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Un esempio di \(\Gamma^-\)-convergenza. (Italian) Zbl 0356.49008
Summary: The \(\Gamma^-\)-convergence is a new kind of convergence proposed by E. De Giorgi and T. Franzoni [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 58, 842–850 (1975; Zbl 0339.49005)]. Here we study the \(\Gamma^-\)-limit of the functionals on \(L^1(\mathbb{R}^n)\)
\[ F_h(u)=\begin{cases} \int_{\mathbb{R}^n} \left[ \frac{| Du| ^2}{h} +h\mathrm{sen}^2(h\pi u) \right] \mathrm{d}x \qquad &\text{if }u\in H^{1,3}(\mathbb{R}^n)\\ +\infty &\text{otherwise} \end{cases} \]
which is related with the total variation \(\int_{\mathbb{R}^n} | Du|\), and then we study the \(\Gamma^-\)-limit of the functionals obtained by \(F_h\) deleting \(h\) in the argument of \(\mathrm{sen}^2\).

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
49J27 Existence theories for problems in abstract spaces
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