Begoña Barrios Barrera, Alessio Figalli, Enrico Valdinoci
We prove that $C^{1,α}$ $s$-minimal surfaces are automatically $C^\infty$. For this, we develop a new bootstrap regularity theory for solutions of integro-differential equations of very general type, which we believe is of independent interest.
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We obtain Schauder estimates for a general class of linear integro-differential equations. The estimates are applied to a scalar non-local Burgers equation and complete the global well-posedness results obtained in \cite{ISV}.
In this paper, we investigate the existence of nonnegative solutions for the problem $$ -\mathcal{L}_{K}u+V(x)u=f(u) $$ in $\mathbb R^n$, where $-\mathcal{L}_{K}$ is a integro-differential operator with measurable kernel $K$ and $V$ is a continuous potential. Under apropriate hypothesis, we prove, using variational methods, that the above equation has solution.
Fredholm Property of Nonlocal Problems for Integro-Differential Hyperbolic Systems
本文研究了非局部时间周期边值问题的Fredholm性质，证明了在特定正则性条件下，此类问题在连续和时间周期函数空间中满足Fredholm替代定理。
This paper deals with the existence and uniqueness of solutions for a nonlinear boundary value problem involving a sequential $ψ$-Hilfer fractional integro-differential equations with nonlocal boundary conditions. The existence and uniqueness of solutions are established for the considered problem by using the Banach contraction principle, Sadovski's fixed point theorem, and Krasnoselskii-Schaefer fixed point theorem due to Burton and Kirk. In addition, the Ulam-Hyers stability of solutions is discussed. Finally, the obtained results are illustrated by examples.
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。
In this paper, we investigate the existence of nontrivial weak solutions to a class of elliptic equations ($\mathscr{P}$) involving a general nonlocal integrodifferential operator $\mathscr{L}_{\mathcal{A}K}$, two real parameters, and two weight functions, which can be sign-changing. Considering different situations concerning the growth of the nonlinearities involved in the problem ($\mathscr{P}$), we prove the existence of two nontrivial distinct solutions and the existence of a continuous family of eigenvalues. The proofs of the main results are based on ground state solutions using the Neh…
We establish new Hoelder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii-Lions's method. We thus extend the Hoelder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the local and nonlocal term, but their overall behavior is driven by the local-nonlocal interaction, e.g. the …
In this paper we are interested in integro-differential elliptic and parabolic equations involving nonlocal operators with order less than one, and a gradient term whose coercivity growth makes it the leading term in the equation. We obtain Lipschitz regularity results for the associated stationary Dirichlet problem in the case when the nonlocality of the operator is confined to the domain, feature which is known in the literature as censored nonlocality. As an application of this result, we obtain strong comparison principles which allow us to prove the well-posedness of both the stationary a…
Unraveling Forward and Backward Source Problems for a Nonlocal Integrodifferential Equation: A Journey through Operational Calculus for Dzherbashian-Nersesian Operator
本文为Dzherbashian-Nersesian算子建立Mikusiński型运算微积，并据此求解带非局部边界条件的正演与反演源问题，给出基于Mittag-Leffler型函数的精确解及存在唯一性条件。
We study integrodifferential operators and regularity estimates for solutions to integrodifferential equations. Our emphasis is on kernels with a critically low singularity which does not allow for standard scaling. For example, we treat operators that have a logarithmic order of differentiability. For corresponding equations we prove a growth lemma and derive a priori estimates. We derive these estimates by classical methods developed for partial differential operators. Since the integrodifferential operators under consideration generate Markov jump processes, we are able to offer an alternat…
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…
