arXiv subjects

Susanna Terracini

Publications and source records attributed to Susanna Terracini.

At least 37 records · Page 2Linked to original sources

arXiv2022

Chaotic dynamics in refraction galactic billiards

We prove the presence of topological chaos at high internal energies for a new class of mechanical refraction billiards coming from Celestial Mechanics. Given a smooth closed domain $D\in\mathbb{R}^2$, a central mass generates a Keplerian potential in it, while, in $\mathbb{R}^2\setminus \overline{D}$, a harmonic oscillator-type potential acts. At the interface, Snell's law of refraction holds. The chaoticity result is obtained by imposing progressive assumptions on the domain, arriving to geometric conditions which hold generically in $C^1$. The workflow starts with the existence of a symbolic dynamics and ends with the proof of topological chaos, passing through the analytic non-integrability and the presence of multiple heteroclinic connections between different equilibrium saddle points. This work can be considered as the final step of the investigation carried on in arXiv:2108.11159 [math.DS] and arXiv:2105.02108 [math.DS].

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arXiv2022

Rotating Spirals in segregated reaction-diffusion systems

We give a complete characterization of the boundary traces $φ_i$ ($i=1,\dots,K$) supporting spiraling waves, rotating with a given angular speed $ω$, which appear as singular limits of competition-diffusion systems of the type \[ \frac{\partial}{\partial t} u_i -Δu_i = μu_i -βu_i \sum_{j \neq i} a_{ij} u_j \text{ in } Ω\times\mathbb{R}^+, \qquad u_i = φ_i \text{ on $\partialΩ\times\mathbb{R}^+$}, \qquad u_i(\mathbf{x},0) = u_{i,0}(\mathbf{x}) \text{ for $\mathbf{x} \in Ω$} \] as $β\to +\infty$. Here $Ω$ is a rotationally invariant planar set and $a_{ij}>0$ for every $i$ and $j$. We tackle also the homogeneous Dirichlet and Neumann boundary conditions, as well as entire solutions in the plane. As a byproduct of our analysis we detect explicit families of eternal, entire solutions of the pure heat equation, parameterized by $ω\in\mathbb{R}$, which reduce to homogeneous harmonic polynomials for $ω=0$.

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arXiv2021

On some Refraction Billiards

The aim of this work is to continue the analysis, started in arXiv:2105.02108, of the dynamics of a point-mass particle $P$ moving in a galaxy with an harmonic biaxial core, in whose center sits a Keplerian attractive center (e.g. a Black Hole). Accordingly, the plane $\mathbb R^2$ is divided into two complementary domains, depending on whether the gravitational effects of the galaxy's mass distribution or of the Black Hole prevail. Thus, solutions alternate arcs of Keplerian hyperbolae with harmonic ellipses; at the interface, the trajectory is refracted according to Snell's law. The model was introduced in arXiv:1501.05577, in view of applications to astrodynamics. In this paper we address the general issue of periodic and quasi-periodic orbits and associated caustics when the domain is a perturbation of the circle, taking advantage of KAM and Aubry-Mather theories.

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arXiv2021

Refraction Periodic Trajectories in Central Mass Galaxies

We consider a new type of dynamical systems of physical interest, where two different forces act in two complementary regions of the space, namely a Keplerian attractive center sits in the inner region, while an harmonic oscillator is acting in the outer one. In addition, the two regions are separated by an interface $Σ$, where a Snell's law of ray refraction holds. Trajectories concatenate arcs of Keplerian hyperbolae with harmonic ellipses, with a refraction at the boundary. When the interface also has a radial symmetry, then the system is integrable, and we are interested in the effect of the geometry of the interface on the stability and bifurcation of periodic orbits from the homotetic collision-ejection ones. We give local condition on the geometry of the interface for the stability and obtain a complete picture of stability and bifurcations in the elliptic case for period one and period two orbits.

