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Using a calibration method we prove that, if $\Gamma \subset \Omega$ is a closed regular hypersurface and if the function $g$ is discontinuous along $\Gamma$ and regular outside, then the function $u_{\beta }$ which solves $$\left\{\begin{array}{cc}\Delta u_{\beta }=\beta (u_{\beta }-g)\hfill & \text{in}\Omega \setminus \Gamma \hfill \\ {\partial }_{\nu }{u}_{\beta }=0\hfill & \text{on}\partial \Omega \cup \Gamma \hfill \end{array}\right.$$ is in turn discontinuous along $\Gamma$ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional $${\int }_{\Omega \setminus S_u}{\left|\nabla u\right|}^{2}dx+{\mathscr{H}}^{n-1}\left(S_u\right)+\beta {\int }_{\Omega \setminus S_u}{(u-g)}^{2}dx,$$ over $SBV\left(\Omega \right)$, for $\beta$ large enough. Applications of the result to the study of the gradient flow by the method of minimizing movements are shown.

Following the $\Gamma$-convergence approach introduced by Müller and Ortiz, the convergence of discrete dynamics for lagrangians with quadratic behavior is established.

We prove necessary and sufficient conditions for the validity of the classical chain rule in the Sobolev space $W_{\text{loc}}^{1,1}({ℝ}^N;{ℝ}^d)$ and in the space $B{V}_{\text{loc}}({ℝ}^N;{ℝ}^d)$ of functions of bounded variation.

Following the -convergence approach introduced by Müller and Ortiz, the convergence of discrete dynamics for Lagrangians with quadratic behavior is established.