In this paper, we study the existence and multiplicity of weak solutions for a general class of elliptic equations (\mathscr{P}_λ) in a smooth bounded domain, driven by a nonlocal integrodifferential operator \mathscr{L}_{\mathcal{A}K} with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz type growth conditions. Using different versions of the Mountain Pass Theorem, as well as, the Fountain Theorem and Dual Fountain Theorem with Cerami condition, we obtain the existence of weak solutions for the problem (\mathscr{P}_λ) and we show that the problem tr…
In this work we prove the uniqueness of solutions to the nonlocal linear equation $L φ- c(x)φ= 0$ in $\mathbb{R}$, where $L$ is an elliptic integro-differential operator, in the presence of a positive solution or of an odd solution vanishing only at zero. As an application, we deduce the nondegeneracy of layer solutions (bounded and monotone solutions) to the semilinear problem $L u = f(u)$ in $\mathbb{R}$ when the nonlinearity is of Allen-Cahn type. To our knowledge, this is the first work where such uniqueness and nondegeneracy results are proven in the nonlocal framework when the Caffarelli…
In this paper we consider the numerical approximation of nonlocal integro differential parabolic equations via neural networks. These equations appear in many recent applications, including finance, biology and others, and have been recently studied in great generality starting from the work of Caffarelli and Silvestre. Based in the work by Hure, Pham and Warin, we generalize their Euler scheme and consistency result for Backward Forward Stochastic Differential Equations to the nonlocal case. We rely on Lèvy processes and a new neural network approximation of the nonlocal part to overcome the …
We consider fully nonlinear integro-differential equations governed by kernels that have different homogeneities in different directions. We prove a nonlocal version of the ABP estimate, a Harnack inequality and the interior $C^{1, γ}$ regularity, extending the results of [Caffarelli and Silvestre, CPAM 62, 2009] to the anisotropic case.
We study existence of principal eigenvalues of fully nonlinear integro-differential elliptic equations with a drift term via the Krein-Rutman theorem which based on regularity up to boundary of viscosity solutions. We also show the simplicity of the eigenfunctions in viscosity sense by a nonlocal version of ABP estimate.
Stability and instability of constant stationary solutions to some integro-differential equations
本文研究具有非局部资源消耗和种内竞争的反应扩散方程，给出了非平凡常数稳态解稳定性与不稳定性的系数精确条件。
We state and prove a general Harnack inequality for minimizers of nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian.
Existence results for Cahn-Hilliard type systems driven by nonlocal integrodifferential operators with singular kernels
本文研究具有奇异势的分数阶Cahn-Hilliard方程，利用极小化运动格式证明了一般积分微分算子（含分数阶q-Laplacian）下弱解的存在唯一性，并推广到区域分数阶算子及分数阶Neumann和周期边界条件。
Semiconvexity estimates for nonlinear integro-differential equations
该研究首次建立带一般核积分微分算子驱动的完全非线性方程与障碍问题的局部半凸性估计，发展适用于非局部算子的Bernstein技巧，解决公开问题，得到非局部障碍问题的最优正则性与自由边界光滑性结果，并将技巧推广到抛物方程与非对称算子情形。
By using the Jensen's maximum principle and the Alexandrov's theorem in the convex analysis, we present the nonlocal version of the Jensen-Ishii lemma, which leads to the precise argument for the comparison principle.
In this paper we will prove the existence of three nontrivial weak solutions of the following problem involving a nonlinear integro-differential operator and a term with critical exponent. \begin{align*} \begin{split} -\mathscr{L}_Φu & = |u|^{{p_{s}^{\ast}}-2}u+λf(x,u)\,\,\mbox{in}\,\,Ω,\\ u & = 0\,\, \mbox{in}\,\, \mathbb{R}^N\setminus Ω, \end{split} \end{align*} Here $q\in(p, p_s^*)$, where $p_s^*$ is the fractional Sobolev conjugate of $p$ and $-\mathscr{L}_Φ$ represents a general nonlocal integro-differential operator of order $s\in(0,1)$. This operator is possibly degenerate and covers th…
We prove existence and uniqueness of nonnegative solutions for a nonlocal in time integrodifferential diffusion system related to angiogenesis descriptions. Fundamental solutions of appropriately chosen parabolic operators with bounded coefficients allow us to generate sequences of approximate solutions. Comparison principles and integral equations provide uniform bounds ensuring some convergence properties for iterative schemes and providing stability bounds. Uniqueness follows from chained integral inequalities.
