arXiv subjects

Susanna Terracini

Publications and source records attributed to Susanna Terracini.

At least 19 recordsLinked to original sources

arXiv2026

Analytic rigidity and symbolic dynamics for two-centre billiards

We establish a sharp rigidity--chaos dichotomy for planar two-centre billiards, motivated by a natural analogue of the Birkhoff--Poritsky conjecture: the only tables integrable at every energy should be ellipses confocal with the two centres. Let $\Omega$ be a bounded domain with $\mathcal C^1$ boundary containing the segment joining the centres. At every fixed energy $h\geq 0$, if $\partial\Omega$ is not a confocal ellipse, we construct billiard trajectories that shadow the stable and unstable manifolds of the collision--reflection orbit and realise arbitrarily prescribed sequences of sufficiently large winding numbers around the segment. This yields an invariant set semiconjugate to the full shift on a countable alphabet, periodic trajectories with prescribed finite itineraries, and compact invariant subsystems with arbitrarily large topological entropy. If, in addition, $\partial\Omega$ is real-analytic, every real-analytic function on the fixed-energy phase space $M_h$ that is invariant under the billiard map is constant. This establishes the real-analytic form of the two-centre Birkhoff--Poritsky conjecture throughout the non-negative-energy regime.

math.DS↗

arXiv2026

Expansive solutions and the boundary at infinity for the homogeneous $N$-body problem

We investigate expansive solutions of the $N$-body problem in $\mathbb{R}^d$ ($d\ge2$) driven by homogeneous Newtonian potentials of degree $-\alpha$. We establish the existence of half-entire expansive motions with prescribed initial configuration and asymptotic direction for a wide range of homogeneity exponents $\alpha$. Our approach is variational and relies on the minimization of a suitably renormalized Lagrangian action, allowing us to treat in a unified framework the hyperbolic, parabolic, and hyperbolic-parabolic regimes in the sense of Chazy's classification. Beyond existence, we derive refined asymptotic expansions for all classes of expansive solutions, identifying higher-order correction terms and improving previously known growth estimates, including the classical Newtonian case $\alpha=1$. In particular, for hyperbolic-parabolic solutions, we provide a detailed description of the interplay between linear escape of cluster centers and internal parabolic dynamics, extending the cluster scattering picture to general homogeneous potentials. Finally, we interpret these solutions within the geometric framework of the Jacobi-Maupertuis metric and the weak KAM theory. In this perspective, expansive motions correspond to geodesic rays and calibrating curves for the associated Hamilton-Jacobi equation, yielding a dynamical characterization of the boundary at infinity and a refined description of global viscosity solutions.

math.DS↗

arXiv2025

Chaotic Boltzmann's Billiard Systems at positive energy

This paper deals with the so-called Boltzmann billiard, that is, a billiard subjected to a central force of the type $V(r)=-\alpha/r-\beta/r^2$, $\alpha$ and $\beta$ being positive constants, and with a straight reflection table. In the particular case of $\alpha$ and $\beta$ positive, we prove the presence of a symbolic dynamics, and hence of positive topological entropy, at positive energy and for $\beta$ sufficiently small.

math.DS↗

arXiv2025

A priori H\"older estimates for equations degenerating on nodal sets

We prove a priori H\"older bounds for continuous solutions to degenerate equations with variable coefficients of type $$ \mathrm{div}\left(u^2 A\nabla w\right)=0\quad\mathrm{in \ }\Omega\subset\mathbb R^n,\qquad \mbox{with}\qquad \mathrm{div}\left(A\nabla u\right)=0, $$ where $A$ is a Lipschitz continuous, uniformly elliptic matrix (possibly $u$ has non-trivial singular nodal set). Such estimates are uniform with respect to $u$ in a class of normalized solutions that have a bounded Almgren frequency. As a consequence, a boundary Harnack principle holds for the quotient of two solutions vanishing on a common set. This analysis relies on a detailed study of the associated weighted Sobolev spaces, including integrability of the weight, capacitary properties of the nodal set, and uniform Sobolev inequalities yielding local boundedness of solutions.

math.AP↗

arXiv2025

On the regularity of solutions to the Hamilton-Jacobi equations for the N-body problem

We prove that certain suitably renormalized value functions associated with the $d$-dimensional ($d\geq2$) $N$-body problem corresponding to different limiting shapes of expanding solutions, under the assumption that the center of mass is at the origin, are viscosity solutions of the associated Hamilton-Jacobi equation. We analyze their singularities, defined as the initial configurations for which the minimizer of the associated variational problem is not unique. Moreover, we estimate the size of the closure of the singular set by proving its $\mathcal{H}^{d(N-1)-1}$-rectifiability, and we provide an upper bound on the Hausdorff dimension of the set of regular conjugate points.

