abstract: It is a joint work with Ilaria Fragalà motivated by the dual formulation of the so called Michell's problem where it appears, like in Monge-Kantorovich theory, a linear problem with constraint on the first order gradient. The Hessian counterpart of this has been recently introduced and studied in 1.
The goal of this lecture is to show how the aboved mentioned Hessian constrained problem can derived through a 3D-2D reduction analysis and how it arises naturally in optimal design problems for thin plates. To that aim, we study the asymptotic behaviour of a sequence of optimal elastic compliance problems, in the double limit when both the maximal height of the design region and the total volume of the material tend to zero. In the vanishing volume limit, a sequence of linear constrained vector first order problem is obtained 2, which in turn - in the vanishing thickness limit -produces a new linear constrained problem where both first and second order gradients appear. When the load is suitably chosen, only the Hessian constraint is active, and we recover exactly the plate optimization problem studied in 1. Some attention is also paid to the possible different approaches to the afore mentioned double limit process, in both the cases of ficticious and non-ficticious materials, which might favour some debate on the modelling of thin plates.
1 G . Bouchitté and I. Fragalà: Optimality conditions for mass design problems and applications to thin plates. Arch. Rat. Mech. Analysis, to appear.
2 G. Bouchitté: Optimization of light structures: the vanishing mass conjecture. Homogenization, 2001 (Naples). GAKUTO Internat.Math. Sci.Appl. 18 Gakkotosho, Tokyo (2003), 131-145.