We consider nonlinear integro-differential equations, like the ones that arise from stochastic control problems with purely jump Lèvy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior $C^{1,α}$ regularity for general fully nonlinear integro-differential equations. Our estimates remain uniform as the degree of the equation approaches two, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations.
Hölder and Harnack estimates for integro-differential operators with kernels of measure
本文针对具有测度核的椭圆型非局部方程，利用De Giorgi型方法建立弱解的Hölder与Harnack估计，推广了Dyda和Kassmann（2020）的已有结果。
We consider a possibly anisotropic integro-differential semilinear equation, run by a nondecreasing and nontrivial nonlinearity. We prove that if the solution grows at infinity less than the order of the operator, then it must be constant.
We investigate the obstacle problem for a class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional $p$-Laplacian operator with measurable coefficients. Amongst other results, we will prove both the existence and uniqueness of the solutions to the obstacle problem, and that these solutions inherit regularity properties, such as boundedness, continuity and Hölder continuity (up to the boundary), from the obstacle.
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。
We construct an example of a one-dimensional parabolic integro-differential equation with nonlocal diffusion which does not have asymptotically finite-dimensional dynamics in the corresponding state space. This example is more natural in the class of evolutionary equations of parabolic type than those known earlier.
A Deep BSDE approximation of nonlinear integro-PDEs with unbounded nonlocal operators
针对带无界非局部算子的高维非线性积分偏微分方程，提出基于正倒向随机微分方程表示的深度BSDE求解格式，结合小跳高斯近似、正向模拟与倒向神经网络回归，证明其在无限活动场景下的收敛性，规避维数灾难。
We consider one-dimensional parabolic free boundary value problem with a nonlocal (integro-differential) condition on the free boundary. Results on $C^m$-regularity of the free boundary are obtained. In particular, a necessary and sufficient condition for infinite differentiability of the free boundary is given.
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.
Nonlinear Hardy-Stein type identities for harmonic functions relative to symmetric integro-differential operators
本文针对含局部与非局部分量的一般对称正则 Dirichlet 型积分–微分算子，建立调和函数及其凸复合、比值的非线性 Hardy–Stein 型恒等式，并用于 Hardy 空间范数刻画与平方函数估计。
Estimates on the asymptotic behaviour of solution to linear integro-differential equations are fundamental in understanding the dynamics occuring in many nonlocal evolution problems. They are usually derived by using precise decay estimates on the heat kernel of the considered diffusion process. In this note, we show that for some generic jump diffusion and particular initial data, one can derive a lower bound of the asymptotic behaviour of the solution using a simple PDE argument. This is viewed as an independant preliminary brick to study invasion phenomena in nonlinear reaction diffusion pr…
We study translation-invariant integrodifferential operators that generate Lévy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula for distributional solutions. Second, we study the question under which assumptions distributional solutions are twice differentiable in the classical sense. Sufficient conditions and counterexamples are provided.
This paper is concerned with the study of the Strong Maximum Principle for semicontinuous viscosity solutions of fully nonlinear, second-order parabolic integro-differential equations. We study separately the propagation of maxima in the horizontal component of the domain and the local vertical propagation in simply connected sets of the domain. We give two types of results for horizontal propagation of maxima: one is the natural extension of the classical results of local propagation of maxima and the other comes from the structure of the nonlocal operator. As an application, we use the Stron…
We study the higher regularity of free boundaries in obstacle problems for integro-differential operators. Our main result establishes that, once free boundaries are $C^{1,α}$, then they are $C^\infty$. This completes the study of regular points, initiated in [5]. In order to achieve this, we need to establish optimal boundary regularity estimates for solutions to linear nonlocal equations in $C^{k,α}$ domains. These new estimates are the core of our paper, and extend previously known results by Grubb (for $k=\infty$) and by the second author and Serra (for $k=1$).
