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Publication 09-CNA-18

Multiple Integrals Under Differential Constraints: Two-Scale Convergence and Homogenization

Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
fonseca@andrew.cmu.edu

Stefan Krömer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
kroemers@andrew.cmu.edu

Abstract: Two-scale techniques are developed for sequences of maps $ \{u_k\}_{k \in \mathbb{N}} \subset L^p(\Omega;\mathbb{R}^M)$ satisfying a linear differential constraint $ Au_k=0$. These, together with $ \Gamma$-convergence arguments and using the unfolding operator, provide a homogenization result for energies of the type

$\displaystyle F_{\epsilon}(u):=\int_{\Omega}f(x, \frac{x}{\epsilon},u(x))dx\ {with}\ u \in L^p(\Omega;\mathbb{R}^M),\ Au=0, $

that generalizes current results in the case where $ A$ = curl.

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