A free boundary problem arising in PDE optimization

A free boundary problem arising from the optimal reinforcement of a membrane or from the reduction of traffic congestion is considered; it is of the form sup ∫ D θ dx=m inf u∈H1 0 (D) ∫ D (1 + θ 2 |∇u| − fu ) dx. We prove the existence of an optimal reinforcement θ and that it has some higher integrability properties. We also provide some numerical computations for θ and u.