*Published Paper*

**Inserted:** 26 may 2015

**Last Updated:** 21 apr 2018

**Journal:** Calc. Var. PDE

**Year:** 2015

**Abstract:**

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

$\displaystyle\sup_{\int_D\theta\,dx=m}\ \inf_{u\in H^1_0(D)}\int_D\Big(\frac{1+\theta}{2}\vert

\nabla u\vert

^2-fu\Big)\,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 $u$.

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