In this paper we consider the homogenization problem for a nonlocal equation that involve different smooth kernels. We assume that the spacial domain is divided into a sequence of two subdomains $A_n \cup B_n$ and we have three different smooth kernels, one that controls the jumps from $A_n$ to $A_n$, a second one that controls the jumps from $B_n$ to $B_n$ and the third one that governs the interactions between $A_n$ and $B_n$.
Assuming that $χ_{A_n} (x) \to X(x)$ weakly-* in $L^\infty$ (and then $χ_{B_n} (x) \to (1-X)(x)$ weakly-* in $L^\infty$) as $n \to \infty$ we show that there is an homogenized limit system in which the three kernels and the limit function $X$ appear.
We deal with both Neumann and Dirichlet boundary conditions. Moreover, we also provide a probabilistic interpretation of our results.
We study the "periodic homogenization" for a class of nonlocal partial differential equations of parabolic-type with rapidly oscillating coefficients, related to stochastic differential equations driven by multiplicative isotropic $α$-stable Lévy noise ($1<α<2$) which is nonlinear in the noise component. Our homogenization method is probabilistic. It turns out that, under suitable regularity assumptions, the limit of the solutions satisfies a nonlocal partial differential equation with constant coefficients, which are associated to a symmetric $α$-stable Lévy process.
In this paper we provide a rate of convergence for periodic homogenization of Hamilton-Jacobi-Bellman equations with nonlocal diffusion. The result is based on the regularity of the associated effective problem, where the convexity plays a crucial role. Such regularity estimates are possible from the available literature once we provide a representation formula for the effective Hamiltonian, a result that has an independent interest.
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.
Our work deals with the systematic study of the coupling between the nonlocal Stokes system and the Vlasov equation. The coupling is due to a drag force generated by the fluid-particles interaction. We establish the existence of global weak solutions for the nonlocal Stokes-Vlasov system in dimensions two and three without resorting to assumptions on higher-order velocity moments of the initial distribution of particles. We then study by the means of the sigma-convergence method, the asymptotic behavior in the general deterministic framework, of the sequence of solutions to the nonlocal Stokes…
This paper studies homogenization of symmetric non-local Dirichlet forms with $α$-stable-like jumping kernels in one-parameter stationary ergodic environment. Under suitable conditions, we establish homogenization results and identify the limiting effective Dirichlet forms explicitly. The coefficients of the jumping kernels of Dirichlet forms and symmetrizing measures are allowed to be degenerate and unbounded; and the coefficients in the effective Dirichlet forms can be degenerate.
We consider a nonlocal version of the quasi-static Navier-Stokes-Korteweg equations with a non-monotone pressure law. This system governs the low-Reynolds number dynamics of a compressible viscous fluid that may take either a liquid or a vapour state. For a porous domain that is perforated by cavities with diameter proportional to their mutual distance the homogenization limit is analyzed. We extend the results for compressible one-phase flow with polytropic pressure laws and prove that the effective motion is governed by a nonlocal version of the Cahn-Hilliard equation. Crucial for the analys…
This work develops a dynamic homogenization approach for metamaterials. It finds an approximate macroscopic homogenized equation with constant coefficients posed in space and time; however, the resulting homogenized equation is higher order in space and time. The homogenized equation can be used to solve initial-boundary-value problems posed on arbitrary non-periodic macroscale geometries with macroscopic heterogeneity, such as bodies composed of several different metamaterials or with external boundaries. First, considering a single band, the higher-order space derivatives lead to additional …
In this paper we consider the homogenization of the evolution problem associated with a jump process that involves three different smooth kernels that govern the jumps to/from different parts of the domain. We assume that the spacial domain is divided into a sequence of two subdomains $A_n \cup B_n$ and we have three different smooth kernels, one that controls the jumps from $A_n$ to $A_n$, a second one that controls the jumps from $B_n$ to $B_n$ and the third one that governs the interactions between $A_n$ and $B_n$.Assuming that $χ_{A_n} (x) \to X(x)$ weakly in $L^\infty$ (and then $χ_{B_n} …
We first consider an elastic thin heterogeneous cylinder of radius of order epsilon: the interior of the cylinder is occupied by a stiff material (fiber) that is surrounded by a soft material (matrix). By assuming that the elasticity tensor of the fiber does not scale with epsilon and that of the matrix scales with epsilon square, we prove that the one dimensional model is a nonlocal system. We then consider a reference configuration domain filled out by periodically distributed rods similar to those described above. We prove that the homogenized model is a second order nonlocal problem. In pa…
The paper deals with the homogenization of a magneto-elastodynamics equation satisfied by the displacement $u\_\varepsilon$ of an elastic body which is subjected to an oscillating magnetic field $B\_\varepsilon$ generating the Lorentz force $\partial\_t u\_\varepsilon\times B\_\varepsilon$.When the magnetic field $B\_\varepsilon$ only depends on time or on space, the oscillations of $B\_\varepsilon$ induce an increase of mass in the homogenized equation. More generally, when the magnetic field is time-space dependent through a uniformly bounded component $G\_\varepsilon(t,x)$ of $B\_\varepsilo…
The paper deals with homogenization of Levy-type operators with rapidly oscillating coefficients. We consider cases of periodic and random statistically homogeneous micro-structures and show that in the limit we obtain a Levy-operator. In the periodic case we study both symmetric and non-symmetric kernels whereas in the random case we only investigate symmetric kernels. We also address a nonlinear version of this homogenization problem.
