Homogenization of Dirichlet problems with convex bounded constraints on the gradient
The paper is concerned with the homogenization, as \(\varepsilon\) decreases to 0, of Dirichlet problems for the functionals \[ F_\varepsilon(u)=\begin{cases}\int_\Omega f\left({x\over\varepsilon},Du(x)\right)dx &\text{if }u\in W^{1,\infty}(\Omega)\cap{\mathcal C}_0(\Omega)\\ +\infty &\text{otherwise,}\end{cases} \] where \(\Omega\) is a bounded open subset of \({\mathbb R}^N\) with \(| \partial\Omega| =0\), \({\mathcal C}_0(\Omega)\) denotes the set of the continuous functions on \(\overline\Omega\) vanishing on \(\partial\Omega\), and \(f:{\mathbb R}^N\times{\mathbb R}^N\to[0,+\infty]\) is a function that is measurable, \((0,1)^N\)-periodic in the first variable, convex and lower semicontinuous in the second one, and such that \(f(\cdot,0)\in L^1((0,1)^N)\) and \(0\in\text{ dom}f(x,\cdot)\subseteq\{z\in{\mathbb R}^N : | z| \leq1\}\) for a.e. \(x\in\Omega\). The variational problems discussed turn out to be pointwise gradient constrained, with convex bounded constraints, since the functionals \(F_\varepsilon\) are finite on those \(u\) satisfying \(Du(x)\in\text{ dom}f({x\over\varepsilon},\cdot)\) for a.e. \(x\in\Omega\). The usual homogenization formula is proved, by showing that the homogenized functional is given by \[ F_{\text{hom}}(u)=\begin{cases} \int_\Omega f_{\text{hom}}(Du(x))dx &\text{if }u\in W^{1,\infty}(\Omega)\cap{\mathcal C}_0(\Omega)\\ +\infty &\text{otherwise,}\end{cases} \] where \[ \begin{multlined} f_{\text{hom}}(z)=\inf\Bigg\{\int_{(0,1)^N}f(y,z+Dv(y))dy : v\in W^{1,\infty}(\Omega),\\ v\;(0,1)^N\text{-periodic and continuous on }[0,1]^N\Bigg\}.\end{multlined} \] As the authors say, the result is not new, and the proof fits in the scheme proposed in [\textit{H. Attouch}, ``Variational convergence for functions and operators'' (1984; Zbl 0561.49012)], but in a more general context, and under one of the additional assumptions \[ 0\text{ belongs to the interior of dom}f_{\text{hom}}, \] or \[ \text{there exists }w\in W^{1,\infty}((0,1)^N)\cap{\mathcal C}_0((0,1)^N)\text{ such that }f_{\text{hom}}(0)=\int_{(0,1)^N}f(y,Dw(y))dy. \] It is based on a suitable inf-convolution approximation of the functionals \(F_\varepsilon\), and on an approximation procedure for Lipschitz functions subject to gradient constraints by means of piecewise affine functions satisfying the same constraint conditions.
scientific article; zbMATH DE number 2068758