L-infinity energies on discontinuous functions

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') V sup ([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.