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.

A 3D-2D dimension reduction for a nonlinear optimal design problem with a perimeter penalization is performed in the realm of Γ-convergence, providing an integral representation for the limit functional. RésuméRéduction dimensionnelle 3D−2D d'un problème non linéaire d'optimisation de forme avec pénalisation sur le périmètre On effectue dans ce travail une réduction dimensionnelle 3D-2D d'un problème non linéaire d'optimisation de forme avec une pénalisation du périmètre. Une représentation intégrale de la fonctionnelle limite est obtenue.

Dedicated to our friend and colleague Graça Carita, who left us far too soon.Abstract. In this paper we apply both the procedure of dimension reduction and the incorporation of structured deformations to a three-dimensional continuum in the form of a thinning domain. We apply the two processes one after the other, exchanging the order, and so obtain for each order both a relaxed bulk and a relaxed interfacial energy. Our implementation requires some substantial modifications of the two relaxation procedures. For the specific choice of an initial energy including only the surface term, we compute the energy densities explicitly and show that they are the same, independent of the order of the relaxation processes. Moreover, we compare our explicit results with those obtained when the limiting process of dimension reduction and of passage to the structured deformation is carried out at the same time. We finally show that, in a portion of the common domain of the relaxed energy densities, the simultaneous procedure gives an energy strictly lower than that obtained in the two-step relaxations.

Abstract. An integral representation result is obtained for the relaxation of a class of energy functionals depending on two vector fields with different behaviors which appear in the context of thermochemical equilibria and are related to image decomposition models and directors theory in nonlinear elasticity.

An integral representation formula is obtained for the relaxation of a class of energy functionals defined in the class of S BV p functions that are constrained to have values on the sphere S d−1 .