Mathematics

The space $SBV(\Omega)$ of special functions of bounded variation, and $GSBV(\Omega)$ of generalised special functions of bounded variation, have been introduced to study the so called free discontinuity problems. In these spaces it is possible to define a trace operator, whose definition coincides with the usual one, when  is a Sobolev function. Unfortunately, due to the fact that a sequence in  $(G)SBV(\Omega)$ may have jump sets getting infinitesimally close to the boundary of $\Omega$, the trace operator is not continuous. This lack of continuity leads for example to free discontinuity problems coupled with prescribed Dirichlet condition, having no solution. In this talk I present a possible way to overcome this problem, by restricting our attention to a smaller class of functions, i.e.  $(G)SBV(\Omega;\Gamma)$ , which are the functions in  $GSBV(\Omega)$  whose jump sets are obliged to lie in the $n-1$-dimensional set $\Gamma$. In these spaces it is possible to introduce a suitable weight function on the $H^{n-1}$ measure of $\partial Omega$, to obtain some continuity results for the trace operator. Finally I will show some applications to the Mumford-Shah minimization problem with Dirichlet condition on the boundary, and to the existence of a solution to the wave equation in a domain with prescribed arbitrary growing cracks with Neumann condition on the boundary.