Periodic homogenization of non-local operators with a convolution type kernel

The paper deals with homogenization problem for a non-local linear operator with a kernel of convolution type in a medium with a periodic structure. We consider the natural diffusive scaling of this operator and study the limit behaviour of the rescaled operators as the scaling parameter tends to 0. More precisely we show that in the topology of resolvent convergence the family of rescaled operators converges to a second order elliptic operator with constant coefficients. We also prove the convergence of the corresponding semigroups both in $L^2$ space and the space of continuous functions, and show that for the related family of Markov processes the invariance principle holds

On operator estimates in homogenization of nonlocal operators of convolution type

The paper studies a bounded symmetric operator ${\mathbf{A}}_\varepsilon$ in $L_2(\mathbf{R}^d)$ with $$({\mathbf{A}}_\varepsilon u) (x) = \varepsilon^{-d-2} \int_{\mathbf{R}^d} a((x-y)/\varepsilon) \mu(x/\varepsilon, y/\varepsilon) \left( u(x) - u(y) \right)\,dy;$$ here $\varepsilon$ is a small positive parameter. It is assumed that $a(x)$ is a non-negative $L_1(\mathbf{R}^d)$ function such that $a(-x)=a(x)$ and the moments $M_k =\int_{\mathbf{R}^d} |x|^k a(x)\,dx$, $k=1,2,3$, are finite. It is also assumed that $\mu(x,y)$ is $\mathbf{Z}^d$-periodic both in $x$ and $y$ function such that $\mu(x,y) = \mu(y,x)$ and $0< \mu_- \leq \mu(x,y) \leq \mu_+< \infty$. Our goal is to study the limit behaviour of the resolvent $({\mathbf{A}}_\varepsilon + I)^{-1}$, as $\varepsilon\to0$. We show that, as $\varepsilon \to 0$, the operator $({\mathbf{A}}_\varepsilon + I)^{-1}$ converges in the operator norm in $L_2(\mathbf{R}^d)$ to the resolvent $({\mathbf{A}}^0 + I)^{-1}$ of the effective operator ${\mathbf{A}}^0$ being a second order elliptic differential operator with constant coefficients of the form ${\mathbf{A}}^0= - \operatorname{div} g^0 \nabla$. We then obtain sharp in order estimates of the rate of convergence

Homogenization of nonlocal convolution type operators: Approximation for the resolvent with corrector

In $L_2(\mathbb{R}^d)$, we consider a selfadjoint bounded operator ${\mathbb A}_\varepsilon$, $\varepsilon >0$, of the form $$({\mathbb A}_\varepsilon u) (\mathbf{x}) = \varepsilon^{-d-2} \int_{\mathbb{R}^d} a((\mathbf{x} - \mathbf{y} )/ \varepsilon ) \mu(\mathbf{x} /\varepsilon, \mathbf{y} /\varepsilon) \left( u(\mathbf{x}) - u(\mathbf{y}) \right)\, d\mathbf{y}.$$ It is assumed that $a(\mathbf{x})$ is a nonnegative function of class $L_1(\mathbb{R}^d)$ such that \hbox{$a(-\mathbf{x}) = a(\mathbf{x})$} and $\mu(\mathbf{x},\mathbf{y})$ is $\mathbb{Z}^d$-periodic in each variable and such that $\mu(\mathbf{x},\mathbf{y}) = \mu(\mathbf{y},\mathbf{x})$ and $0< \mu_- \leqslant \mu(\mathbf{x},\mathbf{y}) \leqslant \mu_+< \infty$. Moreover, it is assumed that the moments $M_k (a)= \int_{\mathbb{R}^d} | \mathbf{x} |^k a(\mathbf{x})\,d\mathbf{x}$, $k=1,2,3,4,$ are finite. We obtain approximation of the resolvent $({\mathbb A}_\varepsilon + I)^{-1}$ for small $\varepsilon$ in the operator norm on $L_2(\mathbb{R}^d)$ with error of order $O(\varepsilon^2)$.Comment: 33 page

A consistent nonlocal scheme based on filters for the homogenization of heterogeneous linear materials with non-separated scales

