear degenerate parabolic pathwise (rough) partial differential equations. This provides an intrinsic method for proving the existence of solutions that relies only on a comparison principle, rather than considering equations driven by smooth approximating paths. The result covers the case of multidimensional geometric rough path noise, where the noise coefficients depend nontrivially on space and on the gradient of the solution. Also included in this note is a discussion of the comparison principle and a summary of the path…
Two-phase problems: Perron solutions and regularity of the Neumann problem in convex cones
研究带Neumann边界条件的完全非线性两相自由边界问题，证明Caffarelli的内部正则性理论可扩展至凸域边界，并建立由不同完全非线性算子驱动的Dirichlet两相问题的一般存在性定理。
In this paper, we prove existence results of a one-dimensional periodic solution to equations with the fractional Laplacian of order $s\in(1/2,1)$, singular nonlinearity, and gradient term under various situations, including nonlocal contra-part of classical Lienard vector equations, as well other nonlocal versions of classical results know only in the context of second-order ODE. Our proofs are based on degree theory and Perron's method, so before that we need to establish a variety of priori estimates under different assumptions on the nonlinearities appearing in the equations. Besides, we o…
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…
Nonlocal Hamilton-Jacobi Equations on a network with Kirchhoff type conditions
本文研究网络上具有Kirchhoff型条件的非局部Hamilton-Jacobi方程，允许非局部项跨多条边贡献，通过消失粘性法和Perron方法证明Lipschitz解的存在性，并建立通量受限解理论，推广到一般网络。
We show that any viscosity solution to a general fully nonlinear nonlocal elliptic equation can be approximated by smooth ($C^\infty$) solutions.
Holder regularity for fully nonlinear nonlocal equations
本综述证明带局部梯度项的完全非线性非局部一致椭圆方程的粘性解具有Hölder正则性，推广了Barles等人2011年的结果至极值非局部算子。
This paper is concerned with interior regularity of viscosity solutions of non-translation invariant nonlocal fully nonlinear equations with Dini continuous terms. We obtain $C^σ$ regularity estimates for the nonlocal equations by perturbative methods and a version of a recursive Evans-Krylov theorem.
In this paper, we establish pointwise Schauder estimates for solutions of nonlocal fully nonlinear elliptic equations by perturbative arguments. A key ingredient is a recursive Evans-Krylov theorem for nonlocal fully nonlinear translation invariant equations.
In this paper we prove the existence of smooth solutions to fully nonlinear and nonlocal parabolic equations with critical index. The proof relies on the apriori Hölder estimate for advection fractional-diffusion equation established by Silvestre \cite{Si2}.
We obtain Dini type estimates for a class of concave fully nonlinear nonlocal elliptic equations of order $σ\in (0,2)$ with rough and non-symmetric kernels. The proof is based on a novel application of Campanato's approach and a refined $C^{σ+α}$ estimate in [8].
We prove the existence and $C^{1,α}$ regularity of solutions to nonlocal fully nonlinear elliptic double obstacle problems. We also obtain boundary regularity for these problems. The obstacles are assumed to be Lipschitz semi-concave/semi-convex functions, and we do not require them to be $C^1$. Our approach is to adapt a penalization method to be applicable to the setting of nonlocal equations and their viscosity solutions.
We establish sharp $C^{2s}$ interior regularity estimates for solutions of fully nonlinear nonlocal equations with bounded right hand side. More precisely, we show that if $I$ is a fully nonlinear nonlocal concave or convex elliptic operator and $f\in L^\infty(B_1)$ then \[ Iu=f\quad\textrm{ in }\quad B_1 \quad \Rightarrow\quad u\in C^{2s}(B_{1/2}). \] This result generalizes the linear counterpart proved by Ros-Oton and Serra and extends previous available results for fully nonlinear nonlocal operators. As an application, we get a basic regularity estimate for the nonlocal two membranes probl…
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…
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…
Regularity of solutions to variable-exponent degenerate mixed fully nonlinear local and nonlocal equations
针对沿梯度零点退化的变指数混合完全非线性局部—非局部椭圆方程，本文结合 Ishii–Lions 黏性方法、消失黏性法与改进平坦性技术，证明解的存在性、Lipschitz 及内部 $C^{1,\delta}$ 正则性。
Hölder Continuity for Fully Fractional Parabolic Equations with Space-time Nonlocal Operators
本文研究含空间分数阶扩散与Marchaud型分数阶时间导数的全分数阶抛物型方程，在不假设解和非齐次项有界的条件下，引入时间依赖非局部尾项，利用非局部De Giorgi迭代技术证明弱解的局部有界性与局部Hölder连续性。
Initial-boundary value problems for second order fully nonlinear PDEs with Caputo time fractional derivatives of order less than one are considered in the framework of viscosity solution theory. Associated boundary conditions are Dirichlet and Neumann, and they are considered in the strong sense and the viscosity sense, respectively. By a comparison principle and Perron's method, unique existence for the Cauchy-Dirichlet and Cauchy-Neumann problems are proved.