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arXiv2021

Symbolic dynamics for the anisotropic $N$-centre problem at negative energies

The planar $N$-centre problem describes the motion of a particle moving in the plane under the action of the force fields of $N$ fixed attractive centres: \[ \ddot{x}(t)=\sum_{j=1}^N\nabla V_j(x-c_j). \] In this paper we prove symbolic dynamics at slightly negative energy for an $N$-centre problem where the potentials $V_j$ are positive, anisotropic and homogeneous of degree $-α_j$: \[ V_j(x)=|x|^{-α_j}V_j\left(\frac{x}{|x|}\right). \] The proof is based on a broken geodesics argument and trajectories are extremals of the Maupertuis' functional. Compared with the classical $N$-centre problem with Kepler potentials, a major difficulty arises from the lack of a regularization of the singularities. We will consider both the collisional dynamics and the non collision one. Symbols describe geometric and topological features of the associated trajectory.

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arXiv2020

Parabolic orbits in Celestial Mechanics: a functional-analytic approach

We prove the existence of half-entire parabolic solutions, asymptotic to a prescribed central configuration, for the equation \begin{equation*} \ddot{x} = \nabla U(x) + \nabla W(t,x), \qquad x \in \mathbb{R}^{d}, \end{equation*} where $d \geq 2$, $U$ is a positive and positively homogeneous potential with homogeneity degree $-α$ with $α\in\mathopen{]}0,2\mathclose{[}$, and $W$ is a (possibly time-dependent) lower order term, for $\vert x \vert \to +\infty$, with respect to $U$. The proof relies on a perturbative argument, after an appropriate formulation of the problem in a suitable functional space. Applications to several problems of Celestial Mechanics (including the $N$-centre problem, the $N$-body problem and the restricted $(N+H)$-body problem) are given.

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arXiv2020

An anisotropic monotoncity formula, with applications to some segregation problems

We prove an Alt-Caffarelli-Friedman montonicity formula for pairs of functions solving elliptic equations driven by different ellipticity matrices in their positivity sets. As application, we derive Liouville-type theorems for subsolutions of some elliptic systems, and we analyze segregation phenomena for systems of equations where the diffusion of each density is described by a different operator.

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arXiv2020

On the nodal set of solutions to a class of nonlocal parabolic equations

We investigate the local properties, including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type: \begin{equation*} \begin{cases} \partial_t \overline{u} - y^{-a} \nabla \cdot(y^a \nabla \overline{u}) = 0 \quad &\text{ in } \mathbb{B}_1^+ \times (-1,0) \\ -\partial_y^a \overline{u} = q(x,t)u \quad &\text{ on } B_1 \times \{0\} \times (-1,0), \end{cases} \end{equation*} where $a\in(-1,1)$, is a fixed parameter $\mathbb{B}_1^+\subset \mathbb{R}^{N+1}$ is the upper unit half ball and $B_1$ is the unit ball in $\mathbb{R}^N$. Our main motivation comes from its relation with a class of nonlocal parabolic equations involving the fractional power of the heat operator \begin{equation*} H^su(x,t) = \frac{1}{|Γ(-s)|} \int_{-\infty}^t \int_{\mathbb{R}^N} \left[u(x,t) - u(z,τ)\right] \frac{G_N(x-z,t-τ)}{(t-τ)^{1+s}} dzdτ. \end{equation*} We characterise the possible blow-ups and we examine the structure of the nodal set of solutions vanishing with a finite order. More precisely, we prove that the nodal set has at least parabolic Hausdorff codimension one in $\mathbb{R}^N\times\mathbb{R}$, and can be written as the union of a locally smooth part and a singular part, which turns out to possess remarkable stratification properties. Moreover, the asymptotic behaviour of general solutions near their nodal points is classified in terms of a class of explicit polynomials of Hermite and Laguerre type, obtained as eigenfunctions to an Ornstein-Uhlenbeck type operator. Our main results are obtained through a fine blow-up analysis which relies on the monotonicity of an Almgren-Poon type quotient and some new Liouville type results for parabolic equations, combined with more classical results including Federer's reduction principle and the parabolic Whitney's extension.

