Homogenizations of integro-differential equations with L{é}vy operators with asymmetric and degenerate densities

某些非线性积分-微分方程的随机均匀化

Stochastic Homogenization for Some Nonlinear Integro-Differential Equations

针对一类具有振荡系数的非线性非局部椭圆型方程，将周期均匀化结果推广到随机平稳遍历情形，证明了有效方程的存在性及原方程解的收敛性。

Cauchy problem and periodic homogenization for nonlocal Hamilton-Jacobi equations with coercive gradient terms

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.

Periodic Homogenization for Weakly Elliptic Hamilton-Jacobi-Bellman Equations with Critical Fractional Diffusion

In this paper we establish periodic homogenization for Hamilton-Jacobi-Bellman (HJB) equations, associated to nonlocal operators of integro-differential type. We consider the case when the fractional diffusion has the same order as the drift term, and is weakly elliptic. The outcome of the paper is twofold. One one hand, we provide Lipschitz regularity results for weakly elliptic non-local HJB, extending the results previously obtained in [8]. On the other hand, we establish a convergence result, based on half relaxed limits and a comparison principle for the effective problem. The latter stro…

非局部哈密顿-雅可比-贝尔曼方程的随机均匀化

Stochastic Homogenization of Non-local Hamilton-Jacobi-Bellman equations

本文应用Kosygina等人2006年提出的方法，建立平稳遍历随机介质中带跳扩散的HJB方程的随机均匀化，将非局部积分-微分算子表示为正则积分算子的散度，拓展了原有方法的适用范围。

分数阶哈密顿-雅可比方程的C^(σ+α)正则性及周期均匀化中的收敛速率

C^(sigma+alpha) regularity for fractional Hamilton-Jacobi equations, and rates of convergence in periodic homogenization

该研究针对非局部贝尔曼型方程，建立其C^(σ+α)正则性理论，结合折扣近似校正子量化扰动检验函数方法，证明分数阶哈密顿-雅可比方程周期均匀化解以ε^(σα/(2+α))的速率收敛到有效解。

具有大非线性漂移和Robin边界数据的反应扩散问题的均匀化

Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data

本文研究无界穿孔域中带大非线性漂移和Robin边界条件的反应扩散问题周期均匀化，利用带漂移的双尺度紧性方法严格推导升尺度模型与有效系数，并给出校正函数的强收敛结果。

Scaling effects on the periodic homogenization of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer

We study the question of periodic homogenization of a variably scaled reaction-diffusion problem with non-linear drift posed for a domain crossed by a flat composite thin layer. The structure of the non-linearity in the drift was obtained in earlier works as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) process for a population of interacting particles crossing a domain with obstacle. Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive the homogenized evolution equation and the corresponding effective mo…

基于均匀化理论的拓扑 Anderson 绝缘体

Topological Anderson Insulators by homogenization theory

本文提出微扰 Dirac 方程模型，证明大幅高频微扰能诱导产生与未微扰状态不同的拓扑相，并在两相交界面处实现稳健的非对称输运，为拓扑 Anderson 绝缘体提供了基于均匀化理论与 Fredholm 算子指数表征的数学理论支撑。

Spatial asymptotics at infinity for heat kernels of integro-differential operators

We study a spatial asymptotic behaviour at infinity of kernels $p_t(x)$ for convolution semigroups of nonlocal pseudo-differential operators. We give general and sharp sufficient conditions under which the limits $$ \lim_{r \to \infty} \frac{p_t(rθ-y)}{t \, ν(rθ)}, \quad t \in T, \ \ θ\in E, \ \ y \in \mathbb R^d, $$ exist and can be effectively computed. Here $ν$ is the corresponding Lévy density, $T \subset (0,\infty)$ is a bounded time-set and $E$ is a subset of the unit sphere in $\mathbb R^d$, $d \geq 1$. Our results are local on the unit sphere. They apply to a wide class of convolution …

Continuous dependence on the coefficients for a class of non-autonomous evolutionary equations

The continuous dependence of solutions to certain (non-autonomous, partial, integro-differential-algebraic, evolutionary) equations on the coefficients is addressed. We give criteria that guarantee that convergence of the coefficients in the weak operator topology implies weak convergence of the respective solutions. We treat three examples: A homogenization problem for a Kelvin-Voigt model for elasticity, the discussion of continuous dependence of the coefficients for acoustic waves with impedance type boundary conditions and a singular perturbation problem for a mixed type equation. By means…

The Solvability of a Strongly-Coupled Nonlocal System of Equations

We prove existence and uniqueness of strong (pointwise) solutions to a linear nonlocal strongly coupled hyperbolic system of equations posed on all of Euclidean space. The system of equations comes from a linearization of a nonlocal model of elasticity in solid mechanics. It is a nonlocal analogue of the Navier-Lamé system of classical elasticity. We use a well-known semigroup technique that hinges on the strong solvability of the corresponding steady-state elliptic system. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to co…

具有随机遍历系数的稳定型算子的均匀化

Homogenization of stable-like operators with random, ergodic coefficients

本文研究具有平稳遍历随机系数的稳定型过程的均匀化，证明其弱收敛到由平均系数决定的α-稳定过程，并给出相应随机积分微分方程解的收敛性解释。

A Fisher-KPP model with a nonlocal weighted free boundary: analysis of how habitat boundaries expand, balance or shrink

In this paper, we propose a novel free boundary problem to model the movement of single species with a range boundary. The spatial movement and birth/death processes of the species found within the range boundary are assumed to be governed by the classic Fisher-KPP reaction-diffusion equation, while the movement of a free boundary describing the range limit is assumed to be influenced by the weighted total population inside the range boundary and is described by an integro-differential equation. Our free boundary equation is a generalization of the classical Stefan problem that allows for nonl…