Searcharxiv⌕ Search

arXiv subjects

Anna Dall'Acqua

Publications and source records attributed to Anna Dall'Acqua.

At least 19 recordsLinked to original sources
arXiv2026

Stability of the free boundary Willmore problem

We study the Willmore problem with free boundary by means of a new Łojasiewicz-Simon gradient inequality for functionals on infinite dimensional manifolds. In contrast to previous works, we do not rely on a gradient-like representation of the Fréchet derivative, but merely on an inequality. For the free boundary Willmore flow, we prove that solutions starting sufficiently close to a local minimizer exist for all times and converge. In the static setting, we prove quantitative stability of free boundary Willmore immersions and a local rigidity result in a neighborhood of free boundary minimal surfaces.

math.AP↗
arXiv2025

The length-preserving elastic flow with free boundary on hypersurfaces in $\mathbb{R}^n$

We study the length-preserving elastic flow of curves in arbitrary codimension with free boundary on hypersurfaces. This constrained gradient flow is given by a nonlocal evolution equation with nonlinear higher-order boundary conditions. We prove global existence and subconvergence to critical points. The proof strategy involves a careful treatment of short-time existence, uniqueness, and parabolic energy estimates.

math.AP↗
arXiv2025

Dimension reduction for Willmore flows of tori: fixed conformal class and analysis of singularities

This work studies Willmore flows of tori and their singularities via a dimension reduction approach. We introduce a Willmore flow that preserves the degenerate constraint of prescribed conformal class and, for rotationally symmetric initial data, we establish a strong relation with the length-preserving elastic flow in the hyperbolic plane. We provide a necessary condition for singularities and a criterion for the initial datum that allows to exclude them. Our results allow for initial data with arbitrarily large energy, in particular exceeding the usual Li-Yau threshold of $8π$. As an application, we obtain existence of a new class of conformally constrained Willmore tori. Moreover, we investigate singularities of the classical Willmore flow. For a class of tori, we identify a non-smooth object, the inverted catenoid, as the limit shape and we show that the flow can be restarted at this singular surface and converges to a round sphere.

math.AP↗
arXiv2024

An obstacle problem for the p-elastic energy

In this paper we consider an obstacle problem for a generalization of the p-elastic energy among graphical curves with fixed ends. Taking into account that the Euler--Lagrange equation has a degeneracy, we address the question whether solutions have a flat part, i.e. an open interval where the curvature vanishes. We also investigate which is the main cause of the loss of regularity, the obstacle or the degeneracy. Moreover, we give several conditions on the obstacle that assure existence and nonexistence of solutions. The analysis can be refined in the special case of the p-elastica functional, where we obtain sharp existence results and uniqueness for symmetric minimizers.

math.AP↗
arXiv2024

Conservation, convergence, and computation for evolving heterogeneous elastic wires

The elastic energy of a bending-resistant interface depends both on its geometry and its material composition. We consider such a heterogeneous interface in the plane, modeled by a curve equipped with an additional density function. The resulting energy captures the complex interplay between curvature and density effects, resembling the Canham-Helfrich functional. We describe the curve by its inclination angle, so that the equilibrium equations reduce to an elliptic system of second order. After a brief variational discussion, we investigate the associated nonlocal $L^2$-gradient flow evolution, a coupled quasilinear parabolic problem. We analyze the (non)preservation of quantities such as convexity, positivity, and symmetry, as well as the asymptotic behavior of the system. The results are illustrated by numerical experiments.

math.AP↗
arXiv2023

The Willmore Flow of Tori of Revolution

We study long-time existence and asymptotic behavior for the $L^2$-gradient flow of the Willmore energy, under the condition that the initial datum is a torus of revolution. We show that if an initial datum has Willmore energy below $8π$ then the solution of the Willmore flow converges for $t \rightarrow \infty$ to the Clifford Torus, possibly rescaled and translated. The energy threshold of $8π$ turns out to be optimal for such a convergence result. We give an application to the conformally constrained Willmore minimization problem.

math.AP↗
arXiv2023

A dynamic approach to heterogeneous elastic wires

We consider closed planar curves with fixed length and arbitrary winding number whose elastic energy depends on an additional density variable and a spontaneous curvature. Working with the inclination angle, the associated $L^2$-gradient flow is a nonlocal quasilinear coupled parabolic system of second order. We show local well-posedness, global existence of solutions, and full convergence of the flow for initial data in a weak regularity class.

math.AP↗
arXiv2021

Minimal elastic networks

We consider planar networks of three curves that meet at two junctions with prescribed equal angles, minimizing a combination of the elastic energy and the length functional. We prove existence and regularity of minimizers, and we show some properties of the minimal configurations.