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arXiv2020

Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions

We consider a class of equations in divergence form with a singular/degenerate weight $$-\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)\; \quad\textrm{or} \ \textrm{div}(|y|^aF(x,y))\;.$$ Under suitable regularity assumptions for the matrix $A$ and $f$ (resp. $F$) we prove Hölder continuity of solutions which are even in $y\in\mathbb{R}$, and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the $C^{0,α}$ and $C^{1,α}$ a priori bounds for approximating problems in the form $$-\mathrm{div}((\varepsilon^2+y^2)^{a/2} A(x,y)\nabla u)=(\varepsilon^2+y^2)^{a/2} f(x,y)\; \quad\textrm{or} \ \textrm{div}((\varepsilon^2+y^2)^{a/2}F(x,y))$$ as $\varepsilon\to 0$. Finally, we derive $C^{0,α}$ and $C^{1,α}$ bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems.

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arXiv2020

Liouville type theorems and regularity of solutions to degenerate or singular problems part II: odd solutions

We consider a class of equations in divergence form with a singular/degenerate weight $$ -\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)+\textrm{div}(|y|^aF(x,y))\;. $$ Under suitable regularity assumptions for the matrix $A$, the forcing term $f$ and the field $F$, we prove Hölder continuity of solutions which are odd in $y\in\mathbb{R}$, and possibly of their derivatives. In addition, we show stability of the $C^{0,α}$ and $C^{1,α}$ a priori bounds for approximating problems in the form $$ -\mathrm{div}((\varepsilon^2+y^2)^{a/2} A(x,y)\nabla u)=(\varepsilon^2+y^2)^{a/2} f(x,y)+\textrm{div}((\varepsilon^2+y^2)^{a/2}F(x,y)) $$ as $\varepsilon\to 0$. Our method is based upon blow-up and appropriate Liouville type theorems.

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arXiv2020

Minimal collision arcs asymptotic to central configurations

We are concerned with the analysis of finite time collision trajectories for a class of singular anisotropic homogeneous potentials of degree $-α$, with $α\in(0,2)$ and their lower order perturbations. It is well known that, under reasonable generic assumptions, the asymptotic normalized configuration converges to a central configuration. Using McGehee coordinates, the flow can be extended to the collision manifold having central configurations as stationary points, endowed with their stable and unstable manifolds. We focus on the case when the asymptotic central configuration is a global minimizer of the potential on the sphere: our main goal is to show that, in a rather general setting, the local stable manifold coincides with that of the initial data of minimal collision arcs. This characterisation may be extremely useful in building complex trajectories with a broken geodesic method. The proof takes advantage of the generalised Sundman's monotonicity formula.

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arXiv2019

Spiraling asymptotic profiles of competition-diffusion systems

This paper describes the structure of the nodal set of segregation profiles arising in the singular limit of planar, stationary, reaction-diffusion systems with strongly competitive interactions of Lotka-Volterra type, when the matrix of the inter-specific competition coefficients is asymmetric and the competition parameter tends to infinity. Unlike the symmetric case, when it is known that the nodal set consists in a locally finite collection of curves meeting with equal angles at a locally finite number of singular points, the asymmetric case shows the emergence of spiraling nodal curves, still meeting at locally isolated points with finite vanishing order.

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arXiv2018

Segregated configurations involving the square root of the laplacian and their free boundaries

We study the local structure and the regularity of free boundaries of segregated critical configurations involving the square root of the laplacian. We develop an improvement of flatness theory and, as a consequence of this and Almgren's monotonicity formula, we obtain partial regularity (up to a small dimensional set) of the nodal set, thus extending the known resultsfor the standard diffusion to some anomalous case.

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arXiv2018

On the nodal set of solutions to degenerate or singular elliptic equations with an application to $s-$harmonic functions

This work is devoted to the geometric-theoretic analysis of the nodal set of solutions to degenerate or singular equations involving a class of operators including $$ L_a = \mbox{div}(\abs{y}^a \nabla), $$ with $a\in(-1,1)$ and their perturbations. As they belong to the Muckenhoupt class $A_2$, these operators appear in the seminal works of Fabes, Kenig, Jerison and Serapioni \cite{fkj,fjk2,fks} and have recently attracted a lot of attention in the last decade due to their link to the localization of the fractional Laplacian via the extension in one more dimension \cite{CS2007}. Our goal in the present paper is to develop a complete theory of the stratification properties for the nodal set of solutions of such equations in the spirit of the seminal works of Hardt, Simon, Han and Lin \cite{MR1010169,MR1305956,MR1090434}.

