Emanuele Paolini

Decomposition of acyclic normal currents in a metric space

Abstract We prove that every acyclic normal one-dimensional real Ambrosio–Kirchheim current in a Polish (i.e. complete separable metric) space can be decomposed in curves, thus generalizing the analogous classical result proven by S. Smirnov in Euclidean space setting. The same assertion is true for every complete metric space under a suitable set-theoretic assumption.

A Short Proof of the Minimality of Simons Cone

In 1969 Bombieri, De Giorgi and Giusti proved that Simons cone is a minimal surface, thus providing the first example of a minimal surface with a singularity. We suggest a simplified proof of the same result. Our proof is based on the use of sub-calibrations, which are unit vector fields extending the normal vector to the surface, and having non-positive divergence. With respect to calibrations (which are divergence free) sub-calibrations are more easy to find and anyway are enough to prove the minimality of the surface.

Optimal transportation networks as flat chains

We provide a model of optimization of transportation networks (e.g. urban traffic lines, subway or railway networks) in a geographical area (e.g. a city) with given density of population and that of services and/or workplaces, the latter being the destinations of everyday movements of the former. The model is formulated in terms of the Federer‐Fleming theory of currents, and allows us to get both the position and the necessary capacity of the optimal network. Existence and some qualitative properties of solutions to the relevant optimization problem are studied. Also, in an important particular case it is shown that the model proposed is equivalent to another known model of optimization of a transportation network, the latter not using the language of currents.

Origami and Partial Differential Equations

Origami is the ancient Japanese art of folding paper and it has well known algebraic and geometrical properties, but it also has unexpected relations with partial differential equations.In this note we describe these relations for a large audience, leaving the technical as pects to other specialized papers.

The Steiner problem for infinitely many points

Let A be a given compact subset of the euclidean space. We consider the problem of finding a compact connected set S of minimal 1dimensional Hausdorff measure, among all compact connected sets containing A. We prove that when A is a finite set any minimizer is a finite tree with straight edges, thus recovery the classical Steiner Problem. Analogously, in the case when A is countable, we prove that every minimizer is a (possibly) countable union of straight segments.

An example of an infinite Steiner tree connecting an uncountable set

Abstract We construct an example of a Steiner tree with an infinite number of branching points connecting an uncountable set of points. Such a tree is proven to be the unique solution to a Steiner problem for the given set of points. As a byproduct we get the whole family of explicitly defined finite Steiner trees, which are unique connected solutions of the Steiner problem for some given finite sets of points, and with growing complexity (i.e. the number of branching points).

STABILITY OF CRYSTALLINE EVOLUTIONS

In this paper we analyze the stability properties of the Wulff-shape in the crystalline flow. It is well known that the Wulff-shape evolves self-similarly, and eventually shrinks to a point. We consider the flow restricted to the set of convex polyhedra, we show that the crystalline evolutions may be viewed, after a proper rescaling, as an integral curve in the space of polyhedra with fixed volume, and we compute the Jacobian matrix of this field. If the eigenvalues of such a matrix have real part different from zero, we can determine if the Wulff-shape is stable or unstable, i.e. if all the evolutions starting close enough to the Wulff-shape converge or not, after rescaling, to the Wulff-shape itself.

Regularity results for some 1-homogeneous functionals

We consider local minimizers for a class of 1-homogeneous integral functionals defined on BVloc(Ω), with Ω ⊆ R2. Under general assumptions on the functional, we prove that the boundary of the subgraph of such minimizers is (locally) a lipschitz graph in a suitable direction. The proof of this statement relies on a regularity result holding for boundaries in R2 which minimize an anisotropic perimeter. This result is applied to the boundary of sublevel sets of a minimizer u ∈ BVloc(Ω).We also provide an example which shows that such regularity result is optimal.

Minimal clusters of four planar regions with the same area

We prove that the optimal way to enclose and separate four planar regions with equal area using the less possible perimeter requires all regions to be connected. Moreover, the topology of such optimal clusters is uniquely determined.

Minimal cluster computation for four planar regions with the same area

Abstract The topology of a minimal cluster of four planar regions with equal areas and smallest possible perimeter was found in [9]. Here we describe the computation used to check that the symmetric cluster with the given topology is indeed the unique minimal cluster.