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 $$\sup_{\int_D \theta dx=m} \inf_{u\in H_0^1(D)} \int_D \left(\frac{1 + \theta}2 |\nabla u|^2 − f u\right) dx.$$ 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 $u$.