AbstractIn this work, the question of homogenizing linear elastic, heterogeneous materials with periodic microstructures in the case of non-separated scales is addressed. A framework if proposed, where the notion of mesoscopic strain and stress fields are defined by appropriate integral operators which act as low-pass filters on the fine scale fluctuations. The present theory extends the classical linear homogenization by substituting averaging operators by integral operators, and localization tensors by nonlocal operators involving appropriate Green functions. As a result, the obtained constitutive relationship at the mesoscale appears to be nonlocal. Compared to nonlocal elastic models introduced from a phenomenological point of view, the nonlocal behavior has been fully derived from the study of the microstructure. A discrete version of the theory is presented, where the mesoscopic strain field is approximated as a linear combination of basis functions. It allows computing the mesoscopic nonlocal operator by means of a finite number of transformation tensors, which can be computed numerically on the unit cell

Variational analysis of integral functionals involving nonlocal gradients on bounded domains

The center of interest in this work are variational problems with integral functionals depending on special nonlocal gradients. The latter correspond to truncated versions of the Riesz fractional gradient, as introduced in [Bellido, Cueto & Mora-Corral 2022] along with the underlying function spaces. We contribute several new aspects to both the existence theory of these problems and the study of their asymptotic behavior. Our overall proof strategy builds on finding suitable translation operators that allow to switch between the three types of gradients: classical, fractional, and nonlocal. These provide useful technical tools for transferring results from one setting to the other. Based on this approach, we show that quasiconvexity, which is the natural convexity notion in the classical -- and as shown in [Kreisbeck & Sch\"onberger 2022] also in the fractional -- calculus of variations, gives a necessary and sufficient condition for the weak lower semicontinuity of the nonlocal functionals as well. As a consequence of a general Gamma-convergence statement, we obtain relaxation and homogenization results. The analysis of the limiting behavior for varying fractional parameters yields, in particular, a rigorous localization with a classical local limit model.Comment: Only the acknowledgements section has been modifie

A variational theory for integral functionals involving finite-horizon fractional gradients

The center of interest in this work are variational problems with integral functionals depending on nonlocal gradients with finite horizon that correspond to truncated versions of the Riesz fractional gradient. We contribute several new aspects to both the existence theory of these problems and the study of their asymptotic behavior. Our overall proof strategy builds on finding suitable translation operators that allow to switch between the three types of gradients: classical, fractional, and nonlocal. These provide useful technical tools for transferring results from one setting to the other. Based on this approach, we show that quasiconvexity, which is the natural convexity notion in the classical calculus of variations, gives a necessary and sufficient condition for the weak lower semicontinuity of the nonlocal functionals as well. As a consequence of a general Γ -convergence statement, we obtain relaxation and homogenization results. The analysis of the limiting behavior for varying fractional parameters yields, in particular, a rigorous localization with a classical local limit model

Stochastic homogenization of nonlinear evolution equations with space-time nonlocality

In this paper we consider the homogenization problem of nonlinear evolution equations with space-time non-locality, the problems are given by Beltritti and Rossi [JMAA, 2017, 455: 1470-1504]. When the integral kernel $J(x,t;y,s)$ is re-scaled in a suitable way and the oscillation coefficient $\nu(x,t;y,s)$ possesses periodic and stationary structure, we show that the solutions $u^{\varepsilon}(x,t)$ to the perturbed equations converge to $u_{0}(x,t)$, the solution of corresponding local nonlinear parabolic equation as scale parameter $\varepsilon\rightarrow 0^{+}$. Then for the nonlocal linear index $p=2$ we give the convergence rate such that $||u^\varepsilon -u_{0}||_{_{L^{2}(\mathbb{R}^{d}\times(0,T))}}\leq C\varepsilon$. Furthermore, we obtain that the normalized difference $\frac{1}{\varepsilon}[u^{\varepsilon}(x,t)-u_{0}(x,t)]-\chi(\frac{x}{\varepsilon}, \frac{t}{\varepsilon^{2}}) \nabla_{x}u_{0}(x,t)$ converges to a solution of an SPDE with additive noise and constant coefficients. Finally, we give some numerical formats for solving non-local space-time homogenization.Comment: 24 pages, 1 figur