Sequences of non-local functionals which approximate free-discontinuity problems (Q1275904)

The aim of the paper is to develop a general method for the approximation, in the sense of \(\Gamma\)-convergence, of free discontinuity problems \[ \min\biggl\{ \int_\Omega g(x,\nabla u) dx+\int_{S_u\cap\Omega} \varphi(x,[u],\nu_u) d{\mathcal H}^{n-1}: u\in \text{SBV}(\Omega)\biggr\} \] (here \(S_u\) is the approximate discontinuity set, \([u]\) is the jump and \(\nu_u\) is the approximate normal to \(S_u\)) by problems defined in Sobolev spaces. It is well known that the nonconvex character of the limit problem prevents the possibility of having an approximation by local (i.e., integral functionals of \(u\) and \(\nabla u\)) functionals. Following a previous work of \textit{G. Dal Maso} and \textit{A. Braides} [Calc. Var. Partial Differ. Equ. 5, No. 4, 293-322 (1997; Zbl 0873.49009)], one looks for nonlocal approximations of the form \[ {1\over\varepsilon}\int_\Omega f_\varepsilon\biggl( \varepsilon\int_\Omega g_\varepsilon(y,\nabla u(y))\psi_\varepsilon(x-y) dy \biggr) dx. \tag{\(*\)} \] In the model case when the limit energy is the Mumford-Shah functional, we have \(f_\varepsilon(t)=1\wedge t\), \(g_\varepsilon(y,p)=| p| ^2\) and \(\varepsilon^n\psi_\varepsilon\) is the characteristic function of the unit ball of \({\mathbb R}^n\). The author exploits the abstract methods of \(\Gamma\)-convergence (localization, fundamental estimate\dots) to prove a rather general convergence result for functionals of the form \((*)\).