On the lower semicontinuity of supremal functional under differential constraints
Dip. di Matematica, Sapienza Università di Roma, P.le Aldo Moro
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.
We study the weak* lower semicontinuity of functionals of the form
where Ω ⊂ ℝN is a bounded open set, and 𝒜 is a constant-rank partial differential operator. The notion of 𝒜-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.
Mathematics Subject Classification: 49J45 / 35E99
Key words: Supremal functionals / Γ-convergence / Lp-approximation / lower semicontinuity / 𝒜-quasiconvexity
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