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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