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On a \(\Gamma\)-limit of a family of anisotropic singular perturbations. (English) Zbl 0885.49006
In the paper the nonconvex energy functional \[ E(u) = \int_{\Omega}W(u(x))\;dx \tag{1} \] and the family of perturbed and rescaled energies \[ F_{\varepsilon}[u] = {1 \over {\varepsilon}} \int_{\Omega} W(u)\;dx + {\varepsilon} \int_{\Omega} h^2(x,\nabla u)\;dx \tag{2} \] are considered (\(\Omega\) being an open and bounded domain in \(R^N\), \(u: \Omega \to R^p\) and \(W\) supporting two phases). Some results of L. Modica (for the scalar valued case of \(h\)) and of A. Barroso and I. Fonseca (for the vector valued case of \(h\)) concerning the characterization of the De Giorgi \(\Gamma\) limit of functionals (2) by means of the so-called interfacial energy density are recalled. Then an example of functions \(W\) and \(h\) is provided (and some of its generalizations are indicated as well) showing that, unlikely to the scalar valued case, in the vector valued case the minimizers of functionals \(F_{\varepsilon}\) given by (2) for small \(\varepsilon\), in general, might not be essentially locally constant along the boundary of an appropriately defined set. This complicates considerably the computation of the interfacial energy density.

49J45 Methods involving semicontinuity and convergence; relaxation
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