HIGHER INTEGRABILITY FOR MINIMIZERS OF THE MUMFORD-SHAH FUNCTIONAL

Abstract. 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 [5]. 1. introduction Free discontinuity problems are a class of variational problems which involve pairs (u,K) where K is some closed set and u is a function which minimizes some energy outside K. One of the most famous examples is given by the Mumford-Shah energy functional, which arises in image segmentation [10]: given a open set Ω ⊂ Rn, for any K ⊂ Ω relatively closed and u ∈ W 1,2(Ω \K), one defines the Mumford-Shah energy of (u,K) in Ω to be MS(u,K)[Ω]: