Local minimality results for the Mumford-Shah functional via monotonicity

Let [math] be a bounded piecewise [math] open set with convex corners, and let ¶ MS ( u ) : = ∫ Ω | ∇ u | 2 d x + α ℋ 1 ( J u ) + β ∫ Ω | u − g | 2 d x ¶ be the Mumford–Shah functional on the space [math] , where [math] and [math] . We prove that the function [math] such that ¶ − Δ u + β u = β g in Ω , ∂ u ∕ ∂ ν = 0 on ∂ Ω ¶ is a local minimizer of [math] with respect to the [math] -topology. This is obtained as an application of interior and boundary monotonicity formulas for a weak notion of quasiminimizers of the Mumford–Shah energy. The local minimality result is then extended to more general free discontinuity problems taking into account also boundary conditions.