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 $\epsilon$-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.