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...
-energy of a curve on LIP-manifolds and on general metric spaces.
Perimeter on Fractal sets.
Phase field method for mean curvature flow with boundary constraints
This paper is concerned with the numerical approximation of mean curvature flow t → Ω(t) satisfying an additional inclusion-exclusion constraint Ω1 ⊂ Ω(t) ⊂ Ω2. Classical phase field model to approximate these evolving interfaces consists in solving the Allen-Cahn equation with Dirichlet boundary conditions. In this work, we introduce a new phase field model, which can be viewed as an Allen Cahn equation with a penalized double well potential. We first justify this method by a Γ-convergence result...
Phase field method for mean curvature flow with boundary constraints
This paper is concerned with the numerical approximation of mean curvature flow t → Ω(t) satisfying an additional inclusion-exclusion constraint Ω1 ⊂ Ω(t) ⊂ Ω2. Classical phase field model to approximate these evolving interfaces consists in solving the Allen-Cahn equation with Dirichlet boundary conditions. In this work, we introduce a new phase field model, which can be viewed as an Allen Cahn equation with a penalized double well potential. We first justify this method by a Γ-convergence result...
Plongements quasiisométriques du groupe de Heisenberg dans , d’après Cheeger, Kleiner, Lee, Naor
Bref survol du théorème de non-plongement de J. Cheeger et B. Kleiner pour le groupe d’Heisenberg dans .
Quadratic tilt-excess decay and strong maximum principle for varifolds
In this paper, we prove that integral -varifolds in codimension 1 with , , have quadratic tilt-excess decay for -almost all , and a strong maximum principle which states that these varifolds cannot be touched by smooth manifolds whose mean curvature is given by the weak mean curvature , unless the smooth manifold is locally contained in the support of .
Quantitative Isoperimetric Inequalities on the Real Line
In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown that, in the Gauss space, a set of given measure and almost minimal Gauss boundary measure is necessarily close to be a half-space.Using only geometric tools, we extend their result to all symmetric log-concave measures on the real line. We give sharp quantitative isoperimetric inequalities and prove that among sets of given measure and given asymmetry (distance to half line, i.e. distance to sets of minimal perimeter),...