[Paper Review] Homogenization of random convolution energies in heterogeneous and perforated domains

Andrea Braides, Andrey Piatnitski|arXiv (Cornell University)|Sep 15, 2019

Advanced Mathematical Modeling in Engineering37 references4 citations

TL;DR

This paper establishes a homogenization theorem for random convolution energies in heterogeneous and perforated domains, proving almost sure $Γ$-convergence to a deterministic quadratic Dirichlet-type functional. The limit functional's integrand is characterized via an asymptotic formula derived from subadditive processes, extending local homogenization techniques to non-local, asymptotically-local settings under ergodicity and stationarity assumptions.

ABSTRACT

We prove a homogenization theorem for a class of quadratic convolution energies with random coefficients. Under suitably stated hypotheses of ergodicity and stationarity we prove that the $Γ$-limit of such energy is almost surely a deterministic quadratic Dirichlet-type integral functional, whose integrand can be characterized through an asymptotic formula. The proof of this characterization relies on results on the asymptotic behaviour of subadditive processes. The proof of the limit theorem uses a blow-up technique common for local energies, that can be extended to this `asymptotically-local' case. As a particular application we derive a homogenization theorem on random perforated domains.

Motivation & Objective

• To establish a $Γ$-limit for random convolution energies with ergodic and stationary coefficients in heterogeneous and perforated domains.
• To characterize the homogenized energy density using an asymptotic formula derived from subadditive processes.
• To extend the blow-up technique—commonly used for local energies—to non-local, asymptotically-local convolution functionals.
• To derive a homogenization result for energies defined on random perforated domains, where interactions occur only within random connected components.
• To show that the $Γ$-limit is a deterministic, quadratic Dirichlet-type integral functional almost surely.

Proposed method

• Utilizes $Γ$-convergence in the $L^2(D)$ topology to analyze the limit of scaled convolution energies.
• Applies a blow-up technique adapted to non-local energies by exploiting their asymptotically-local structure.
• Employs subadditive process theory to characterize the homogenized energy density via a two-scale limit involving large-scale averages.
• Uses a random dynamical system $(\tau_x)$ on a probability space to model stationarity and ergodicity of the environment.
• Applies an extension theorem (Theorem 7.2) to control the energy on perforated domains by extending functions to the full domain with controlled energy bounds.
• Derives the homogenized matrix $A_{\text{hom}}$ via a variational formula over large cubes $Q_R$ with boundary layers of size $K$.

Experimental results

Research questions

• RQ1 What is the $Γ$-limit of a random convolution energy with ergodic and stationary coefficients in a heterogeneous domain?
• RQ2 How can the homogenized energy density be characterized when the interaction kernel is non-local and the medium is random?
• RQ3 Can the blow-up method used for local energies be extended to non-local convolution energies with random coefficients?
• RQ4 What is the homogenized limit of an energy defined on a random perforated domain where interactions are restricted to the connected components?
• RQ5 Under what conditions does the $Γ$-limit of such random energies converge to a deterministic, quadratic Dirichlet-type functional?

Key findings

• The $Γ$-limit of the energy functional $F^{ω}_{\varepsilon}(u)$ is almost surely a deterministic quadratic Dirichlet-type integral functional on $H^1(D)$.
• The homogenized energy density is characterized by the matrix $A_{\text{hom}}$, whose quadratic form is given by $\langle A_{\text{hom}}z,z\rangle = \lim_{K\to\infty}\lim_{R\to\infty}\frac{1}{R^d}\inf\left\{\int_{Q_R\cap E^{\omega}}\int_{E^{\omega}}a(x-y)(v(x)-v(y))^2\,dx\,dy : v(x)=\langle z,x\rangle \text{ near }\partial Q_R \right\}$.
• The limit is independent of the random realization $\omega$ almost surely, due to the ergodicity and stationarity assumptions.
• The energy on random perforated domains satisfies a uniform extension estimate (Theorem 7.2), ensuring control of the energy via a full-domain extension with bounded energy increase.
• The homogenization result holds under the integrability condition $\int_{\mathbb{R}^d} a(\xi)(1+|\xi|^2)\,d\xi < \infty$ and the lower bound $a(\xi) \geq c > 0$ for $|\xi| \leq r_0$.
• The convergence is established in the $L^2(D)$ topology, and the limit functional is continuous and coercive on $H^1(D)$.