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.
In his celebrated article, Aronson established Gaussian bounds for the fundamental solution to the Cauchy problem governed by a second order divergence form operator with uniformly elliptic coefficients. We extend Aronson's proof of upper heat kernel estimates to nonlocal operators whose jumping kernel satisfies a pointwise upper bound and whose energy form is coercive. A detailed proof is given in the Euclidean space and extensions to doubling metric measure spaces are discussed.
Homogenization of Wilson-Cowan type of nonlocal neural field models is investigated. Motivated by the presence of a convolution terms in this type of models, we first prove some general convergence results related to convolution sequences. We then apply theses results to the homogenization problem of the Wilson-Cowan type model in a general deterministic setting. Key ingredients in this study are the notion of algebras with mean value and the related concept of sigma-convergence.
We prove sharp two-sided bounds of the fundamental solution for an integro-differential operator of order $α\in (0,2)$ that generates a $d$-dimensional Markov process. The corresponding Dirichlet form is comparable to that of $d$ independent copies of one-dimensional jump processes, i.e., the jumping measure is singular with respect to the $d$-dimensional Lebesgue measure.
C^(sigma+alpha) regularity for fractional Hamilton-Jacobi equations, and rates of convergence in periodic homogenization
Université Paris Cité, CNRS, Sorbonne Université, Laboratoire Jacques-Louis Lions (LJLL)（巴黎西岱大学，法国国家科学研究中心，索邦大学，雅克-路易·利翁斯实验室）；O. Mayer Mathematics Institute, Romanian Academy - Iaşi Branch（罗马尼亚科学院雅西分院奥·迈尔数学研究所）；Departamento de Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción（康塞普西翁大学物理与数学院数学系）；Instituto de Matemática, Universidade Federal do Rio de Janeiro（里约热内卢联邦大学数学研究所）
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…
The paper is mainly concerned with an approximate three-ball inequality for solutions in elliptic periodic homogenization. We consider a family of second order operators $\mathcal{L}_ε$ in divergence form with rapidly oscillating and periodic coefficients. It is the first time such an approximate three-ball inequality for homogenization theory is obtained. It implies an approximate quantitative propagation of smallness. The proof relies on a representation of the solution by the Poisson kernel and the Lagrange interpolation technique. Another full propagation of smallness result is also shown.
We study a nonlocal capillarity problem with interaction kernels that are possibly anisotropic and not necessarily invariant under scaling. In particular, the lack of scale invariance will be modeled via two different fractional exponents $s_1, s_2\in (0,1)$ which take into account the possibility that the container and the environment present different features with respect to particle interactions. We determine a nonlocal Young's law for the contact angle and discuss the unique solvability of the corresponding equation in terms of the interaction kernels and of the relative adhesion coeffici…
We prove Hölder regularity results for a class of nonlinear parabolic problem with fractional-time derivative with nonlocal and Mittag-Leffler nonsingular kernel. Existence of weak solutions via approximating solutions is proved. Moreover, the Hölder continuity of viscosity solutions is obtained. We get the similar results as those obtained by Allen (see {\url{https://arxiv.org/abs/1610.10073}})
In this paper, the global dynamics of two-species Lotka-Volterra competition models with nonlocal dispersals is studied. Under the assumption that dispersal kernels are symmetric, we prove that except for very special situations, local stability of semi-trivial steady states implies global stability, while when both semi-trivial steady states are locally unstable, the positive steady state exists and is globally stable. Moreover, our results cover the case that competition coefficients are location-dependent and dispersal strategies are mixture of local and nonlocal dispersals.
