Homogenization of integral functionals with linear growth defined on vector-valued functions
In the paper functionals of the form \[ F_\varepsilon(u)= \int_\Omega f(x/\varepsilon, Du)dx,\quad u\in u_0+ W^{1, 1}_0(\Omega, \mathbb{R}^m) \] are considered, where \(f(x, z)\) is a Carathéodory function, periodic in \(x\), and with a linear growth, that is \(|z|\leq f(x, z)\leq C(1+ |z|)\). It is shown that the \(\Gamma\)-limit of \(F_\varepsilon\) as \(\varepsilon\) goes to zero is given by \[ F(u)= \int_\Omega f(\nabla u) dx+ \int_\Omega f^\infty(D^s u)+ \int_{\partial\Omega} f^\infty((u_0- u)\otimes \nu)dH^{n- 1}, \] where \(u\in \text{BV}(\Omega; \mathbb{R}^m)\), \(Du= \nabla u dx+ D^s u\) is the Lebesgue-Nikodým decomposition of the measure \(Du\), and \(f^\infty\) is the recession function associated to \(f\). Here, the limit integrand \(f\) is given by the usual homogenization formula \[ f(z)= \lim_{\varepsilon\to 0} \inf\Biggl\{\varepsilon^n \int_{Y/\varepsilon} f(x, z+ Dw)dx: w\in W^{1,1}_0(Y/\varepsilon; \mathbb{R}^m)\Biggr\}. \]
scientific article; zbMATH DE number 784447
scientific article