Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford-Shah functional

The paper is concerned with the higher regularity properties of the minimizers of the Mumford-Shah functional. It is shown that, near to singular points where the scaled Dirichlet integral tends to 0, the discontinuity set is close to an Almgren area minimizing set. As a byproduct, the set of singular points of this type has Hausdorff dimension at most N - 2, N being the dimension of the ambient space. Assuming higher integrability of the gradient this leads to an optimal estimate of the Hausdorff dimension of the full singular set.