math.DS↗

arXiv2025

On the Birkhoff conjecture for Kepler billiards

We investigate the integrability of Kepler billiards-mechanical billiard systems in which a particle moves under the influence of a Keplerian potential and reflects elastically at the boundary of a strictly convex planar domain. Our main result establishes that, except possibly for one location of the gravitational center, analytic integrability at high energies occurs only when the domain is an ellipse and the center is placed at one of its foci. This provides a partial affirmative answer to a Keplerian analogue of the classical Birkhoff-Poritsky Conjecture. Our approach is based on the construction of symbolic dynamics arising from chaotic subsystems that emerge in the high-energy regime. Depending on the geometric configuration of the boundary and the location of the attraction center, we construct three types of symbolic dynamics by shadowing chains of punctured Birkhoff-type trajectories. These constructions yield subsystems conjugated to Bernoulli shifts, implying positive topological entropy and precluding analytic integrability. We further analyze the notion of focal points of the second kind, showing that in real-analytic, non-elliptic domains there can be at most one such point, while ellipses are the only domains admitting two-coinciding with their classical foci. Finally, we demonstrate the existence of an infinite-dimensional family of non-elliptic domains possessing a focal point of the second kind, and conclude with numerical simulations illustrating chaotic behavior in such cases.

math.DS↗

arXiv2025

New solutions for the symmetrical n-body problem through variational approach and optimisation techniques

Advances in the variational approach to the $n$-body problem have led to significant progress in celestial mechanics, uncovering new types of possible orbits. In this paper, critical points of the Lagrangian action associated with the $n$-body problem are analysed using evolutionary algorithms to identify periodic and symmetrical solutions of the discretised system. A key objective is to locate minimum points of the action functional, as these correspond to feasible periodic solutions that satisfy the system's differential equations. By employing both stochastic and deterministic algorithms, we explore the solution space and obtain numerical representations of these orbits. Next, we examine the stability of these orbits by treating them as critical points. One approach is to compute their discrete Morse index to distinguish between minimum points and saddle points. Another is to classify them based on their action levels. Finally, analysing the boundaries of their attraction basins allows us to identify non-minimal critical points via the Ambrosetti-Rabinowitz Mountain Pass Theorem. This leads to an updated version of the algorithm that provides a constructive proof of the theorem, yielding new orbits in specific cases. This paper builds upon and extends the results presented in \cite{nostro}, providing a more detailed theoretical framework and deeper insights into the formulation. Additionally, we present new numerical results and an extended analysis of the critical points found, further enhancing the findings of the previous study.

math.DS↗

arXiv2024

On partially segregated harmonic maps: optimal regularity and structure of the free boundary

We consider triplets of densities $(u_1,u_2,u_3)$ minimizing the Dirichlet energy \[\sum_{j=1}^3 \int_{\Omega} |\nabla u_j|^2\,dx \] over a bounded domain $\Omega\subset \mathbb{R}^N$, subject to the partial segregation condition: \[ u_1\,u_2\,u_3 \equiv 0 \ \text{in $\Omega$.} \] We prove optimal regularity of the minimizers in spaces of H\"older continuous functions of exponent $3/4$; furthermore we prove that the free boundary is a collection of a locally finite number of smooth codimension one manifolds up to a residual set of Hausdorff dimension at most $N-2$. Finally we prove uniform-in-$\beta$ a priori bounds for minimal solutions to the penalized energy: \[ J_\beta(\mathbf{u}, \Omega) = \int_{\Omega} \sum_{i=1}^3 |\nabla u_i|^2 \,dx+ \beta \int_{\Omega} \prod_{j=1}^3 u_j^2\,dx, \] in spaces of H\"older continuous functions of exponent less than $3/4$. The proofs make use of an Almgren-type monotonicity formula, blow-up analysis together with some new Liouville-type theorems.

math.AP↗

arXiv2024

Frozen planet orbits for the $n$-electron atom

We seek periodic trajectories of a system of multiple mutually repelling electrons on a half-line, with an attractive nucleus sitting at the origin. We adopt a variational viewpoint and study critical points of the associated Lagrange-action functional, by means of a modified Lusternik-Schnirelmann theory for manifolds with boundary. Additionally, when the charges of the electrons tend to zero, we show that frozen planet orbits converge to segments of a brake orbit for a Kepler-type problem, establishing a strong analogy with the Schubart orbits of the gravitational $n$-body problem.

