arXiv subjects

Marco Pozzetta

Publications and source records attributed to Marco Pozzetta.

28 records · Page 2Linked to original sources

arXiv2021

Isoperimetric sets in spaces with lower bounds on the Ricci curvature

In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in $\sf RCD$ spaces where the reference measure is the Hausdorff measure. A quasi-perimeter is a functional given by the sum of the usual perimeter and of a suitable continuous term. In particular, isoperimetric sets are a particular case of our study. We prove that on an ${\sf RCD}(K,N)$ space $({\rm X},{\sf d},\mathcal{H}^N)$, with $K\in\mathbb R$, $N\geq 2$, and a uniform bound from below on the volume of unit balls, volume constrained minimizers of quasi-perimeters are open bounded sets with $(N-1)$-Ahlfors regular topological boundary coinciding with the essential boundary. The proof is based on a new Deformation Lemma for sets of finite perimeter in ${\sf RCD}(K,N)$ spaces $({\rm X},{\sf d},\mathfrak m)$ and on the study of interior and exterior points of volume constrained minimizers of quasi-perimeters. The theory applies to volume constrained minimizers in smooth Riemannian manifolds, possibly with boundary, providing a general regularity result for such minimizers in the smooth setting.

math.DG↗

arXiv2021

Convergence of elastic flows of curves into manifolds

For a given $p\in[2,+\infty)$, we define the $p$-elastic energy $\mathscr{E}$ of a closed curve $γ:\mathbb{S}^1\to M$ immersed in a complete Riemannian manifold $(M,g)$ as the sum of the length of the curve and the $L^p$--norm of its curvature (with respect to the length measure). We are interested in the convergence of the $(L^p,L^{p'})$--gradient flow of these energies to critical points. By means of parabolic estimates, it is usually possible to prove sub-convergence of the flow, that is, convergence to critical points up to reparametrizations and, more importantly, up to isometry of the ambient. Assuming that the flow sub-converges, we are interested in proving the smooth convergence of the flow, that is, the existence of the full limit of the evolving flow. We first give an overview of the general strategy one can apply for proving such a statement. The crucial step is the application of a Lojasiewicz-Simon gradient inequality, of which we present a versatile version. Then we apply such strategy to the flow of $\mathscr{E}$ of curves into manifolds, proving the desired improvement of sub-convergence to full smooth convergence of the flow to critical points. As corollaries, we obtain the smooth convergence of the flow for $p=2$ in the Euclidean space $\mathbb{R}^n$, in the hyperbolic plane $\mathbb{H}^2$, and in the two-dimensional sphere $\mathbb{S}^2$. In particular, the result implies that such flow in $\mathbb{R}^n$ or $\mathbb{H}^2$ remains in a bounded region of the space for any time.

math.AP↗

arXiv2021

On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of the isoperimetric profile. The latter is new in the generality of noncollapsed manifolds with Ricci curvature bounded below.

math.DG↗

arXiv2021

A varifold perspective on the $p$-elastic energy of planar sets

Under suitable regularity assumptions the $p$-elastic energy of a planar set $E\subset\mathbb{R}^2$ is defined to be $\int_{\partial E} 1 + |k_{\partial E}|^p \,\, d\mathcal{H}^1,$ where $k_{\partial E}$ is the curvature of the boundary $\partial E$. In this work we use a varifold approach to investigate this energy, that can be well defined on varifolds with curvature. First we show new tools for the study of 1-dimensional curvature varifolds, such as existence and uniform bounds on the density of varifolds with finite elastic energy. Then we characterize a new notion of $L^1$-relaxation of this energy by extending the definition of regular sets by an intrinsic varifold perspective, also comparing this relaxation with the classical ones. Finally we discuss an application to the inpainting problem, examples and qualitative properties of sets with finite relaxed energy.

math.AP↗

arXiv2020

The Lojasiewicz-Simon inequality for the elastic flow

We define the elastic energy of smooth immersed closed curves in $\mathbb{R}^n$ as the sum of the length and the $L^2$-norm of the curvature, with respect to the length measure. We prove that the $L^2$-gradient flow of this energy smoothly converges asymptotically to a critical point. One of our aims was to the present the application of a Lojasiewicz-Simon inequality, which is at the core of the proof, in a quite concise and versatile way.

