Local minimality results for the Mumford-Shah functional via monotonicity

Let omega ⊆ R² be a bounded piecewise C¹,¹ open set with convex corners, and let MS(u):= ∫ omega [pipe]∇u[pipe]² dx +alphaH¹(Ju)beta ∫ omega [pipe]u-g[pipe]² dx be the Mumford-Shah functional on the space SBV(omega), where g ∈ L∞(omega) and alpha,beta> 0. We prove that the function u ∈ H¹(omega) such that [-[increment]u +beta u = beta g in omega, ∂u/∂v = 0 on ∂omega is a local minimizer of MS with respect to the L¹-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.