We obtain Schauder estimates for a class of concave fully nonlinear nonlocal parabolic equations of order $σ\in (0,2)$ with rough and non-symmetric kernels. As a application, we prove that the solution to a translation invariant equation with merely bounded data is $C^σ$ in $x$ variable and $Λ^1$ in $t$ variable, where $Λ^1$ is the Zygmund space.
This paper is concerned with a general class of fully nonlinear parabolic equations with monotone nonlocal terms. We investigate the quasiconvexity preserving property of positive, spatially coercive viscosity solutions. We prove that if the initial value is quasiconvex, the viscosity solution to the Cauchy problem stays quasiconvex in space for all time. Our proof can be regarded as a limit version of that for power convexity preservation as the exponent tends to infinity. We also present several concrete examples to show applications of our result.
We show that a certain class of fully nonlinear nonlocal equations have smooth solutions as long as the right-hand side is nice and the boundary datum is bounded. To this end we follow the classical strategy. We first show that solutions are $C^{σ+α}$ continuous, then develop a bootstrap argument robust enough for operators that may depend on the solution and its σ-derivatives.
Nonlinear fractional equations in the Heisenberg group
该研究针对海森堡群中以分数阶次拉普拉斯为原型的$(s,p)$阶积分-微分算子主导的非线性非局部方程，给出弱解的有界性与边界正则性估计，引入非局部Perron方法并列出相关开放问题。
We establish $C^{σ+α}$ interior estimates for concave nonlocal fully nonlinear equations of order $σ\in(0,2)$ with rough kernels. Namely, we prove that if $u\in C^α(\mathbb R^n)$ solves in $B_1$ a concave translation invariant equation with kernels in $\mathcal L_0(σ)$, then $u$ belongs to $C^{σ+α}(\overline{ B_{1/2}})$, with an estimate. More generally, our results allow the equation to depend on $x$ in a $C^α$ fashion. Our method of proof combines a Liouville theorem and a blow-up (compactness) procedure. Due to its flexibility, the same method can be useful in different regularity proofs fo…
Spectral gap for the growth-fragmentation equation via Harris's Theorem
本文利用哈里斯定理并结合h-变换与对偶Perron特征值问题，在推广条件下证明了生长-碎裂方程谱隙的存在性及指数收敛，并给出可量化的收敛速率。
Optimal Regularity for Fully Nonlinear Nonlocal Equations with Unbounded Source Terms
本文研究带无界源项的完全非线性非局部方程，利用逼近技术与 Liouville 型论证，依据源项的 $L^p$ 可积性建立粘性解在临界阈值下的最优正则性估计。
We obtain Dini and Schauder type estimates for concave fully nonlinear nonlocal parabolic equations of order $σ\in (0,2)$ with rough and non-symmetric kernels, and drift terms. We also study such linear equations with only measurable coefficients in the time variable, and obtain Dini type estimates in the spacial variable. This is a continuation of the work [10, 11] by the first and last authors.
