Inserted: 8 may 2003
Last Updated: 10 may 2005
Journal: Discrete and Continuous Dynamical Systems
\noindent We study necessary and sufficient conditions for the lower-semicontinuity of one-dimensional energies defined on ($BV$ and) $SBV$ of the model form $F(u)= \sup f(u')\vee\sup g([u])$, and prove a relaxation theorem. We apply these results to the study of problems with Dirichlet boundary conditions, highlighting a complex behaviour of solutions. We draw a comparison with the parallel theory for integral energies on $SBV$.