Weak lower semicontinuity for non coercive polyconvex integrals

Abstract

Abstract. We prove a lower semicontinuity theorem for a polyconvex functional of integral form, related to maps u W R n ! R m in W 1;n . I R m / with n m 2, with respect to the weak W 1;p -convergence for p > m 1, without assuming any coercivity condition.

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

“…with Theorem 1.4 below for a sharper result in case m = n). In particular, in Theorem 1.1 we prove the following Serrin type result building upon the chain-rule argument introduced in [4], that extends to this setting the ideas of [32]. However, contrary to the above mentioned results, we also need to assume Lipschitz continuity in the variable u. Theorem 1.1.…”

“…Re-writing the functional through the chain-rule. Using that (u k ) k ⊂ W 1,m , that c is Lipschitz continuous and the multi-linearity and antisimmetry of the determinant, following [4] we can write…”

“…In this section we prove theorem 1.1. The argument is based on the chain-rule formula in the spirit of [4], and on proposition 2.7.…”

“…In this case weak lower semicontinuity holds under weaker assumptions concerning both the growth of the integrands and the underlying topology following the results of Dacorogna & Marcellini [8] (cp. [1,3,4,6,9,11,13,14,15,21,22,24,25]). …”

Lower semicontinuity of variational integrals on elliptic complexes

“…In fact, if N = 2, taking Pα = ∇α, α ∈ C ∞ (R 2 , R), Qγ = curl γ = ∂γ 2 /∂x−∂γ 1 /∂y, γ ∈ C ∞ (R 2 , R 2 ) then one has an elliptic complex and Pα, Q * β is equal to the determinant of the matrix whose rows are given by ∇α, ∇β. In this case, the lower semicontinuity has been studied in a series of papers (see for instance [AD,ADMM,CD,DM2,DS,FH,G,M1,M2,Ma1,Ma2]).…”