On a general coarea inequality and applications.
On a variational property of integral functionals and related conjectures
The aim of this paper is to give the proofs of those results that in [4] were only announced, and, at the same time, to propose some possible developments, indicating some of the most significant open problems.
On a variational theory of image amodal completion
On Conditions for Unrectifiability of a Metric Space
We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.
On embedded minimal disks in convex bodies
On minimal hypersurfaces of nonnegatively Ricci curved manifolds.
On some recent developments of the theory of sets of finite perimeter
In this paper we describe some recent progress on the theory of sets of finite perimeter, currents, and rectifiability in metric spaces. We discuss the relation between intrinsic and extrinsic theories for rectifiability
On stability of minimal submanifolds in compact symmetric spaces
On the Equality of two Stieltjes Integrals in the Complex Plane
On the Existence of Integral Currents with Prescribed Mean Curvature Vector.
On the extremals of a subsidiary capillary problem.
On the propagation of singularities of semi-convex functions
On the rectifiability of domains with finite perimeter
On the representation of effective energy densities
On the Representation of Effective Energy Densities
We consider the question raised in [1] of whether relaxed energy densities involving both bulk and surface energies can be written as a sum of two functions, one depending on the net gradient of admissible functions, and the other on net singular part. We show that, in general, they cannot. In particular, if the bulk density is quasiconvex but not convex, there exists a convex and homogeneous of degree 1 function of the jump such that there is no such representation.
On the Singular Structure of Two-Dimensional Area Minimizing Surfaces in Rn.
Optimal regularity for codimension one minimal surfaces with a free boundary.
Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem
In the paper the problem of constructing an optimal urban transportation network in a city with given densities of population and of workplaces is studied. The network is modeled by a closed connected set of assigned length, while the optimality condition consists in minimizing the Monge-Kantorovich functional representing the total transportation cost. The cost of trasporting a unit mass between two points is assumed to be proportional to the distance between them when the transportation is carried...