Functionals defined on partitions in sets of finite perimeter. II: Semicontinuity, relaxation and homogenization
By using the results of the first paper [the authors, ibid.; Zbl 0676.49028], we find general necessary and sufficient conditions on f(x,i,j,\(\nu)\) which ensure the lower semicontinuity, with respect to convergence in measure, of the functional \[ F(E_ 1,...,E_ m)=\sum_{i<j}\int_{\partial^*E_ i\cap \partial^*E_ j}f(x,i,j,\nu (x))d{\mathcal H}^{n-1}(x), \] defined on all partitions of a domain \(\Omega\) in m sets of finite perimeter \(E_ 1,...,E_ m\). In particular, we show that F has a minimum on the class of partitions with prescribed Lebesgue measure. Moreover, we deal with the homogenization problem \[ \Gamma \lim_{\epsilon \to 0+}F_{\epsilon},\quad F_{\epsilon}(E_ 1,...,E_ m)\quad =\quad \sum_{i<j}\int_{\partial^*E_ i\cap \partial^*E_ j}f(x/\epsilon,i,j,\nu (x))d{\mathcal H}^{n-1}(x), \] finding an integral representation of the limit functional in case f is almost-periodic in x.
scientific article; zbMATH DE number 4107556