*Submitted Paper*

**Inserted:** 9 oct 2018

**Last Updated:** 9 oct 2018

**Year:** 2018

**Abstract:**

Let $\Omega\subseteq R^2$ be a bounded piecewise $C^{1,1}$ open set with convex corners, and let $ MS(u):=\int_\Omega \vert \nabla u\vert^2\,dx+\alpha {\mathcal H}^1(J_u)+\beta\int_\Omega(u-g)^2 dx $ be the Mumford-Shah functional on the space $SBV(\Omega)$, where $g\in L^\infty(\Omega)$ and $\alpha,\beta>0$. We prove that the function $u\in H^1(\Omega)$ such that $$ \begin{cases} -\Delta u+\beta u=\beta g&\text{in }\Omega\\ \frac{\partial u}{\partial \nu}=0 &\text{on }\partial\Omega \end{cases} $$ is a local minimizer of $MS$ with respect to the $L^1$-topology. This is obtained as an application of interior and boundary monotonicity formulas for a weak notion of quasi minimizers of the Mumford-Shah energy. The local minimality result is then extended to more general free discontinuity problems taking into account also boundary conditions.

**Download:**