𝒜$\mathcal{A}$-quasiconvexity at the boundary and weak lower semicontinuity of integral functionals

Abstract

We state necessary and su cient conditions for weak lower semicontinuity of integral functionals of the form u → ∫ Ω h (x, u(x)) dx, where h is continuous and possesses a positively p-homogeneous recession function, p > , and u ∈ L p (Ω; ℝ m ) lives in the kernel of a constant-rank rst-order di erential operator A which admits an extension property. In the special case A = curl, apart from the quasiconvexity of the integrand, the recession function's quasiconvexity at the boundary in the sense of Ball and Marsden is known to play a crucial role. Our newly de ned notions of A-quasiconvexity at the boundary, generalize this result. Moreover, we give an equivalent condition for the weak lower semicontinuity of the above functional along sequences weakly converging in L p (Ω; ℝ m ) and approaching the kernel of A even if A does not have the extension property.

A variational approach to nonlinear electro-magneto-elasticity: Convexity conditions and existence theorems

“…where the deformation gradient outside is defined by (12) using the extensionỹ on c which is fixed, and J = det F. Following [39,Chapter 8], we note that the last term in (23) is independent of the choice of the fictitious 'deformation'ỹ since it can be transformed into the vacuum energy…”

“…Closely related is the compensated compactness theory [9,10]. The reader is referred to [11][12][13] for more recent developments and additional literature. This section discusses these notions from a general point of view; the specialization to electro-magneto-elastic materials is the subject of the succeeding sections.…”

“…for the deformation gradient F . We make a full use of (1) and (2) by adopting the convexity theory under differential constraints known as the scriptA -quasiconvexity theory [7–13]. Central to the theory are the notions of scriptA -quasiconvex function, scriptA -quasiaffine function and scriptA -polyconvex function, see Definitions 5.1 and 5.4.…”

“…Our choice of the basic variables in the constitutive equations are the deformation gradient F, the (lagrangean) electric displacement d and the (lagrangean) magnetic induction b. Among the new issues that we encounter is that the electromagnetic variables satisfy div d = 0, div b = 0 ( 1 ) identically as a counterpart of curl F = 0 ( 2 ) for the deformation gradient F. We make a full use of (1) and (2) by adopting the convexity theory under differential constraints known as the A-quasiconvexity theory [7][8][9][10][11][12][13]. Central to the theory are the notions of A-quasiconvex function, A-quasiaffine function and A-polyconvex function, see Definitions 5.1 and 5.4.…”

A note on $3$d-$1$d dimension reduction with differential constraints

“…Remark 3. Lemma 2.2 essentially requires ω to be an A -free extension domain in the sense of [17,Definition 4.3]. The example of the differential operator associated with the Cauchy-Riemann equations (d = m = 2, C ∼ = R 2 ) makes clear that an extension operator as in Lemma 2.2 may not always exist; for more details see [17,Section 4.2].…”