arXiv subjects

F. Giannoni

Publications and source records attributed to F. Giannoni.

12 recordsLinked to original sources

arXiv2022

A non-autonomous variational problem describing a nonlinear Timoshenko beam

We study the non-autonomous variational problem: \begin{equation*} \inf_{(ϕ,θ)} \bigg\{\int_0^1 \bigg(\frac{k}{2}ϕ'^2 + \frac{(ϕ-θ)^2}{2}-V(x,θ)\bigg)\text{d}x\bigg\} \end{equation*} where $k>0$, $V$ is a bounded continuous function, $(ϕ,θ)\in H^1([0,1])\times L^2([0,1])$ and $ϕ(0)=0$ in the sense of traces. The peculiarity of the problem is its setting in the product of spaces of different regularity order. Problems with this form arise in elastostatics, when studying the equilibria of a nonlinear Timoshenko beam under distributed load, and in classical dynamics of coupled particles in time-depending external fields. We prove the existence and qualitative properties of global minimizers and study, under additional assumptions on $V$, the existence and regularity of local minimizers.

math.AP↗

arXiv2015

Functions on the sphere with critical points in pairs and orthogonal geodesic chords

Using an estimate on the number of critical points for a Morse-even function on the sphere $\mathbb S^m$, $m\ge1$, we prove a multiplicity result for orthogonal geodesic chords in Riemannian manifolds with boundary that are diffeomorphic to Euclidean balls. This yields also a multiplicity result for brake orbits in a potential well.

math.DS↗

arXiv2015

On the normal exponential map in singular conformal metrics

Brake orbits and homoclinics of autonomous dynamical systems correspond, via Maupertuis principle, to geodesics in Riemannian manifolds endowed with a metric which is singular on the boundary (Jacobi metric). Motivated by the classical, yet still intriguing in many aspects, problem of establishing multiplicity results for brake orbits and homoclinics, as done in [6, 7, 10], and by the development of a Morse theory in [8] for geodesics in such kind of metric, in this paper we study the related normal exponential map from a global perspective.

math.DS↗

arXiv2015

Multiple brake orbits in $\mathbf m$-dimensional disks

Let $(M,g)$ be a (complete) Riemannian surface, and let $Ω\subset M$ be an open subset whose closure is homeomorphic to a disk. We prove that if $\partialΩ$ is smooth and it satisfies a strong concavity assumption, then there are at least two distinct orthogonal geodesics in $\overlineΩ=Ω\bigcup\partialΩ$. Using the results given in [6], we then obtain a proof of the existence of two distinct brake orbits for a class of Hamiltonian systems. In our proof we shall use recent deformation results proved in [7].

math.DS↗

arXiv2014

Morse Theory for geodesics in singular conformal metrics

Motivated by the use of degenerate Jacobi metrics for the study of brake orbits and homoclinics, we develop a Morse theory for geodesics in conformal metrics having conformal factors vanishing on a regular hypersurface of a Riemannian manifold.

math.DS↗

arXiv2014

Examples with minimal number of brake orbits and homoclinics in annular potential regions

We use a geometric construction to exhibit examples of autonomous Lagrangian systems admitting exactly two homoclinics emanating from a nondegenerate maximum of the potential energy and reaching a regular level of the potential having the same value of the maximum point. Similarly, we show examples of Hamiltonian systems that admit exactly two brake orbits in an annular potential region connecting the two connected components of the boundary of the potential well. These examples show that the estimates proven in [R. Giambò, F. Giannoni, P. Piccione, Arch. Ration. Mech. Anal. 200, (2011) 691-724] are sharp.

math.DS↗

arXiv2012

Potential wells with a unique brake orbit. Counterexamples to a conjecture by H. Seifert

In this paper we prove the existence of real-analytic natural Hamiltonian systems - i.e. where H(q,p)=T(q,p)+V(q) in the 2N-dimensional real space, where N is any integer greater than 1 - with non critical energy levels E for the potential V such that the sublevel E of V is homeomorphic to the N-dimensional disk, and that only one brake orbit of energy E exists. A famous conjecture formulated by H. Seifert in 1948 claimed the existence of at least N distinct brake orbits for this situation.

math.DS↗

arXiv2010

Existence of orthogonal geodesic chords on Riemannian manifolds with concave boundary and homeomorphic to the N-dimensional disk

In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link of the multiplicity problem with the famous Seifert conjecture (formulated in 1948) about multiple brake orbits for a class of Hamiltonian systems at a fixed energy level.

math.DS↗

arXiv2008

Genericity of blackhole formation in the gravitational collapse of homogeneous self-interacting scalar fields

The gravitational collapse of a wide class of self-interacting homogeneous scalar fields models is analyzed. The class is characterized by certain general conditions on the scalar field potential, which, in particular, include both asymptotically polynomial and exponential behaviors. Within this class, we show that the generic evolution is always divergent in a finite time, and then make use of this result to construct radiating star models of the Vaidya type. It turns out that blackholes are generically formed in such models.

gr-qc↗

arXiv2007

Dynamics of homogeneous scalar fields with general self-interaction potentials: cosmological and gravitational collapse models

The general relativistic dynamics of a wide class of self-interacting, self-gravitating homogeneous scalar fields models is analyzed. The class is characterized by certain general conditions on the scalar field potential, which include both asymptotically polynomial and exponential behaviors. Within this class, we show that the generic evolution is always divergent in a finite time, and then make use of this result to construct cosmological models as well as radiating collapsing star models of the Vaidya type. It turns out that blackholes are generically formed in such models.

math.CA↗

arXiv2006

A variational approach to homogeneous scalar fields in General Relativity

A result of existence of homogeneous scalar field solutions between prescribed configurations is given, using a modified version of Euler--Maupertuis least action variational principle. Solutions are obtained as limit of approximating variational problems, solved using techniques introduced by Rabinowitz.

gr-qc↗

arXiv1999

A Generalized Index Theorem for Morse-Sturm Systems and Applications to semi-Riemannian Geometry

We prove an extension of the Index Theorem for Morse-Sturm systems of the form $-V''+RV=0$, where R is symmetric with respect to a (non positive) symmetric bilinear form, and thus the corresponding differential operator is not self-adjoint. The result is then applied to the case of a Jacobi equation along a geodesic in a Lorentzian manifold, obtaining an extension of the Morse Index Theorem for Lorentzian geodesics with variable initial endpoints. Given a Lorentzian manifold (M,g), we consider a geodesic $γ$ in M starting orthogonally to a smooth submanifold P of M. Under suitable hypotheses, satisfied, for instance, if (M,g) is stationary, the theorem gives an equality between the index of the second variation of the action functional f at $γ$ and the sum of the {\em Maslov index} of $γ$ with the index of the metric g on P. Under generic circumstances, the Maslov index of $γ$ is given by an algebraic count of the P-focal points along $γ$. Using the Maslov index, we obtain the global Morse relations for geodesics between two fixed points in a stationary Lorentzian manifold.

math.DG↗