Abstract: "Models involving bulk and interfacial energies have been used to describe phenomena in fracture mechanics, phase transitions, and image segmentation. One of the main mathematical questions involved concerns the regularity of the crack site, or discontinuity set S[subscript u], for local minimizers u of energy functionals of the type G(v):=[integral subscript omega]F([delta]v) dx + H[superscript N-1](Sv[intersection][omega]). The existence of a classical solution in the case where F([xi]):=[absolute value of xi][superscript p], p>1, was proven recently by means of compactness results in a somewhat large functional space, followed by a thorough regularity analysis of the jump set S[subscript u] of a local minimizer u of G thus obtained. Here these regularity properties are extended to a class of anisotropic, non homogeneous densities F, with p-growth."