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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.

We approximate, in the sense of Γ-convergence, one-dimensional free- discontinuity functionals with linear growth in the gradient by means of a sequence of non-local integral functionals depending on the averages of the gradient on small intervals.