Spectral Optimization Problems

In this survey paper we present a class of shape optimization problems where the cost function involves the solution of a PDE of elliptic type in the unknown domain. In particular, we consider cost functions which depend on the spectrum of an elliptic operator and we focus on the existence of an optimal domain. The known results are presented as well as a list of still open problems. Related fields as optimal partition problems, evolution flows, Cheeger-type problems, are also considered.Comment: 42 pages with 8 figure

Shape optimization problems on metric measure spaces

We consider shape optimization problems of the form $$\min\big\{J(\Omega)\ :\ \Omega\subset X,\ m(\Omega)\le c\big\},$$ where $X$ is a metric measure space and $J$ is a suitable shape functional. We adapt the notions of $\gamma$-convergence and weak $\gamma$-convergence to this new general abstract setting to prove the existence of an optimal domain. Several examples are pointed out and discussed.Comment: 27 pages, the final publication is available at http://www.journals.elsevier.com/journal-of-functional-analysis

Improved energy bounds for Schr\"odinger operators

Given a potential $V$ and the associated Schr\"odinger operator $-\Delta+V$, we consider the problem of providing sharp upper and lower bound on the energy of the operator. It is known that if for example $V$ or $V^{-1}$ enjoys suitable summability properties, the problem has a positive answer. In this paper we show that the corresponding isoperimetric-like inequalities can be improved by means of quantitative stability estimates.Comment: 31 page

On some systems controlled by the structure of their memory

We consider an optimal control problem governed by an ODE with memory playing the role of a control. We show the existence of an optimal solution and derive some necessary optimality conditions. Some examples are then discussed

Optimal spatial pricing strategies with transportation costs.

We consider an optimization problem in a given region Q where an agent has to decide the price p(x) of a product for every x ∈ Q. The customers know the pricing pattern p and may shop at any place y, paying the cost p(y) and additionally a transportation cost c(x, y) for a given trans- portation cost function c. We will study two models: the first one where the agent operates everywhere on Q and a second one where the agent op- erates only in a subregion. For both models we discuss the mathematical framework and we obtain an existence result for a pricing strategy which maximizes the total profit of the agent. We also present some particular cases where more detailed computations can be made, as the case of con- cave costs, the case of quadratic cost, and the onedimensional case. Finally we discuss possible extensions and developments, as for instance the case of Nash equilibria when more agents operate on the same market.Optimization; Pricing; Transportation costs;