Related papers: On the Steiner property for planar minimizing clus…
We consider the isoperimetric problem for clusters in the plane with a double density, that is, perimeter and volume depend on two weights. In this paper we consider the isotropic case, in the parallel paper "On the Steiner property for…
(N, M)-clusters are partitions of $\mathbb{R}^d$ into N+M regions, where N chambers have prescribed finite measure and M chambers have infinite measure. Locally minimizing clusters are the configurations which minimize the perimeter among…
In this paper we consider the isoperimetric problem with double density in an Euclidean space, that is, we study the minimisation of the perimeter among subsets of $\mathbb{R}^n$ with fixed volume, where volume and perimeter are relative to…
We show the existence of generalized clusters of a finite or even infinite number of sets, with minimal total perimeter and given total masses, in metric measure spaces homogeneous with respect to a group acting by measure preserving…
An infinite cluster $\mathbf E$ in $\mathbb R^d$ is a sequence of disjoint measurable sets $E_k\subset \mathbb R^d$, $k\in \mathbb N$, called regions of the cluster. Given the volumes $a_k\ge 0$ of the regions $E_k$, a natural question is…
We study a variational problem for the perimeter associated with the Grushin plane, called minimal partition problem with trace constraint. This consists in studying how to enclose three prescribed areas in the Grushin plane, using the…
In this article we consider the isoperimetric problem for partitioning the plane into three disjoint domains, one having unit area and the remaining two having infinite area. We show that the only solution, up to rigid motions of the plane,…
We consider the variational problem of minimizing an anisotropic perimeter functional under a volume constraint in a Euclidean convex domain. We extend to this setting analytical properties of the isoperimetric profile, topological features…
In 1993 Foisy et al. proved that the optimal Euclidean planar double bubble---the least-perimeter way to enclose and separate two given areas---is three circular arcs meeting at 120 degrees. We consider the plane with density $r^p$, joining…
We describe a quantitative construction of almost-normal diffeomorphisms between embedded orientable manifolds with boundary to be used in the study of geometric variational problems with stratified singular sets. We then apply this…
The aim of this work is to study the rigidity problem for Steiner's inequality for the anisotropic perimeter, that is, the situation in which the only extremals of the inequality are vertical translations of the Steiner symmetral that we…
We study the isoperimetric problem for anisotropic left-invariant perimeter measures on $\mathbb R^3$, endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm $\phi$ on the horizontal…
We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also…
We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive…
The existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved, together with a partial regularity result.
We perform direct numerical simulations of sub-Kolmogorov, inertial spheroids settling under gravity in homogeneous, isotropic turbulence and find that small-scale clustering, measured via the correlation dimension, depends sensitively on…
Magnetic systems with frustration often have large classical degeneracy. We show that their low-energy physics can be understood as dynamics within the space of classical ground states. We demonstrate this mapping in a family of quantum…
This article is devoted to extend the "$\eps-\eps^\beta$ property" to the case of clusters in an Euclidean space with a double density.
The classical definitions of the Incipient Infinite Cluster (IIC) of percolation consist in conditioning the origin on being connected to radius $n$ and letting $n$ go to infinity. We provide a short proof of that convergence in the planar…
In this paper we suggest new effective criteria for the density property. This enables us to give a trivial proof of the original Anders\'en-Lempert result and to establish (almost free of charge) the algebraic density property for all…