The terminology "free-discontinuity problems" refers to a class of variational problems characterized by a competition between volume and surface energies. Among them the Mumford-Shah functional represents an outstanding example, arising in the framework of image segmentation.
Scope of this seminar is to give an overview of the Mumford-Shah problem. After a brief introduction to the spaces of BV and SBV functions, we will focus on the problem of the existence of minimizers for the MS functional. Then, elliptic variational approximations (via Gamma-convergence) of the MS functional will be discussed.
L. Ambrosio, N. Fusco, D. Pallara: Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford, 2000.
A. Braides: Gamma-convergence for beginners. Oxford Lecture Series in Mathematics and its Applications, 22. Oxford University Press, Oxford, 2002.