Existence and relaxation problems in optimal shape design

Existence and Approximationin Optimal Shape

We consider a system given by a second order elliptic equation with jumps in the coeecients. This models a body made of two diierent materials and we study the question of the material distribution that minimizes a certain cost functional. We introduce a local compactness condition for a class of characteristic functions to obtain the existence of the optimum. We also indicate a new approximati...

Optimal design problems in rough inhomogeneous media. Existence theory∗

This paper settles the existence question for a rather general class of convex optimal design problems with a volume constraint. In low dimensions, we prove the existence of an optimal configuration for general convex minimization problems ruled by bounded measurable degenerate elliptic operators. Under a mild continuity assumption on the medium, the free boundary is proven to enjoy the appropr...

Relaxation through Homogenization for Optimal Design Problems with Gradient Constraints

The problem of relaxation of optimal design problems for multiphase composite structures in the presence of constraints on the gradient of the state variable is addressed. A relaxed formulation for the problem is given in the presence of a finite or infinite number of constraints. The relaxed formulation is used to identify minimizing sequences of configurations of phases.

Control/fictitious domain method for solving optimal shape design problems

— Combining a fictitious domain and an optimal control approach, we present a new method for the numerical realization of optimal shape design problems. This approach enables us to perform ail calculations on a fixed domain with a fixed grid. Finite element approximation is studied. Résumé. — On présente une méthode nouvelle pour la résolution numérique des problèmes de Voptimisation de la form...

Shape-Measure Method for Solving Elliptic Optimal Shape Problems (Fixed Control Case)