We deal with a variational approach to the inverse crack problem, that is the detection and reconstruction of cracks, and other defects, inside a conducting body by performing boundary
measurements of current and voltage type. We formulate such an inverse problem in a free-discontinuity problems framework and propose a novel method for the numerical reconstruction
of the cracks by the available boundary data. The proposed method is amenable to numerical computations and it is justified by a convergence analysis, as the error on the measurements
goes to zero. We further notice that we use the Gamma-convergence approximation of the Mumford–Shah functional due to Ambrosio and Tortorelli as the required regularization term.
