Title: Multiple integrals under differential constraints: two-scale convergence and homogenization

Authors: Irene Fonseca and Stefan Kroemer

Issue: Volume 59 (2010), Issue 2, 427-458

Abstract: Two-scale techniques are developed for sequences of maps $\{u_k\} \subset L^{p}(\Omega;\mathbb{R}^{M})$ satisfying a linear differential constraint $\mathcal{A}u_k = 0$. These, together with $\Gamma$-convergence arguments and using the unfolding operator, provide a homogenization result for energies of the type \begin{align*} \MoveEqLeft[5] F_{epsilon}(u) \coloneqq \int_{\Omega}f \left(x,\frac{x}{epsilon},u(x)\right)\, \mathrm{d}x\\ &\mbox{with }u \in L^{p}(\Omega;\mathbb{R}^M),\ \mathcal{A}u = 0, \end{align*} that generalizes current results in the case where $\mathcal{A} = \mbox{curl}$.