Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in ${ℝ}^n$, then in metric spaces, and finally in the space of probability measures endowed with the Wasserstein distance (induced by the quadratic transport cost). Differently from the usual theory by Jordan-Kinderlehrer-Otto and Ambrosio-Gigli-Savaré, we propose an approach where the optimality conditions for the minimizers of the optimization problems that one solves at every time step...

Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in ${ℝ}^n$, then in metric spaces, and finally in the space of probability measures endowed with the Wasserstein distance (induced by the quadratic transport cost). Differently from the usual theory by Jordan-Kinderlehrer-Otto and Ambrosio-Gigli-Savaré, we propose an approach where the optimality conditions for the minimizers of the optimization problems that one solves at every time step...

Si prova la maggior sommabilità del gradiente dei minimi locali di funzionali integrali della forma $${\int }_{\mathrm{\Omega }}f\left(Du\right)dx,$$ dove $f$ soddisfa l'ipotesi di crescita $${\left|\xi \right|}^p-c_1\le f\left(\xi \right)\le c\left(1+{\left|\xi \right|}^q\right),$$ con $1<p<q\le 2$. L'integrando $f$ è $C^2$ e $DDf$ ha crescita $p-2$ dal basso e $q-2$ dall'alto.

A two-person zero-sum differential game with unbounded controls is considered. Under proper coercivity conditions, the upper and lower value functions are characterized as the unique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacs equations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upper and lower value functions coincide, leading to the existence of the value function of the differential game. Due to the unboundedness of the controls,...

Effective, simulation-based trajectory optimization algorithms adapted to heterogeneous computers are studied with reference to the problem taken from alpine ski racing (the presented solution is probably the most general one published so far). The key idea behind these algorithms is to use a grid-based discretization scheme to transform the continuous optimization problem into a search problem over a specially constructed finite graph, and then to apply dynamic programming to find an approximation...

We prove regularity results for real valued minimizers of the integral functional $\int f\left(x,u,Du\right)$ under non-standard growth conditions of $p\left(x\right)$-type, i.e. $$L^{-1}{\left|z\right|}^{p\left(x\right)}\le f\left(x,s,z\right)\le L\left(1+{\left|z\right|}^{p\left(x\right)}\right)$$ under sharp assumptions on the continuous function $p\left(x\right)>1$.

We give a different and probably more elementary proof of a good part of Jean Taylor’s regularity theorem for Almgren almost-minimal sets of dimension $2$ in ${ℝ}^3$. We use this opportunity to settle some details about almost-minimal sets, extend a part of Taylor’s result to almost-minimal sets of dimension $2$ in ${ℝ}^n$, and give the expected characterization of the closed sets $E$ of dimension $2$ in ${ℝ}^3$ that are minimal, in the sense that $H^2(E\setminus F)\le H^2(F\setminus E)$ for every closed set $F$ such that there is a bounded set $B$ so that $F=E$ out...