Closure result for $$ \Gamma $$-limits of functionals with linear growth

Abstract

AbstractWe consider integral functionals$$ \mathcal {F}^{(j)}_{\varepsilon } $$Fε(j), doubly indexed by$$ \varepsilon > 0 $$ε>0and$$j \in \mathbb N\cup \{ \infty \}$$j∈N∪{∞}, satisfying a standard linear growth condition. We investigate the question of$$ \Gamma $$Γ-closure, i.e., when the$$ \Gamma $$Γ-convergence of all families$$ \{ \mathcal {F}^{(j)}_{\varepsilon } \}_{\varepsilon }$${Fε(j)}εwith finitejimplies$$ \Gamma $$Γ-convergence of$$\{ \mathcal {F}^{(\infty )}_{\varepsilon } \}_{\varepsilon }$${Fε(∞)}ε. This has already been explored forp-growth with$$ p > 1 $$p>1. We show by an explicit counterexample that due to the differences between the spaces$$ W^{1,1} $$W1,1and$$ W^{1,p} $$W1,pwith$$ p > 1 $$p>1, an analog cannot hold. Moreover, we find a sufficient condition for a positive answer.