Partial Regularity for Minimizers of Free Discontinuity Problems with p-th Growth

In the last ten years many problems have been investigated which were denoted by Ennio De Giorgi in [Gio9lb] as free discontinuity problems.In such variational problems we want to minimize a functional which depends on a closed setKand on a function u suitably smooth outside ofK.A functional widely studied in this field was proposed by Mumford and Shah [MS89b] (see also [BZ87]) for a variational approach to the segmentation problem in computer vision theory. Other functionals have been considered in connection with fracture mechanics, liquid crystals theory, immiscible fluids, elastic-plastic plates (see [AFF93], [Amb90], [ACM97], [Car95], [CLT92], [CLT94], [CLT96], [CT91] and the references therein).