Many techniques developed for free-discontinuity problems, arising for example in imaging or in fracture mechanics, may be successfully applied to reconstruction methods for inverse problems whose unknowns may be characterized by discontinuous functions.

As an example we consider the inverse problem of determining insulating cracks or cavities by performing few electrostatic measurements on the boundary. We show the validity of this approach both from the theoretical point of view, by a convergence analysis, and from the numerical point of view.