We investigate the regularity of the solutions for a class of degenerate/singular fully nonlinear nonlocal equations. In the degenerate scenario, we establish that there exists at least one viscosity solution of class $C_{loc}^{1, α}$, for some constant $α\in (0, 1)$. In addition, under suitable conditions on $σ$, we prove regularity estimates in Hölder spaces for any viscosity solution. We also examine the singular setting and prove Hölder regularity estimates for the gradient of the solutions.
We deal with existence, uniqueness, and regularity for solutions of the boundary value problem $$ \begin{cases} {\mathcal L}^s u = μ&\quad \text{in $Ω$}, u(x)=0 \quad &\text{on} \ \ \mathbb{R}^N\backslashΩ, \end{cases} $$ where $Ω$ is a bounded domain of $\mathbb{R}^N$, $μ$ is a bounded radon measure on $Ω$, and ${\mathcal L}^s$ is a nonlocal operator of fractional order $s$ whose kernel $K$ is comparable with the one of the factional laplacian.
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.
In this note we describe how the Neumann homogenization of fully nonlinear elliptic equations can be recast as the study of nonlocal (integro-differential) equations involving elliptic integro-differential operators on the boundary. This is motivated by a new integro-differential representation for nonlinear operators with a comparison principle which we also introduce. In the simple case that the original domain is an infinite strip with almost periodic Neumann data, this leads to an almost periodic homogenization problem involving a fully nonlinear integro-differential operator on the Neuman…
Weak solutions to parabolic integro-differential operators of order $α\in (α_0, 2)$ are studied. Local a priori estimates of Hölder norms and a weak Harnack inequality are proved. These results are robust with respect to $α\nearrow 2$. In this sense, the presentation is an extension of Moser's result in 1971.
Nonlocal gradient, the nonlocal Laplacian and maximum principles
研究非局部\(\rho -\)拉普拉斯算子，通过确定核\(\rho\)条件，建立其与积分 - 微分椭圆算子类的联系，并在更弱\(\rho\)条件下建立最大值和比较原理，涵盖更广泛算子族，且对核假设最小。
The Wiener criterion for nonlocal Dirichlet problems
该研究针对一类积分微分算子的Dirichlet问题，建立了非局部版本的Wiener准则，用于通过非局部非线性位势理论刻画边界点的正则性。
Consistently fitting vanilla option surfaces is an important issue when it comes to modelling in finance. Local volatility models introduced by Dupire in 1994 are widely used to price and manage the risks of structured products. However, the inconsistencies observed between the dynamics of the smile in those models and in real markets motivate researches for stochastic volatility modelling. Combining both those ideas to form Local and Stochastic Volatility models is of interest for practitioners. In this paper, we study the calibration of the vanillas in those models. This problem can be writt…
We study nonlocal operators acting on functions in the Euclidean space. The operators under consideration generate anisotropic jump processes, e.g., a jump process that behaves like a stable process in each direction but with a different index of stability. Its generator is the sum of one-dimensional fractional Laplace operators with different orders of differentiability. We study such operators in the general framework of bounded measurable coefficients. We prove a weak Harnack inequality and Hölder regularity results for solutions to corresponding integro-differential equations.
The aim of this article is to derive some Lewy-Stampacchia estimates and existence of solutions for equations driven by a nonlocal integro-differential operator on the Heisenberg group.
We construct and estimate the fundamental solution of highly anisotropic space-inhomogeneous integro-differential operators. We use the Levi method. We give applications to the Cauchy problem for such operators.
