A Global Method for Relaxation

Abstract.

A new method for the identification of the integral representation of a class of functionals defined on $BV(\Omega ;{\Bbb R}^d)\times {\cal A}(\Omega)$(where ${\cal A}(\Omega)$ represents the family of open subsets of Ω) is presented. Applications are derived, such as the integral representation of the relaxed energy in $BV(\Omega;{\Bbb R}^d)$ corresponding to a functional defined in $W^{1,1}(\Omega;{\Bbb R}^d)$ with a discontinuous integrand with linear growth; relaxation and homogenization results in $SBV(\Omega;{\Bbb R}^d)$ are recovered in the case where bulk and surface energies are present.