Homogenization of non-local integral functionals via two-scale Young measures
Giacomo Bertazzoni, Andrea Torricelli, Elvira Zappale
TL;DR
The paper develops a rigorous homogenization theory for non-local integral functionals $I_\varepsilon(u)=\iint_{\Omega\times\Omega} W\big(x,x', \langle x/\varepsilon\rangle, \langle x'/\varepsilon\rangle, u(x),u(x')\big) dx dx'$, showing that it $\Gamma$-converges to a homogenized energy $I_{hom}$ defined by a minimization over two-scale Young measures subject to the macroscopic constraint $u(x)=\int_Q u_1(x,y) dy$. The main technical tool is a full characterization of two-scale Young measures: a family $\{\nu_{(x,y)}\}$ is a two-scale Young measure with underlying deformation $u$ if (i) $\int_Q\int_{\mathbb{R}^d} |\xi|^p d\nu_{(x,y)}(\xi) dy \in L^1(\Omega)$, (ii) there exists $u_1 \in L^p(\Omega;L^p_{per}(Q))$ with $\int_{\mathbb{R}^d} \xi d\nu_{(x,y)}(\xi) = u_1(x,y)$ and $u(x)=\int_Q u_1(x,y) dy$, and (iii) a homogenization inequality $\int_Q\int_{\mathbb{R}^d} f(y,\xi) d\nu_{(x,y)}(\xi) dy \ge f_{hom}(u(x))$ for all admissible $f \in \mathcal{E}_p$, where $f_{hom}$ is given by a cell-type limit. This framework yields the homogenized energy formula $I_{hom}(u) = \min_{\nu \in \mathcal{M}_u} \iint_{\Omega\times\Omega}\iint_{Q\times Q}\iint_{\mathbb{R}^d\times\mathbb{R}^d} W(x,x',y,y',\xi,\xi') d\nu_{(x,y)}(\xi)d\nu_{(x',y')}(\xi') dy dy' dx dx'$, providing a rigorous bridge between microstructure and macro-scale behavior for non-local models such as peridynamics. The results connect with and extend existing relaxation and homogenization theories, including the classical isotropic relaxation when microscale dependence is removed, and offer a solid foundation for nonlocal materials modeling in heterogeneous media.
Abstract
We prove a homogenization result in terms of two-scale Young measures for non-local integral functionals. The result is obtained by means of a characterization of two-scale Young measures.