Higher Integrability for Minimizers of the Mumford-Shah Functional

We prove higher integrability for the gradient of local minimizers of the Mumford-Shah energy functional, providing a positive answer to a conjecture of De Giorgi (Free discontinuity problems in calculus of variations. Frontiers in pure and applied mathematics, North-Holland, Amsterdam, pp 55-62, 1991). © 2014 Springer-Verlag Berlin Heidelberg.