Existence and regularity in the fully nonlinear one-phase free boundary problem
本文研究一般完全非线性算子及法向量依赖自由边界条件下单相问题的粘性解，用Perron方法证明存在性，通过平坦度二次改进和高度图变换分别得到$C^{2,α}$及更高正则性。
Regularity theory for degenerate fully nonlinear nonlocal equations with a Hamiltonian term
针对带哈密顿项的退化完全非线性非局部椭圆方程，通过刻画退化规律与哈密顿项的作用关系，结合Ishii-Lions方法与扰动技术，建立解的Lipschitz正则性、梯度Hölder连续性并探究其$C^1$可微性。
Boundedness through nonlocal dampening effects in a fully parabolic chemotaxis model with sub and superquadratic growth
本文研究了包含次和超二次增长外部源及非局部阻尼反应的趋化模型，证明了在有界不可渗透区域中，只要阻尼效应足够强，细胞就不会随时间出现聚集效应。
Inverse coefficient problem for a fully fractional diffusion equation with nonlinear and source nonlocal initial condition
本文研究了完全分数扩散方程中非线性及源非局部初始条件的反系数问题，通过傅里叶方法和固定点论证证明了解的适定性。
Fine boundary regularity for fully nonlinear mixed local-nonlocal problems
本文针对完全非线性混合局部-非局部非平移不变算子的Dirichlet问题，通过构造次解和超解并结合Harnack不等式，证明了粘性解的全局Lipschitz正则性及梯度直到边界的Hölder正则性。
In this paper we consider the system involving fully nonlinear nonlocal operators: $$ \left\{ \begin{array}{ll} F_α(u(x)) = C_{n,α} PV \int_{{R}^n} \frac{G(u(x)-u(y))}{|x-y|^{n+α}} dy=f(v(x)), F_β(v(x)) = C_{n,β} PV \int_{{R}^n} \frac{G(v(x)-v(y))}{|x-y|^{n+β}} dy=g(u(x)). \end{array} \right. $$ A \textit{narrow region principle} and a \textit{decay at infinity} for the system for carrying on the method of moving planes are established. Then we prove the radial symmetry and monotonicity for positive solutions to the nonlinear system in the whole space. Non-existence of positive solutions to th…
On fully nonlinear parabolic mean field games with nonlocal and local diffusions
研究定义在 $[0,T]\times\mathbb{R}^d$ 的全非线性平均场博弈，提出涉及随机过程时间变化率的新控制解释，证明了解的存在性与唯一性，并应用于非退化方程。
We prove space and time regularity for solutions of fully nonlinear parabolic integro-differential equations with rough kernels. We consider parabolic equations $u_t = \I u$, where $\I$ is translation invariant and elliptic with respect to the class $\mathcal L_0(σ)$ of Caffarelli and Silvestre, $σ\in(0,2)$ being the order of $\I$. We prove that if $u$ is a viscosity solution in $B_1 \times (-1,0]$ which is merely bounded in $\R^n \times (-1,0]$, then $u$ is $C^β$ in space and $C^{β/σ}$ in time in $\overline{B_{1/2}} \times [-1/2,0]$, for all $β< \min\{σ, 1+α\}$, where $α>0$. Our proof combine…
On a family of fully nonlinear integro-differential operators: From fractional Laplacian to nonlocal Monge-Ampère
本文提出了一类介于分数阶拉普拉斯算子和非局部 Monge-Ampère 之间的完全非线性积分微分算子，利用重排技术给出了其表示公式与最优传输的联系，并证明了全空间中全局 Poisson 问题解的存在性、唯一性与 $C^{1,1}$-正则性。
Strong maximum principle for fully nonlinear nonlocal problems
本文研究了非局部问题的强最大原理，通过引入新的非局部假设和 Liouville 定理，探讨了死核心的形成与解的性质。
In this paper, we prove the Evans-Krylov theorem for nonlocal parabolic fully nonlinear equations.
We study the large-time behavior in all $L^p$ norms and in different space-time scales of solutions to a nonlocal heat equation in $\mathbb{R}^N$ involving a Caputo $α$-time derivative and a power of the Laplacian $(-Δ)^s$, $s\in (0,1)$, extending recent results by the authors for the case $s=1$. The initial data are assumed to be integrable, and, when required, to be also in $L^p$. The main novelty with respect to the case $s=1$ comes from the behaviour in fast scales, for which, thanks to the fat tails of the fundamental solution of the equation, we are able to give results that are not avai…
A strongly degenerate fully nonlinear mean field game with nonlocal diffusion
本文研究同时具有完全非线性和强退化扩散的MFG系统，利用先前建立的适定性理论首次在退化情形下证明其存在唯一性，关键难点在于处理低阶非局部Lévy扩散下退化非光滑Fokker-Planck方程的唯一性。
In this paper, we propose quasilinearization methods that convert nonlocal fully-nonlinear parabolic systems into the nonlocal quasilinear parabolic systems. The nonlocal parabolic systems serve as important mathematical tools for modelling the subgame perfect equilibrium solutions to time-inconsistent dynamic choice problems, which are motivated by the study of behavioral economics. Different types of nonlocal parabolic systems were studied but left behind the fully-nonlinear case and the connections among them. This paper shows the equivalence in solvability between nonlocal fully-nonlinear …
Nonlinear dynamics of reaction-diffusion wave trains under large and fully nonlocalized modulations
本文研究了实线上反应扩散系统周期波列在完全非局部大调制下的动力学，通过扩展 $L^\infty$-稳定性理论消除了局部化要求，证明了解在增强扩散速率下收敛于由粘性 Hamilton-Jacobi 方程支配的调制波列，实现了全局稳定性。
Generalized Evans--Krylov and Schauder type estimates for nonlocal fully nonlinear equations with rough kernels of variable orders
该研究针对带变阶粗糙核的非局部完全非线性椭圆方程，引入描述可变可微阶数的函数与广义Hölder空间，建立了推广标准结果的广义Evans--Krylov型与Schauder型$C^{\phi\psi}$估计。
Regularity for a class of degenerate fully nonlinear nonlocal elliptic equations
本文研究一类在解梯度为零时退化的完全非线性积分微分方程，利用紧致性和扰动论证，根据不同的分数阶扩散阶数完整刻画了粘性解的正则性。
In this paper we prove a nonlocal version of the Cordes-Niremberg estimates. We use it to extend our previous regularity results for fully nonlinear integro-differential equations to the variable coefficient case and several other settings where the equation can be thought of as a perturbation of the translation invariant case.
