Functionals defined on partitions in sets of finite perimeter. I: Integral representation and Gamma-convergence (Q1122775)

The aim of this paper is to study the convergence of functionals \(F_ h\) defined on partitions of a given domain \(\Omega\) in a finite number of sets of finite perimeter \(E_ 1,...,E_ m\). The functionals are defined as follows \[ F_ h(E_ 1,...,E_ m)=\sum_{i<j}\int_{\partial^*E_ i\cap \partial^*E_ j}f_ h(x,i,j,\nu (x))d{\mathcal H}^{n-1}(x), \] where \(f_ h(x,i,j,\nu)\) is an energy depending on \(x\in \Omega\), and the unitary vector \(\nu\) normal to the interface. We prove by means of an integral representation theorem a closure property of this class of functionals with respect to \(\Gamma\)- convergence. [For part II see the authors, ibid.; Zbl 0676.49029.]