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 p
rove the existence of an optimal reinforcement $theta$ and that it has some higher integrability properties. We also provide some numerical computations for $theta$ and $u$.
We consider optimization problems for cost functionals which depend on the negative spectrum of Schrodinger operators of the form $-Delta+V(x)$, where $V$ is a potential, with prescribed compact support, which has to be determined. Under suitable ass
umptions the existence of an optimal potential is shown. This can be applied to interesting cases such as costs functions involving finitely many negative eigenvalues.
We consider a given region $Omega$ where the traffic flows according to two regimes: in a region $C$ we have a low congestion, where in the remaining part $Omegasetminus C$ the congestion is higher. The two congestion functions $H_1$ and $H_2$ are gi
ven, 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.
We consider Cheeger-like shape optimization problems of the form $$minbig{|Omega|^alpha J(Omega) : Omegasubset Dbig}$$ where $D$ is a given bounded domain and $alpha$ is above the natural scaling. We show the existence of a solution and analyze as $J
(Omega)$ the particular cases of the compliance functional $C(Omega)$ and of the first eigenvalue $lambda_1(Omega)$ of the Dirichlet Laplacian. We prove that optimal sets are open and we obtain some necessary conditions of optimality.
