A note on the integral representation of functionals in the space SBD(O)
François Ebobisse, Rodica Toader · 2003
In this paper we study the integral representation in the space SBD of special functions with bounded deformation of some L 1 -norm lower semicon- Several phenomena in phase transition, fracture mechanics, liquid crystals, can be mo- delled as energy minimization problems where the natural energy has both volume and surface terms. In many cases the energy functional is obtained as a limit of approximating functionals and some of its properties can be deduced from the approximation process. A basic step is then to obtain, starting from these properties, an integral representation of the energy. We consider here this problem for local functionals F defined on the space BD of functions with bounded deformation, which are lower semicontinuous with respect to the L 1 -topology, satisfy linear growth and coercivity conditions, as set functions are (restrictions of) Radon measures, and are invariant with respect to rigid motions. In order to identify the volume and the surface densities we follow the global method for relaxation introduced by Bouchitte, Fonseca and Mascarenhas in (7) for functionals defined on the space BV of functions with bounded variation, which is characterized by the identification of both bulk and surface densities from a local Dirichlet problem. This kind of approach has already been used in some other contexts, as, for instance, homogenization, where the homogenized density is obtained from a Dirichlet problem in the cell. An example of functional in the class we consider is given by the relaxed functional F of the bulk energy