A-quasiconvexity: Relaxation and homogenization (Q2701814)

The paper deals with problems in the calculus of variations where minimization of the energy functional NEWLINE\[NEWLINE (u,v)\mapsto\int_\Omega f\bigl(x,u(x),v(x)\bigr) dx NEWLINE\]NEWLINE (with \(\Omega\subset{\mathbb{R}}^N\) open, \(u:\Omega\to{\mathbb{R}}^m\), \(v:\Omega\to{\mathbb{R}}^d\)) is performed with a differential constraint \({\mathcal A}v=0\), where NEWLINE\[NEWLINE {\mathcal A}v:=\sum_{i=1}^N A^{(i)}{\partial v\over\partial x_i},NEWLINE\]NEWLINE and \(A^{(i)}:{\mathbb{R}}^d\to{\mathbb{R}}^l\) are linear transformations satisfying the property that the rank of the linear operator \(\sum_iw_iA^{(i)}w\) is independent of \(w\in{\mathbb{R}}^N\setminus\{0\}\). This general framework contains the classical problems of the calculus of variations (with \({\mathcal A}\) equal to the curl operator, so that \(v\) is a gradient) but also several other cases of interest: divergence free fields, magnetostatics, higher-order gradients and others. One of the main results of the paper is the integral representation of the relaxed energy, in the case when convergence in measure of \(u\) and weak convergence of \(v\) in \(L^q\) are taken into account, with \(q>1\). The relaxed energy functional has the same form of the original function, with \(f\) replaced by the \({\mathcal A}\)-quasiconvexification of \(f(x,u,\cdot)\), defined by the infimum of NEWLINE\[NEWLINE \int_{(0,1)^N}f\bigl(x,u(x),v+w(y)\bigr) dy NEWLINE\]NEWLINE among all smooth and 1-periodic functions \(w\) with zero mean and in the kernel of \({\mathcal A}\). This result extends and puts in a unified framework of several relaxation results previously available in the literature. Applications to relaxation problems for higher-order gradients are also discussed in detail. The final part of the paper deals with homogenization results for periodic integrand in the context of \({\mathcal A}\)-quasiconvexity. The proofs are bases on the blow-up method of Fonseca and Muller and on the theory of Young measures.