math.DS↗

arXiv2024

Equivariant optimisation for the gravitational $n$-body problem: a computational factory of symmetric orbits

In this paper we present \texttt{SymOrb.jl}, a software which combines group representation theory and variational methods to provide numerical solutions of singular dynamical systems of paramount relevance in Celestial Mechanics and other interacting particles models. Among all, it prepares for large-scale search of symmetric periodic orbits for the classical $n$-body problem and their classification, paving the way towards a computational validation of Poincar\'e conjecture about the density of periodic orbits. Through the accessible language of Julia, \texttt{SymOrb.jl} offers a unified implementation of an earlier version. This paper provides theoretical and practical guidelines for the specific approach we adopt, complemented with examples.

math.DS↗

arXiv2024

On some singularly perturbed elliptic systems modeling partial segregation: uniform H\"older estimates and basic properties of the limits

We prove uniform H\"older estimates in a class of singularly perturbed competition-diffusion elliptic systems, with the particular feature that the interactions between the components occur three by three (ternary interactions). These systems are associated to the minimization of Gross-Pitaevski energies modeling ternary mixture of ultracold gases and other multicomponent liquids and gases. We address the question whether this regularity holds uniformly throughout the approximation process up to the limiting profiles, answering positively. A very relevant feature of limiting profiles in this process is that they are only partially segregated, giving rise to new phenomena of geometric pattern formation and optimal regularity.

math.AP↗

arXiv2024

Bifurcation for the Lotka-Volterra competition model

We analyze the bifurcation phenomenon for the following two-component competition system: \begin{equation*} \begin{cases} -Δu_1=μu_1(1-u_1)-βαu_1u_2,& \text{in}\ B_1\subset \mathbb{R}^N, -Δu_2=σu_2(1-u_2)-βγu_1u_2,& \text{in}\ B_1\subset \mathbb{R}^N, \frac{\partial u_1}{\partial n}= \frac{\partial u_2}{\partial n} =0,&\text{on}\ \partial B_1, \end{cases} \end{equation*} where $N\ge 2$, $α>γ>0$, $σ\geμ>0$ and $β>\fracσγ$. More precisely, treating $β$ as the bifurcation parameter, we initially perform a local bifurcation analysis around the positive constant solutions, obtaining precise information of where bifurcation could occur, and determine the direction of bifurcation. As a byproduct, the instability of the constant solution is provided. Furthermore, we extend our exploration to the global bifurcation analysis. Lastly, under the condition $σ=μ$, we demonstrate the limiting configuration on each bifurcation branch as the competition rate $β\rightarrow+\infty$.

math.AP↗

arXiv2024

A priori regularity estimates for equations degenerating on nodal sets

We prove a priori and a posteriori H\"older bounds and Schauder $C^{1,\alpha}$ estimates for continuous solutions of degenerate elliptic equations with variable coefficients of the form $$ \mathrm{div}\left(|u|^a A\nabla w\right)=0\qquad\mathrm{in \ }\Omega\subset\mathbb R^2,\quad a\in\mathbb R, $$ where the weight $u$ is itself a solution to an elliptic equation of the type $\mathrm{div}(A \nabla u) = 0$, with $A$ a Lipschitz-continuous, uniformly elliptic matrix. The function $u$ is allowed to have a nontrivial, possibly singular nodal set. The estimates are uniform with respect to $u$ within a class of normalized solutions having bounded Almgren frequency. In the special case $a = 2$, our results apply to the ratio of two solutions to the same elliptic equation sharing a common zero set. Precisely, we prove higher-order boundary Harnack principles on nodal domains, via the derived Schauder estimates for the associated degenerate equations. The results are based upon a fine blow-up argument, a Liouville theorem, and quasiconformal maps.

math.AP↗

arXiv2024

Mountain pass frozen planet orbits in the helium atom model

We seek frozen planet orbits for the helium atom through an application of the Mountain Pass Lemma to the Lagrangian action functional. Our method applies to a wide class of gravitational-like interaction potentials thus generalising the results in [7] (Cieliebak, Frauenfelder and Volkov - 2023). We also let the charge of the two electrons tend to zero and perform the asymptotic analysis to prove convergence to a limit trajectory having a collision-reflection singularity between the electrons.

