Regularity of the singular set for Mumford-Shah minimizers in R3 near a minimal cone

Abstract

We show that if (u;K) is a minimizer of the Mumford-Shah functional in an open set of R 3 , and if x2 K and r > 0 are such that K is close enough to a minimal cone of type P (a plane), Y (three half planes meeting with 120 angles) or T (cone over a regular tetrahedron centered at the origin) in terms of Hausdor distance in B(x;r), then K is C 1; equivalent to the minimal cone in B(x;cr) where c < 1 is an universal constant.