Relaxation for bulk and interfacial energies

Several physical phenomena in phase transitions, fracture mechanics, image segmentation can be investigated through the study of minimum problems associated to energy functionals of the form F(u)=\int_\Omega f(\nabla u) dx +\int_{S_u\cap\Omega}g(u^+,u^-,\nu) dH^{n-1}, (1) where u is a ``smooth" function outside a (n-1)-dimensional discontinuity set S_u on which the traces u^+ and u^- and the normal \nu are well defined. In order to perform this study in the spirit of the direct method of the Calculus of Variations, it is crucial (i) to search for (necessary and) sufficient conditions on f and g which will guarantee lower semicontinuity of the functional F in some natural function space; (ii) to study the relaxed energy, and the identification of the relaxed functional in some integral form, when the lower semicontinuity of F fails. The aim of these notes is to present some lower semicontinuity and relaxation results for the class of functionals of the form (1).