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13.
Closure result for ▫$\Gamma$▫-limits of functionals with linear growth
Jesenko, Martin, 1977-
We consider integral functionals ▫${\mathcal F}^{(j)}_{\varepsilon }$▫, doubly indexed by ▫$\varepsilon > 0$▫ and ▫$j \in {\mathbb N}\cup \{ \infty \}$▫, satisfying a standard linear growth ...condition. We investigate the question of ▫$\Gamma$▫-closure, i.e., when the ▫$\Gamma$▫-convergence of all families ▫$\{ {\mathcal F}^{(j)}_{\varepsilon} \}_{\varepsilon }$▫ with finite ▫$j$▫ implies ▫$\Gamma$▫-convergence of ▫$\{ {\mathcal F}^{(\infty )}_{\varepsilon } \}_{\varepsilon }$▫. This has already been explored for ▫$p$▫-growth with ▫$p > 1$▫. We show by an explicit counterexample that due to the differences between the spaces ▫$W^{1,1}$▫ and ▫$W^{1,p}$▫ with ▫$p > 1$▫, an analog cannot hold. Moreover, we find a sufficient condition for a positive answer.
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