We consider a thin multidomain of ℝN, N ≥ 2, consisting (e.g. in a 3D setting) of a vertical rod upon a horizontal disk. In this thin multidomain we introduce a bulk energy density of the kind W(D2U), where W is a convex function with growth p ∈ ]1,+∞[, and D2U denotes the Hessian tensor of a scalar (or vector-valued) function U. By assuming that the two volumes tend to zero with the same rate, under suitable boundary conditions, we prove that the limit model is well-posed in the union of the limit domains, with dimensions, respectively, 1 and N - 1. Moreover, we show that the limit problem is uncoupled if [Formula: see text], "partially" coupled if [Formula: see text], and coupled if N - 1 < p. The main result is applied in order to derive the equilibrium configuration of two joint beams, T-shaped, clamped at the three endpoints and subject to transverse loads. The main result is also applied in order to describe the equilibrium configuration of a wire upon a thin film with contact at the origin, when the thin structure is filled with a martensitic material.

Abstract\ud A Γ -convergence analysis is used to perform a 3D–2D dimension reduction of variational problems with linear growth. The\ud adopted scaling gives rise to a nonlinear membrane model which, because of the presence of higher order external loadings inducing\ud a bending moment, may depend on the average in the transverse direction of a Cosserat vector field, as well as on the deformation\ud of the mid-plane. The assumption of linear growth on the energy leads to an asymptotic analysis in the spaces of measures and of\ud functions with bounded variation

We study variational problems involving nonlocal supremal functionalswhere ⊂ R n is a bounded, open set and W : R m × R m → R is a suitable function. Motivated by existence theory via the direct method, we identify a necessary and sufficient condition for L ∞ -weak * lower semicontinuity of these functionals, namely, separate level convexity of a symmetrized and suitably diagonalized version of the supremands. More generally, we show that the supremal structure of the functionals is preserved during the process of relaxation. The analogous statement in the related context of double-integral functionals was recently shown to be false. Our proof relies substantially on the connection between supremal and indicator functionals. This allows us to recast the relaxation problem into characterizing weak * closures of a class of nonlocal inclusions, which is of independent interest. To illustrate the theory, we determine explicit relaxation formulas for examples of functionals with different multi-well supremands.

A lower semicontinuity result is proved in the space of special vector fields with bounded deformation for a fracture energetic model of the typeA representation of the energy density Ψ , which ensures lower semicontinuity, is also given.

The paper is devoted to determine necessary and sufficient conditions for existence of solutions to the problem inf ess sup(Ω) , when the supremand f is not necessarily level convex. These conditions are obtained through a comparison with the related level convex problem and are written in terms of a differential inclusion involving the boundary datum. Several conditions of convexity for the supremand f are also investigated. RésuméDans cet article onétudie des conditions nécessaires et suffisantes pour l'existence de solutions pour le problème de minimisation inf ess sup x∈Ω f (∇u(x)) : u ∈ u0 + W 1,∞ 0 (Ω) lorsque la fonction f n'est pas une fonction convexe par niveaux. La stratégie utilisée pour obtenir ces conditions est celle de comparer ce problème avec son problème relaxé. On obtient comme condition nécessaire et suffisante une inclusion différentielle sur la donnée au bord. Onétudie aussi plusieurs conditions de convexité.

It is studied the lower semicontinuity of functionals of the type Ω f (x, u, v, ∇u)dx with respect to the (W 1,1 × L p )-weak * topology. Moreover in absence of lower semicontinuity, it is also provided an integral representation in W 1,1 × L p for the lower semicontinuous envelope.We are interested in studying the lower semicontinuity and relaxation of (1) with respect to the L 1 -strong ×L p -weak convergence. Clearly, bounded sequences {u n } ⊂ W 1,1 (Ω; R n ) may converge in L 1 , up to a subsequence, to a BV function. In this paper we restrict our analysis to limits u which are in W 1,1 (Ω; R n ). Thus, our results can be considered as a step towards the study of relaxation in BV (Ω; R n ) × L p (Ω; R m ) of functionals (1).We will consider separately the cases 1 < p < ∞ and p = ∞. To this end we introduce for 1 < p < +∞ the functionalfor any pair (u, v) ∈ W 1,1 (Ω; R n ) × L p (Ω; R m ), and for p = ∞ the functional J ∞ (u, v) := inf lim inf J(u n , v n ) :

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The paper is devoted to the relaxation and integral representation in the space of functions of bounded variation for an integral energy arising from optimal design problems. The presence of a perimeter penalization is also considered in order to avoid non existence of admissible solutions, besides this leads to an interaction in the limit energy. Also more general models have been taken into account.