math.AP↗

arXiv2020

Confined Willmore energy and the Area functional

We consider minimization problems of functionals given by the difference between the Willmore functional of a closed surface and its area, when the latter is multiplied by a positive constant weight $Λ$ and when the surfaces are confined in the closure of a bounded open set $Ω\subset\mathbb{R}^3$. We explicitly solve the minimization problem in the case $Ω=B_1$. We give a description of the value of the infima and of the convergence of minimizing sequences to integer rectifiable varifolds, depending on the parameter $Λ$. We also analyze some properties of these functionals and we provide some examples. Finally we prove the existence of a $C^{1,α}\cap W^{2,2}$ embedded surface that is also $C^\infty$ inside $Ω$ and such that it achieves the infimum of the problem when the weight $Λ$ is sufficiently small.

math.AP↗

arXiv2020

Degenerate elastic networks

We minimize a linear combination of the Willmore and the length functional among networks in $\mathbb{R}^d$ belonging to a given class determined by the number of curves, the order of the junctions and the angles between curves at the junctions. Since this class lacks compactness, we characterize the set of limits of sequences of networks bounded in energy, providing an explicit representation of the relaxed problem. This is expressed in terms of the new notion of degenerate elastic networks that, rather surprisingly, involves only the properties of the given class, without reference to the curvature. In the case of $d=2$ we also give an equivalent description of degenerate elastic networks by means of a combinatorial definition easy to validate by a finite algorithm. Moreover we provide examples, counterexamples, and additional results that motivate our study and show the sharpness of our characterization.

math.OC↗

arXiv2020

On the Plateau-Douglas problem for the Willmore energy of surfaces with planar boundary curves

For a smooth closed embedded planar curve $Γ$, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus $\mathfrak{g}\geq1$ having the curve $Γ$ as boundary, without any prescription on the conormal. By general lower bound estimates, in case $Γ$ is a circle we prove that such problem is equivalent if restricted to embedded surfaces, we prove that do not exist minimizers, and the infimum equals $β_\mathfrak{g}-4π$, where $β_\mathfrak{g}$ is the energy of the closed minimizing surface of genus $\mathfrak{g}$. We also prove that the same result also holds if $Γ$ is a straight line for the suitable analogously defined minimization problem on asymptotically flat surfaces.\\ Then we study the case in which $Γ$ is compact, $\mathfrak{g}=1$ and the competitors are restricted to a suitable class $\mathcal{C}$ of varifolds including embedded surfaces. We prove that under suitable assumptions minimizers exists in this class of generalized surfaces.

math.AP↗

arXiv2020

Connected perimeter of planar sets

We introduce a notion of connected perimeter for planar sets defined as the lower semi-continuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied. We prove a representation formula which links the connected perimeter, the classical perimeter, and the length of suitable Steiner trees. We also discuss the application of this notion to the existence of solutions to a nonlocal minimization problem.

math.FA↗

arXiv2019

Connected surfaces with boundary minimizing the Willmore energy

For a given family of smooth closed curves $γ^1,...,γ^α\subset\mathbb{R}^3$ we consider the problem of finding an elastic \emph{connected} compact surface $M$ with boundary $γ=γ^1\cup...\cupγ^α$. This is realized by minimizing the Willmore energy $\mathcal{W}$ on a suitable class of competitors. While the direct minimization of the Area functional may lead to limits that are disconnected, we prove that, if the infimum of the problem is $<4π$, there exists a connected compact minimizer of $\mathcal{W}$ in the class of integer rectifiable curvature varifolds with the assigned boundary conditions. This is done by proving that varifold convergence of bounded varifolds with boundary with uniformly bounded Willmore energy implies the convergence of their supports in Hausdorff distance. Hence, in the cases in which a small perturbation of the boundary conditions causes the non-existence of Area-minimizing connected surfaces, our minimization process models the existence of optimal elastic connected compact generalized surfaces with such boundary data. We also study the asymptotic regime in which the diameter of the optimal connected surfaces is arbitrarily large. Under suitable boundedness assumptions, we show that rescalings of such surfaces converge to round spheres. The study of both the perturbative and the asymptotic regime is motivated by the remarkable case of elastic surfaces connecting two parallel circles located at any possible distance one from the other. The main tool we use is the monotonicity formula for curvature varifolds, that we extend to varifolds with boundary, together with its consequences on the structure of varifolds with bounded Willmore energy.

math.OC↗