Partial Hölder regularity for fully nonlinear nonlocal parabolic equations with integrable kernels
本文研究了具有可积核的非线性非局部抛物方程的局部Hölder正则性，通过截断核的处理，得出了稳健的正则性估计。
Nonlinear stability of periodic wave trains in the FitzHugh-Nagumo system against fully nonlocalized perturbations
本文针对FitzHugh-Nagumo系统中周期波列对完全非局域扰动的稳定性问题，提出一种适用于一般耗散半线性问题的新方案，通过逆Laplace表示揭示高频阻尼，并将非线性阻尼估计扩展到非局域扰动，从而克服了缺乏抛物性和局域化带来的困难。
We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form $$f(λ(D^2u))=g(x),$$ with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli--Nirenberg--Spruck \cite{Caffarelli1985}, Trudinger \cite{Trudinger1995} and many others, and there had been significant discussions on the solvability of the classical Dirichlet problem via the continuity method, under the assumption that $f$ is a concave function. In this paper, based on the Perron's method, we establish an exterior existence and uniquen…
We prove the existence of a unique viscosity solution to certain systems of fully nonlinear parabolic partial differential equations with interconnected obstacles in the setting of Neumann boundary conditions. The method of proof builds on the classical viscosity solution technique adapted to the setting of interconnected obstacles and construction of explicit viscosity sub- and supersolutions as bounds for Perron's method. Our motivation stems from so called optimal switching problems on bounded domains.
Mixed fully nonlinear local and nonlocal elliptic operators in Heisenberg group
本文研究了在海森堡群中混合全非线性局部和非局部椭圆算子的粘性解的存在性、正则性和比较原理。
We prove that the fractional derivatives of solutions to a class of nonlocal fully nonlinear elliptic equations are epsilon-integrable. We follow Fanghua Lin's original approach to the analogous problem for second order equations, by first proving a potential estimate, then combining this estimate with the ABP-type estimate by N. Guillen and R. Schwab to control the size of the superlevel sets of the fractional derivatives of solutions.
We prove a $C^{1,α}$ interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity of a special class of nonlinear operators.
We study the Cauchy problem for a nonlocal heat equation, which is of fractional order both in space and time. We prove four main theorems: (i) a representation formula for classical solutions, (ii) a quantitative decay rate at which the solution tends to the fundamental solution, (iii) optimal $L^2$-decay of mild solutions in all dimensions, (iv) $L^2$-decay of weak solutions via energy methods. The first result relies on a delicate analysis of the definition of classical solutions. After proving the representation formula we carefully analyze the integral representation to obtain the quantit…
We study existence, uniqueness and boundary blow-up profile for fractional harmonic functions on a bounded smooth domain $Ω\subset \mathbb R^N$. We deal with harmonic functions associated to uniformly elliptic, fully nonlinear nonlocal operators, including the linear case $$ (-Δ)^s u = 0 \quad \mbox{in} \ Ω, $$ where $(-Δ)^s$ denotes the fractional Laplacian of order $2s \in (0,2)$. We use the viscosity solution's theory and Perron's method to construct harmonic functions with zero exterior condition in $\bar Ω^c$, and boundary blow-up profile $$ \lim_{x\to x_0, x \in Ω}\mathrm{dist}(x, \parti…
We prove the well-posedness results, i.e. existence, uniqueness, and stability, of the solutions to a class of nonlocal fully nonlinear parabolic partial differential equations (PDEs), where there is an external time parameter $t$ on top of the temporal and spatial variables $(s,y)$ and thus the problem could be considered as a flow of equations. The nonlocality comes from the dependence on the unknown function and its first- and second-order derivatives evaluated at not only the local point $(t,s,y)$ but also at the diagonal line of the time domain $(s,s,y)$. Such equations arise from time-in…
In this paper, the positive solutions to Choquard equation involving fully nonlinear nonlocal operator are shown to be symmetric and monotone by using the moving plane method which has been introduced by Chen, Li and Li in 2015. The key ingredients are to obtain the "narrow region principle" and "decay at infinity" for the corresponding problems. Similar ideas can be easily applied to various nonlocal problems with more general nonlinearities.
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