A monotonicity formula for minimizers of the Mumford-Shah functional in 2d and a sharp lower bound on the energy density

The Mumford-Shah functional - being roughly speaking defined as the sum of the Dirichlet energy of a function plus the (d-1)-dimensional Hausdorff measure of its set of discontinuities - has been originally proposed by Mumford and Shah in the context of image segmentation; it is also an important prototypical model for the energy arising in mathematical models for fracture mechanics. Its minimization problem is one of the most important instances of a free discontinuity problem, a problem class in the calculus of variations for which the discontinuity set is obtained as a result of the energy minimization process. The Mumford-Shah functional is the subject of a number of intriguing conjectures, most famously the Mumford-Shah conjecture on the dimension and structure of the singular set of minimizers in the 2d case.

We establish a new monotonicity formula for minimizers of the Mumford-Shah functional in 2d. Our monotonicity formula is formulated in terms of a truncation of an entropy introduced by David and Leger. It is in particular able to discriminate between singular points that are part of a C^1 interface and any other type of singularity in terms of a finite gap in the entropy. As a corollary, we prove an optimal lower bound on the energy density around any nonsmooth point for minimizers of the Mumford-Shah functional.