Dipartimento di Matematica, Universita di Pavia, 27100 Pavia, Italy, amar@dragon.ian.pv.cnr.it, and Dipartimento di Matematica, Universita di Pavia, 27100 Pavia, Italy, vitali@dimat.unipv.it
Abstract: We prove an homogenization result in $W^{1,1}$ and in $BV$ for a sequence $(F_\varepsilon)$ of functionals of the form
F_\varepsilon (u) = \int_0^1 f(\frac{u}{\varepsilon}, u^\prime) dt
where $\varepsilon$ is a positive parameter which tends to zero, $f\colon\mathbb{R}^n \times \mathbb{R}^n \rightarrow [0,+\infty)$ is $[0,1)^n$-periodic in the first variable, convex in the second variable and satisfies a suitable growth condition of order one.
Under the additional assumption that $f(x,\cdot)$ is positively $1$-homogeneous, we show how our result is equivalent to the analogous homogenization result (dealt with by Acerbi and Buttazzo) in which growth conditions of order $p>1$ are considered.
Keywords: Homogenization, $\Gamma$-convergence, BV-functions
Classification (MSC2000): 49N20, 35B27
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