math.AP↗
arXiv2020

Elastic flow of networks: short-time existence result

In this paper we study the $L^2$-gradient flow of the penalized elastic energy on networks of $q$-curves in $\R^{n}$ for $q \geq 3$. Each curve is fixed at one end-point and at the other is joint to the other curves at a movable $q$-junction. For this geometric evolution problem with natural boundary condition we show the existence of smooth solutions for a (possibly) short interval of time. Since the geometric problem is not well-posed, due to the freedom in reparametrization of curves, we consider a fourth-order non-degenerate parabolic quasilinear system, called the analytic problem, and show first a short-time existence result for this parabolic system. The proof relies on applying Solonnikov's theory on linear parabolic systems and Banach fixed point theorem in proper Hölder spaces. Then the original geometric problem is solved by establishing the relation between the analytical solutions and the solutions to the geometrical problem.

math.AP↗
arXiv2018

Flow of elastic networks: long-time existence result

We provide a long-time existence and sub-convergence result for the elastic flow of a three network in $\mathbb{R}^{n}$ under some mild topological assumptions. The evolution is such that the sum of the elastic energies of the three curves plus their weighted lengths decrease in time. Natural boundary conditions are considered at the boundary of the curves and at the triple junction.

math.AP↗
arXiv2017

Some minimization problems for planar networks of elastic curve

In this note we announce some results that will appear in [6] (joint work with also Matteo Novaga) on the minimization of the functional $F(Γ)=\int_Γk^2+1\,\mathrm{d}s$, where $Γ$ is a network of three curves with fixed equal angles at the two junctions. The informal description of the results is accompanied by a partial review of the theory of elasticae and a diffuse discussion about the onset of interesting variants of the original problem passing from curves to networks. The considered energy functional $F$ is given by the elastic energy and a term that penalize the total length of the network. We will show that penalizing the length is tantamount to fix it. The paper is concluded with the explicit computation of the penalized elastic energy of the 'Figure Eight', namely the unique closed elastica with self--intersections.

math.AP↗
arXiv2017

The elastic flow of curves in the hyperbolic plane

We consider closed curves in the hyperbolic space moving by the $L^2$-gradient flow of the elastic energy and prove well-posedness and long time existence. Under the additional penalisation of the length we show subconvergence to critical points. A motivation for the study of this flow is given by the relation between elastic curves in the hyperbolic plane and Willmore surfaces of revolution.

math.AP↗
arXiv2016

The Łojasiewicz-Simon gradient inequality for open elastic curves

In this paper we consider the elastic energy for open curves in Euclidean space subject to clamped boundary conditions and obtain the Łojasiewicz-Simon gradient inequality for this energy functional. Thanks to this inequality we can prove that a (suitably reparametrized) solution to the associated $L^2$-gradient flow converges for large time to an elastica, that is to a critical point of the functional.

math.AP↗
arXiv2014

A new Gershgorin-type result for the localisation of the spectrum of matrices

We present a Gershgorin's type result on the localisation of the spectrum of a matrix. Our method is elementary and relies upon the method of Schur complements, furthermore it outperforms the one based on the Cassini ovals of Ostrovski and Brauer. Furthermore, it yields estimates that hold without major differences in the cases of both scalar and operator matrices. Several refinements of known results are obtained.

math.SP↗
arXiv2013

Willmore-Helfrich L^{2}-flow of curves with natural boundary conditions

We consider regular open curves in R^n with fixed boundary points and moving according to the L^{2}-gradient flow for a generalisation of the Helfrich functional. Natural boundary conditions are imposed along the evolution. More precisely, at the boundary the curvature vector is equal to the normal projection of a fixed given vector. A long-time existence result together with subconvergence to critical points is proven.

math.AP↗
arXiv2011

Real analyticity away from the nucleus of pseudorelativistic Hartree-Fock orbitals

We prove that the Hartree--Fock orbitals of pseudorelativistic atoms, that is, atoms where the kinetic energy of the electrons is given by the pseudorelativistic operator sqrt{-Delta+1}-1, are real analytic away from the origin. As a consequence, the quantum mechanical ground state of such atoms is never a Hartree-Fock state. Our proof is inspired by the classical proof of analyticity by nested balls of Morrey and Nirenberg. However, the technique has to be adapted to take care of the non-local pseudodifferential operator, the singularity of the potential at the origin, and the non-linear terms in the equation.

math-ph↗
arXiv2010

Excess charge for pseudo-relativistic atoms in Hartree-Fock theory

We prove within the Hartree-Fock theory of pseudo-relativistic atoms that the maximal negative ionization charge and the ionization energy of an atom remain bounded independently of the nuclear charge Z and the fine structure constant αas long as Zαis bounded.

math-ph↗