Optimal regions for congested transport

We consider a given region $Ω$ where the traffic flows according to two regimes: in a region $C$ we have a low congestion, where in the remaining part $Ω\setminus C$ the congestion is higher. The two congestion functions $H_1$ and $H_2$ are given, but the region $C$ has to be determined in an optimal way in order to minimize the total transportation cost. Various penalization terms on $C$ are considered and some numerical computations are shown.

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