A-quasiconvexity: Relaxation and homogenization.(English)Zbl 0971.35010

The paper deals with problems in the calculus of variations where minimization of the energy functional $(u,v)\mapsto\int_\Omega f\bigl(x,u(x),v(x)\bigr) dx$ (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 ${\mathcal A}v:=\sum_{i=1}^N A^{(i)}{\partial v\over\partial x_i},$ 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 $\int_{(0,1)^N}f\bigl(x,u(x),v+w(y)\bigr) dy$ 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.
Reviewer: L.Ambrosio (Pisa)

MSC:

 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 49J45 Methods involving semicontinuity and convergence; relaxation
Full Text:

References:

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