*Submitted Paper*

**Inserted:** 14 dec 2006

**Year:** 2006

**Abstract:**

Our main goal is to give a rigorous justification for the Hessian-constrained problems introduced in * below, and to show how they are linked to the optimal design of 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 first order vector problems is obtained, 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 *. Some attention is also paid to the possible different approaches to the afore mentioned double limit process, in both the cases of real and ficticious materials, which might favour some debate on the modelling of thin plates.

* G. Bouchitté, I.Fragalà: Optimality conditions for mass design problems and applications to thin plates, to appear on Arch. Rat. Mech. Analysis.