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 $\sup_{\int_Dθ\,dx=m}\ \inf_{u\in H^1_0(D)}\int_D\Big(\frac{1+θ}{2}|\nabla u|^2-fu\Big)\,dx.$ 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 $u$.

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