Smooth and broken minimizers of some free discontinuity problems

We show that minimizers of free discontinuity problems with energy dependent on jump integrals and Dirichlet boundary conditions are smooth provided a smallness condition is imposed on data. We examine in detail two examples: the elastic-plastic beam and the elastic-plastic plate with free yield lines. In both examples there is a gap between the condition for solvability (safe load condition) and this smallness condition (load regularity condition) which imply regularity and uniqueness of minimizers. Such gap allows the existence of damaged/creased minimizers. Eventually we produce explicit examples of irregular solutions when the load is in the gap.