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arXiv2018

An Index theory for asymptotic motions under singular potentials

We develop an index theory for parabolic and collision solutions to the classical n-body problem and we prove sufficient conditions for the finiteness of the spectral index valid in a large class of trajectories ending with a total collapse or expanding with vanishing limiting velocities. Both problems suffer from a lack of compactness and can be brought in a similar form of a Lagrangian System on the half time line by a regularising change of coordinates which preserve the Lagrangian structure. We then introduce a Maslov-type index which is suitable to capture the asymptotic nature of these trajectories as half-clinic orbits: by taking into account the underlying Hamiltonian structure we define the appropriate notion of geometric index for this class of solutions and we develop the relative index theory.

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arXiv2018

Regularity of the free boundary for the vectorial Bernoulli problem

In this paper we study the regularity of the free boundary for a vector-valued Bernoulli problem, with no sign assumptions on the boundary data. More precisely, given an open, smooth set of finite measure $D\subset \mathbb{R}^d$, $Λ>0$ and $φ_i\in H^{1/2}(\partial D)$, we deal with \[ \min{\left\{\sum_{i=1}^k\int_D|\nabla v_i|^2+Λ\Big|\bigcup_{i=1}^k\{v_i\not=0\}\Big|\;:\;v_i=φ_i\;\mbox{on }\partial D\right\}}. \] We prove that, for any optimal vector $U=(u_1,\dots, u_k)$, the free boundary $\partial (\cup_{i=1}^k\{u_i\not=0\})\cap D$ is made of a regular part, which is relatively open and locally the graph of a $C^\infty$ function, a singular part, which is relatively closed and has Hausdorff dimension at most $d-d^*$, for a $d^*\in\{5,6,7\}$ and by a set of branching (two-phase) points, which is relatively closed and of finite $\mathcal{H}^{d-1}$ measure. Our arguments are based on the NTA structure of the regular part of the free boundary.

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arXiv2018

The nodal set of solutions to some elliptic problems: singular nonlinearities

This paper deals with solutions to the equation \begin{equation*} -Δu = λ_+ \left(u^+\right)^{q-1} - λ_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $λ_+,λ_- > 0$, $q \in (0,1)$, $B_1=B_1(0)$ is the unit ball in $\mathbb{R}^N$, $N \ge 2$, and $u^+:= \max\{u,0\}$, $u^-:= \max\{-u,0\}$ are the positive and the negative part of $u$, respectively. We extend to this class of \emph{singular} equations the results recently obtained in \cite{SoTe2018} for \emph{sublinear and discontinuous} equations, $1\leq q<2$, namely: (a) the finiteness of the vanishing order at every point and the complete characterization of the order spectrum; (b) a weak non-degeneracy property; (c) regularity of the nodal set of any solution: the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having Hausdorff dimension at most $N-2$ (locally finite when $N=2$). As an intermediate step, we establish the regularity of a class of \emph{not necessarily minimal} solutions. The proofs are based on a priori bounds, monotonicity formulæ\ for a $2$-parameter family of Weiss-type functionals, blow-up arguments, and the classification of homogenous solutions.

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arXiv2018

On s-harmonic functions on cones

We deal with non negative functions satisfying \[ \left\{ \begin{array}{ll} (-Δ)^s u_s=0 & \mathrm{in}\quad C, u_s=0 & \mathrm{in}\quad \mathbb{R}^n\setminus C, \end{array}\right. \] where $s\in(0,1)$ and $C$ is a given cone on $\mathbb R^n$ with vertex at zero. We consider the case when $s$ approaches $1$, wondering whether solutions of the problem do converge to harmonic functions in the same cone or not. Surprisingly, the answer will depend on the opening of the cone through an auxiliary eigenvalue problem on the upper half sphere. These conic functions are involved in the study of the nodal regions in the case of optimal partitions and other free boundary problems and play a crucial role in the extension of the Alt-Caffarelli-Friedman monotonicity formula to the case of fractional diffusions.

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