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On the nonlocal approximation of free-discontinuity problems. (English) Zbl 0907.49009
The paper deals with the nonlocal approximation, in the sense of $$\Gamma$$-convergence, of a variant of the Mumford–Shah functional in which also the width of the jump (and not only the area of the jump set) is taken into account. Precisely, the authors consider a family of functionals $$F_\varepsilon$$ in $$H^1(\Omega)$$ of the form ${1\over\varepsilon}\int_\Omega f_\varepsilon\bigl(\varepsilon M_\varepsilon|\nabla u|^2(x)\bigr) dx$ where $$M_\varepsilon v(x)$$ denotes the mean value of $$v$$ on $$B_\varepsilon(x)$$. If $$f_\varepsilon$$ converge pointwise to the identity and the rescaled functions $$g_\varepsilon(x)=f_\varepsilon(x/\varepsilon)$$ converge to some non constant function $$g$$, the authors prove that the $$\Gamma$$-limit of $$F_\varepsilon$$ is representable in $$\text{SBV} (\Omega)$$ by $\int_\Omega |\nabla u|^2 dx+\int_{S(u)}\varphi(u^+-u^-) d{\mathcal H}^{n-1}$ where $$\varphi$$ is defined by $\varphi(z):=\inf\left\{\int_{{\mathbb R}}g\Biggl( \int_{x-1}^{x+1}| u'|^2 dt\Biggr) dx:\;u(-\infty)=0, u(+\infty)=z\right\} .$
Reviewer: L.Ambrosio (Pisa)

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 49Q20 Variational problems in a geometric measure-theoretic setting
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