Geodesic
On the lower semicontinuity of supremal functional under differential constraints
Ansini, Nadia1, 2  ; Prinari, Francesca3
1 Dip. di Matematica, Sapienza Università di Roma, P.le Aldo Moro 2, 00185 Rome, Italy.
2 Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK
3 Dip. di Matematica e Informatica, Università di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 1053-1075

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We study the weak* lower semicontinuity of functionals of the form

F(V)= x∈ f(x,V(x)),

where Ω⊂ℝ N is a bounded open set, V∈L ∞ (Ω;)∩ Ker and �� is a constant-rank partial differential operator. The notion of ��-Young quasiconvexity, which is introduced here, provides a sufficient condition when f(x,·) is only lower semicontinuous. We also establish necessary conditions for weak* lower semicontinuity. Finally, we discuss the divergence and curl-free cases and, as an application, we characterise the strength set in the context of electrical resistivity.

Reçu le :
DOI : 10.1051/cocv/2014058
Classification : 49J45, 35E99
Keywords: Supremal functionals, Γ-convergence, Lp-approximation, lower semicontinuity, 𝒜-quasiconvexity

Ansini, Nadia 1, 2 ; Prinari, Francesca 3

1 Dip. di Matematica, Sapienza Università di Roma, P.le Aldo Moro 2, 00185 Rome, Italy.
2 Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK
3 Dip. di Matematica e Informatica, Università di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy. 
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     title = {On the lower semicontinuity of supremal functional under differential constraints},
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Ansini, Nadia; Prinari, Francesca. On the lower semicontinuity of supremal functional under differential constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 1053-1075. doi: 10.1051/cocv/2014058

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