Lower semi-continuity for non-coercive polyconvex integrals in the limit case

Abstract

Abstract. Lower semicontinuity results for polyconvex functionals of the Calculus of Variations along sequences of maps u : Ω ⊂ R n → R m in W 1,m , 2 ≤ m ≤ n, weakly converging in W 1,m−1 are established.In addition, for m = n + 1, we also consider the autonomous case for weakly converging maps in W 1,n−1 .

Vector variational problem with knitting boundary conditions

“…because in such case we deduce from (10) that the sequence of (continuous) functions {v n (σ (x))} converges to Tr u (σ (x)) for all x ∈ Γ + up to the negligible set of points E + 0 := σ −1 E − 0 . On the other hand, (11) follows easily from the left inequality in (8) and from the definition of Hausdorff measure (see [13, p. 171]):…”

“…The limit case p = N − 1 was adressed in [1,10,6,15]. Very recently (see [14] and [11]) the limit case was studied for polyconvex integrands depending on x and/or on u.…”