C-1-Arcs for minimizers of the Mumford-Shah functional

Let Omega be a bounded domain in R(2) and g be a bounded measurable function on Omega. The Mumford-Shah functional is defined by J(u, K) = integral integral(Omega\K)\u - g\(2) + integral integral(Omega\K)\del u\(2) + H-1(K), where K is a closed subset of Omega, H-1(K) denotes the one-dimensional Hausdorff measure of K, and u is a function on Omega\K with a derivative in L(2). The main result of this paper is that there is a constant C for which, whenever (u, K) is an irreducible minimizer for J and B(x, r) is a disk centered on K and contained in Omega, there is a disk D centered on K, contained in B(x, r), with radius greater than or equal to C(-1)r and such that K boolean AND D is a C-7/6-curve that crosses D. In particular, the set of points K around which K is not a C-7/6-curve has zero H-1-measure.