Publisher's Synopsis
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.
