A priori bounds and regularization techniques for functionals with free discontinuities

Some "a priori" estimates for minimizers of functionals with free discontinuities can be improved in such a way to give the initial assumptions which make some bootstrap regularity arguments start. Such arguments have been employed so far thanks to generic properties of the singular set and have led to partial regularity results. Their systematic use allows stability methods which lead to prove new properties of the singular set, in particular for the case of two dimentions.