No abstract

We study a new class of distances between Radon measures similar to those studied in [13]. These distances (more correctly pseudo-distances because can assume the value +∞) are defined generalizing the dynamical formulation of the Wasserstein distance by means of a concave mobility function. We are mainly interested in the physical interesting case (not considered in [13]) of a concave mobility function defined in a bounded interval. We state the basic properties of the space of measures endowed with this pseudo-distance. Finally, we study in detail two cases: the set of measures defined in R d with finite moments and the set of measures defined in a bounded convex set. In the two cases we give sufficient conditions for the convergence of sequences with respect to the distance and we prove a property of boundedness.

Closed sets $K\subset \mathbb R^{n}$ satisfying an external sphere condition with uniform radius (called $\varphi$-convexity, proximal smoothness, or positive reach) are considered. It is shown that for $\mathcal H^{n-1}$-a.e. $x\in \partial K$ the proximal normal cone to $K$ at $x$ has dimension one. Moreover if $K$ is the closure of an open set satisfying a (sharp) nondegeneracy condition, then the De Giorgi reduced boundary is equivalent to $\partial K$ and the unit proximal normal equals $\mathcal H^{n-1}$-a.e. the (De Giorgi) external normal. Then lower semicontinuous functions $f:\mathbb R^{n}\rightarrow \mathbb R\cup\{ +\infty\}$ with $\varphi$-convex epigraph are shown, among other results, to be locally $BV$ and twice $\mathcal L^{n}$-a.e. differentiable; furthermore, the lower dimensional rectifiability of the singular set where $f$ is not differentiable is studied. Finally we show that for $\mathcal L^{n}$-a.e. $x$ there exists $\delta (x )>0$ such that $f$ is semiconvex on $B(x,\delta(x))$. We remark that such functions are neither convex nor locally Lipschitz, in general. Methods of nonsmooth analysis and of geometric measure theory are used

A minimal time problem with linear dynamics and convex target is considered. It is shown, essentially, that the epigraph of the minimal time function T (•) is ϕ-convex (i.e., it satisfies a kind of exterior sphere condition with locally uniform radius), provided T (•) is continuous. Several regularity properties are derived from results in [G. Colombo and A. Marigonda, Calc. Var. Partial Differential Equations, 25 (2005), pp. 1-31], including twice a.e. differentiability of T (•) and local estimates on the total variation of DT .

No abstract