math.DS↗

arXiv2024

Rotating spirals for three-component competition systems

We investigate the existence of rotating spirals for three-component competition-diffusion systems in $B_1\subset \mathbb{R}^2$: \begin{equation*} \begin{cases} \partial_tu_1-Δu_1=f(u_1)-βαu_1u_2-βγu_1 u_3,& \text{in}\ B_1\times \mathbb{R}^+, \partial_tu_2-Δu_2=f(u_2)-βγu_1u_2-βαu_2 u_3,& \text{in}\ B_1\times \mathbb{R}^+, \partial_tu_3-Δu_3=f(u_3)-βαu_1u_3-βγu_2 u_3,& \text{in}\ B_1\times \mathbb{R}^+, u_i(\textbf{x},0)=u_{i,0}(\textbf{x}), i=1,2,3, &\text{in} \ B_1, \end{cases} \end{equation*} with Neumann or Dirichlet boundary conditions, where $f(s)=μs(1-s)$, $μ, β>0$, $α>γ>0$. For the Neumann problem, we establish the existence of rotating spirals by applying the multi-parameter bifurcation theorem. As a byproduct, the instability of the constant positive solution is proved. In addition, for the non-homogeneous Dirichlet problem, the Rothe fixed point theorem is employed to prove the existence of rotating spirals.

math.AP↗

arXiv2024

Higher order boundary Harnack principle via degenerate equations

As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type: \[ -\mathrm{div}\left(ρ^aA\nabla w\right)=ρ^af+\mathrm{div}\left(ρ^aF\right) \quad\textrm{in}\; Ω\] for exponents $a>-1$, where the weight $ρ$ vanishes in a non degenerate manner on a regular hypersurface $Γ$ which can be either a part of the boundary of $Ω$ or mostly contained in its interior. As an application, we extend such estimates to the ratio $v/u$ of two solutions to a second order elliptic equation in divergence form when the zero set of $v$ includes the zero set of $u$ which is not singular in the domain (in this case $ρ=u$, $a=2$ and $w=v/u$). We prove first $C^{k,α}$-regularity of the ratio from one side of the regular part of the nodal set of $u$ in the spirit of the higher order boundary Harnack principle established by De Silva and Savin. Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension $n=2$, we provide local gradient estimates for the ratio which hold also across the singular set.

math.AP↗

arXiv2023

On the existence of minimal expansive solutions to the $N$-body problem

We deal, for the classical $N$-body problem, with the existence of action minimizing half entire expansive solutions with prescribed asymptotic direction and initial configuration of the bodies. We tackle the cases of hyperbolic, hyperbolic-parabolic and parabolic arcs in a unitary manner. Our approach is based on the minimization of a renormalized Lagrangian action, on a suitable functional space. With this new strategy, we are able to confirm the already-known results of the existence of both hyperbolic and parabolic solutions, and we prove for the first time the existence of hyperbolic-parabolic solutions for any prescribed asymptotic expansion in a suitable class. Associated with each element of this class we find a viscosity solution of the Hamilton-Jacobi equation as a linear correction of the value function. Besides, we also manage to give a better description of the growth of parabolic and hyperbolic-parabolic solutions.

math.DS↗

arXiv2022

Oscillatory Motions in the Restricted 3-body Problem: A functional analytic approach

A fundamental question in Celestial Mechanics is to analyze the possible final motions of the Restricted $3$-body Problem, that is, to provide the qualitative description of its complete (i.e. defined for all time) orbits as time goes to infinity. According to the classification given by Chazy back in 1922, a remarkable possible behaviour is that of oscillatory motions, where the motion $q$ of the massless body is unbounded but returns infinitely often inside some bounded region: \[ \limsup_{t\to\pm\infty} |q(t)|=\infty\qquad\qquad\text{and}\qquad\qquad \liminf_{t\to\pm\infty} |q(t)|<\infty. \] In contrast with the other possible final motions in Chazy's classification, oscillatory motions do not occur in the $2$-body Problem, while they do for larger numbers of bodies. A further point of interest is their appearance in connection with the existence of chaotic dynamics. In this paper we introduce new tools to study the existence of oscillatory motions and prove that oscillatory motions exist in a particular configuration known as the Restricted Isosceles $3$-body Problem (RI3BP) for almost all values of the angular momentum. Our method, which is global and not limited to nearly integrable settings, extends the previous results \cite{guardia2021symbolic} by blending variational and geometric techniques with tools from nonlinear analysis such as topological degree theory. To the best of our knowledge, the present work constitutes the first complete analytic proof of existence of oscillatory motions in a non perturbative regime.

math.DS↗