The Regularity Theory for the Mumford-Shah Functional on the Plane

This book provides a thorough and up-to-date account of the state of art of the regularity theory for minimizers of the Mumford-Shah functional of image segmentation in the 2-dimensional setting. Starting with some classical preliminary results, which settle the issue of existence of minimizing couples (u,K), the structure of the set K is then analyzed by using \varepsilon-regularity theorems. Several consequences of the latter are also investigated, in particular leading to different characterizations of the Mumford-Shah conjecture.

The proofs are given with full details, often revisiting the relevant literature and introducing new arguments.