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 $$sup_{int_Dtheta,dx=m} inf_{uin H^1_0(D)}int_DBig(frac{1+theta}{2}| abla u|^2-fuBig),dx.$$ We prove the existence of an optimal reinforcement $theta$ and that it has some higher integrability properties. We also provide some numerical computations for $theta$ and $u$.