We prove a higher regularity result for weak solutions to nonlinear nonlocal equations along the integrability scale of Bessel potential spaces $H^{s,p}$ under a mild continuity assumption on the kernel. By embedding, this also yields regularity in Sobolev-Slobodeckij spaces $W^{s,p}$. Our approach is based on a characterization of Bessel potential spaces in terms of a certain nonlocal gradient-type operator and a perturbation approach commonly used in the context of local elliptic equations in divergence form.
We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. This enables us to obtain a total variation bound on the nonlocal term. By using this, we prove that the (unique) weak solution of the nonlocal problem converges strongly in $C(L^{1}_{\text{loc}})$ to the entropy solution of the local conservation law. We conclude with several numerical illustrations which underline the main results and, in particular, the difference bet…
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 prove existence of solutions and study the nonlocal-to-local asymptotics for nonlocal, convective, Cahn-Hilliard equations in the case of a W^{1,1} convolution kernel and under homogeneous Neumann conditions. Any type of potential, possibly also of double-obstacle or logarithmic type, is included. Additionally, we highlight variants and extensions to the setting of periodic boundary conditions and viscosity contributions, as well as connections with the general theory of evolutionary convergence of gradient flows.
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…
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.
We study the higher Hölder regularity of local weak solutions to a class of nonlinear nonlocal elliptic equations with kernels that satisfy a mild continuity assumption. An interesting feature of our main result is that the obtained regularity is better than one might expect when considering corresponding results for local elliptic equations in divergence form with continuous coefficients. Therefore, in some sense our result can be considered to be of purely nonlocal type, following the trend of various such purely nonlocal phenomena observed in recent years. Our approach can be summarized as …
The paper provides two versions of nonlocal Poincaré-type inequalities for integral operators with a convolution-type structure and functions satisfying a zero-Dirichlet like condition. The inequalities extend existing results to a large class of nonhomogeneous kernels with supports that can vary discontinuously and need not contain a common set throughout the domain. The measure of the supports may even vanish allowing the zero-Dirichlet condition to be imposed on only a lower-dimensional manifold, with or without boundary. The conditions may be imposed on sets with co-dimension larger than o…
We consider a linear nonlocal heat equation in a bounded domain $Ω\subset\mathbb{R}^d$ with Dirichlet boundary conditions. The non-locality is given by the presence of an integral kernel. We analyze the problem of controllability when the control acts on an open subset of the domain. It is by now known that the system is null-controllable when the kernel is time-independent and analytic or, in the one-dimensional case, in separated variables. In this paper, we relax this assumption and we extend the result to a more general class of kernels. Moreover, we get explicit estimates on the cost of n…
We study $s$-fractional $p$-Laplacian type equations with discontinuous kernel coefficients in divergence form to establish $W^{s+σ,q}$ estimates for any choice of pairs $( σ,q)$ with $q\in(p,\infty)$ and $σ\in\left(0,\min\left\{\frac{s}{p-1},1-s\right\}\right)$ under the assumption that the associated kernel coefficients have small BMO seminorms near the diagonal. As a consequence, we find in the literature an optimal fractional Sobolev regularity of such a non-homogeneous nonlocal equation when the right-hand side is presented by a suitable fractional operator. Our results are new even in th…
We study the prescribed constant mean curvature problem in the nonlocal setting where the nonlocal curvature has been defined as $$ H^J_Ω(x):=\int_{\mathbb{R} ^n} J(x-y)(χ_{Ω^c}(y)-χ_Ω(y))dy, $$ where $x \in \mathbb{R}^n$, $Ω\subset \mathbb{R}^n$, $χ$ is the characteristic function for a set, $J$ is a radially symmetric, nonegative, nonincreasing convolution kernel. Several papers have studied the case of nonlocal curvature with nonintegrable singularity, a generalization of the classical curvature concept, which requires the regularity of the boundary to be above $C^2$. Nonlocal curvature of …
This work addresses the regularity of solutions for a nonlocal diffusion equation over the space of periodic distributions. The spatial operator for the nonlocal diffusion equation is given by a nonlocal Laplace operator with a compactly supported integral kernel. We follow a unified approach based on the Fourier multipliers of the nonlocal Laplace operator, which allows the study of regular as well as distributional solutions of the nonlocal diffusion equation, integrable as well as singular kernels, in any spatial dimension. In addition, the results extend beyond operators with singular kern…
In this paper, we gives a complete classification of the global dynamics of two- species Lotka-Volterra competition models with nonlocal dispersals: where K, P represent nonlocal operators, under the assumptions that the nonlo- cal operators are symmetric, the models admit two semi-trivial steady states and 0<bc<1. In particular, when both semi-trivial steady states are locally stable, it is proved that there exist infinitely many steady states and the solution with non- negative and nontrivial initial data converges to some steady state. Furthermore, we generalize these results to the case th…
