In 1993 Fuglede showed a sharp quantitative isoperimetric inequality, namely, that for convex sets in R^N the isoperimetric deficit controls the square of the barycentric asymmetry. The convexity assumption is used because the result is clearly false for general sets. Estimates of this form have then then studied in the following years, for instance by Cicalese-Leonardi, and by...
The Steiner problem, in its classical formulation, is to find the 1-dimensional connected set in the plane with minimal length that contains a finite collection of points.
Although existence and regularity of minimizers is well known, in general finding explicitly a solution is extremely challenging, even numerically.
A possible tool to validate the minimality of a certain candidate is the...
The aim of this seminar is to present some results about the isoperimetric problem for clusters in the plane with double density. This amounts to finding the best configuration of $m$ regions in the plane enclosing given volumes, in order to minimize their total perimeter, in the case where volume and perimeter are weighted by suitable densities.
We focus on the so-called ``Steiner''...
Motivated by the crystallization issue, we focus on the minimization of Heitman-Radin potential energies for configurations of $N$ particles in a periodic lattice, and in particular on the connection with anisotropic isoperimetric problems in the suitably rescaled limit as $N\to\infty$. Besides identifying the asymptotic Wulff shapes through Gamma-convergence, we obtain fluctuation estimates...
Given a set $\Omega\subset \mathbb{R}^N$, the Cheeger constant is a purely geometrical quantity defined as the infimum
$$
h(\Omega):= \inf \left\{\,\frac{P(E)}{|E|}\,:\, E\subset \Omega,\, |E|>0 \,\right\}.
$$
Despite seeming unassuming, it pops up in many contexts that apparently have nothing in common. To name a few, under some mild regularity assumptions on $\Omega$: bounds on the...
In this introductory talk, I will recall the main issues in dealing with the isoperimetric problem on Riemannian manifolds.
I will then focus on manifolds with lower bounds on the Ricci curvature, and discuss some fundamental tools in this setting.
Their rigorous derivation and their applications will naturally call for the analysis on nonsmooth RCD structures, and will be the object of the...
In this talk, we want to give an overview of a result of generalized existence and compactness of isoperimetric regions in the context of smooth (possibly noncompact) Riemannian manifolds without boundaries and of bounded geometry together with metric theoretic proofs that for almost-isoperimetric regions small volumes implies small diameters always in the context of smooth Riemannian...
We present stability results for some functional inequalities (such as the Faber-Krahn and the isocapacitary inequality) in the nonlocal setting. The proof is based on some ideas by Hansen and Nadirashvili (who considered the classical local case) and uses the so-called Caffarelli-Silvestre extension.
In the Heisenberg group $H^1=\mathbb R^3$ we consider the
perimeter associated with a norm on the horizontal distribution.
The existence of isoperimetric sets is well-known. Assuming the $C^2_+$ regularity of the norm, we are able to classify isoperimetric sets of class $C^2$.
This is an extension to the Finsler case of a result by Ritorè and Rosales. Isoperimetric sets turn out to be...
Geometrical properties of Cheeger sets have been deeply studied by many authors since their introduction, as a way of bounding from below the first Dirichlet (p)-Laplacian eigenvualue. They represent, in some sense, the first eigenfunction of the Dirichlet (1)-Laplacian of a domain. In this talk we will introduce a recent property, studied in collaboration with Simone Ciani, concerning their...
We study the motion of charged liquid drop in three dimensions where the equations of motions are given by the Euler equations with free boundary with an electric field. This is a well-known problem in physics going back to the famous work by Rayleigh. Due to experiments and numerical simulations one may expect the charged drop to form conical singularities called Taylor cones, which we...
In the first part of the lecture, I provide an overview on the differential calculus in the setting of RCD spaces, with a particular focus on the theory of sets of finite perimeter. In the second part of the lecture, I show how to apply this differential calculus in order to obtain a Deformation Lemma, which in turn can be used (under suitable non-collapsing assumptions) to prove that the...
The first and second variation of the area are cornerstones in classical Geometric Measure Theory and they are at the heart of its connections with Ricci curvature. In this lecture I will illustrate how they can be estimated for perimeter minimizers and isoperimetric sets on RCD spaces, avoiding any regularity theory.
In this talk, we will show a generalized compactness theorem for sequences of clusters with uniformly bounded perimeter and volume in a Riemannian manifold with bounded geometry. The arguments presented in the proof of this generalized compactness theorem when applied to minimizing sequences of clusters give a generalized existence theorem for isoperimetric clusters. To achieve this goal, we...
In this talk we will explore two consequences of the results explained in the previous lectures.
First, we shall prove a sharp differential inequality for the isoperimetric profile of N-dimensional RCD(K,N) spaces with uniform lower bounds on the volume of unit balls. This inequality is new even in the smooth noncompact setting. We will discuss some consequences of this inequality.
Second,...
I will present a new existence result for isoperimetric sets of large volume on manifolds with nonnegative Ricci curvature and Euclidean volume growth, under an additional assumption on the structure of tangent cones at infinity. After a brief discussion on the sharpness of the additional assumption, I will show that it is always verified on manifolds with nonnegative sectional curvature. I...
I will consider the volume preserving fractional mean curvature flow of a nearly spherical set, showing long time existence and exponential convergence to a ball. The main technical tool used in the proof is a quantitative Alexandrov type estimate for nearly spherical sets.
The result applies in particular to convex initial data under the assumption of global existence.
After recalling the classical variational formulation of the capillarity problem and some related results, we consider a model for vapor-liquid-solid growth of nanowires proposed in the physical literature. In this model, liquid drops are described as local or global volume-constrained minimizers of the capillarity energy outside a semi-infinite convex obstacle modeling the nanowire. We...
The goal of the talk is to discuss a quantitative version of the Levy-Gromov isoperimetric inequality (joint with Cavalletti and Maggi) as well as a quantitative form of Obata’s rigidity theorem (joint with Cavalletti and Semola). Given a closed Riemannian manifold with strictly positive Ricci tensor, one estimates the measure of the symmetric difference of a set with a metric ball with the...
The classical isoperimetric inequality in Euclidean space $\mathbb{R}^n$ states that among all sets ("bubbles") of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems for more general metric-measure spaces, such as on the sphere $\mathbb{S}^n$ and on Gauss space $\mathbb G^n$. Furthermore, one may consider the "multi-bubble"...
We describe through some selected examples an approach based on potential theory toward the proof of relevant geometric inequalities, holding both in classical and curved frameworks. Time permitting, we also discuss some applications of interest in general relativity, including the positive mass theorem and the Riemannian Penrose inequality, which - according to G. Gibbons - can be understood...
I will consider the large mass limit of a nonlocal isoperimetric problem in two dimensions with screened Coulomb repulsion, so that to leading order the nonlocal interaction localizes on the boundary of the sets. For an appropriate choice of screening constant, the perimeter is exactly cancelled out, requiring an analysis of the next order contribution. It turns out that then the nature of the...
