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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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