Simplified Excision Techniques for Free Discontinuity Problems in Several Variables

One of the methods proposed for the study of the problems with a free discontinuity set in two variables has been considered here under slightly different abstract assumptions from those in the previous papers. The technical difficulties are in this new setting simplified and less background is required to the reader. Moreover, applications to problems in several variables can be found thanks to a Sobolev-type theorem which ensures the Hِlder continuity of certain functions out of a suitable neighborhood of their discontinuity set. Many of the results established so far for the case of two dimensions can be so extended to the general case. A new property is also proved.