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...

Si studia il comportamento asintotico di una classe di funzionali integrali che possono dipendere da misure concentrate su strutture periodiche multidimensionali, quando tale periodo tende a 0. Il problema viene ambientato in spazi di Sobolev rispetto a misure periodiche. Si dimostra, sotto ipotesi generali, che un appropriato limite può venire definito su uno spazio di Sobolev usuale usando tecniche di $\mathrm{\Gamma }$-convergenza. Il limite viene espresso come un funzionale integrale il cui integrando è caratterizzato...

The paper is a continuation of a previous work of the same authors dealing with homogenization processes for some energies of integral type arising in the modeling of rubber-like elastomers. The previous paper took into account the general case of the homogenization of energies in presence of pointwise oscillating constraints on the admissible deformations. In the present paper homogenization processes are treated in the particular case of fixed constraints set, in which minimal coerciveness hypotheses...

The paper is a continuation of a previous work of the same authors dealing with homogenization processes for some energies of integral type arising in the modeling of rubber-like elastomers. The previous paper took into account the general case of the homogenization of energies in presence of pointwise oscillating constraints on the admissible deformations. In the present paper homogenization processes are treated in the particular case of fixed constraints set, in which minimal coerciveness hypotheses...