Partial regularity in nonlocal systems I
研究非线性积分 - 微分系统解的正则性，通过非局部过剩泛函的能量阈值刻画可忽略闭子集，利用非线性势分析，在严格假设下得出解的精细性质。
We study weak solutions to nonlocal equations governed by integrodifferential operators. Solutions are defined with the help of symmetric nonlocal bilinear forms. Throughout this work, our main emphasis is on operators with general, possibly singular, measurable kernels. We obtain regularity results which are robust with respect to the differentiability order of the equation. Furthermore, we provide a general tool for the derivation of Hölder a-priori estimates from the weak Harnack inequality. This tool is applicable for several local and nonlocal, linear and nonlinear problems on metric spac…
We prove trace and extension results for fractional Sobolev spaces of order $s\in(0,1)$. These spaces are used in the study of nonlocal Dirichlet and Neumann problems on bounded domains. The results are robust in the sense that the continuity of the trace and extension operators is uniform as $s$ approaches $1$ and our trace spaces converge to $H^{1/2}(\partial Ω)$. We apply these results in order to study the convergence of solutions of nonlocal Neumann problems as the integro-differential operators localize to a symmetric, second order operator in divergence form.
We study growth lemmas and questions of regularity for generators of Markov processes. The generators are allowed to have an arbitrary order of differentiability less than 2. In general, this order is represented by a function and not by a number. The approach enables a careful study of regularity issues up to the phase boundary between integro-differential (positive order of differentiability) and integral operators (nonnegative order of differentiability). The proof is based on intrinsic scaling properties of the underlying operators and stochastic processes.
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…
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。
Optimal boundary regularity for mixed local and nonlocal equations
本文研究了混合局部和非局部方程的最优边界正则性，通过加权霍尔德空间和固定点论证，提供精确的估计并用反例证明其最优性。
Hamilton-Jacobi-Bellman equations on graphs
本文研究图上的Hamilton-Jacobi-Bellman方程，给出其解的存在唯一性条件并分析满足假设的例子，旨在补充统一早期假设、引入非局部思想并论证算子共同结构，多数例子具Bellman-Isaacs表示。
Continuity of solutions to equations with weakly singular nonlocal operators
本文研究涉及几乎无可微性阶积分微分算子的线性非局部方程，证明了有界弱解的连续性并给出了一致的连续性模先验估计，同时允许算子高度各向异性、弱奇异且核函数在奇点附近消失。
Obstacle problems for nonlocal operators with singular kernels
本文研究具有奇异核的非局部障碍问题，针对一类非常一般的积分微分算子建立了最优正则性估计与自由边界平滑性，突破了以往要求核与分数阶拉普拉斯算子核逐点可比的限制。
In this paper, we provide suitable adaptations of the "weak version of Bernstein method" introduced by the first author in 1991, in order to obtain Lipschitz regularity results and Lipschitz estimates for nonlinear integro-differential elliptic and parabolic equations set in the whole space. Our interest is to obtain such Lipschitz results to possibly degenerate equations, or to equations which are indeed "uniformly el-liptic" (maybe in the nonlocal sense) but which do not satisfy the usual "growth condition" on the gradient term allowing to use (for example) the Ishii-Lions' method. We treat …
The aim of this paper is to deal with the elliptic pdes involving a nonlinear integrodifferential operator, which are possibly degenerate and covers the case of fractional $p$-Laplacian operator. We prove the existence of a solution in the weak sense to the problem \begin{align*} \begin{split} -\mathscr{L}_Φu & = λ|u|^{q-2}u\,\,\mbox{in}\,\,Ω,\\ u & = 0\,\, \mbox{in}\,\, \mathbb{R}^N\setminus Ω\end{split} \end{align*} if and only if a weak solution to \begin{align*} \begin{split} -\mathscr{L}_Φu & = λ|u|^{q-2}u +f,\,\,\,f\in L^{p'}(Ω),\\ u & = 0\,\, \mbox{on}\,\, \mathbb{R}^N\setminus Ω\end{sp…
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