Susanna Terracini

University of Turin | UNITO · Dipartimento di Matematica "Giuseppe Peano"

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...

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,...

In this paper we present \emph{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...

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 Cieliebak, Frauenfelder, and Volkov [Ann. Inst. H. Poincaré C Anal. Non Linéaire 40 (2023), 379–455]...

In recent years, a variational approach to the n-body problem has brought significant progress in the field of celestial mechanics and new possible orbits have been determined. In this case, one studies the critical points of the Lagrangian action associated to the n-body problem. In particular cases and with the use of evolutionary algorithms, one...

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...

We deal, for the classical N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N$\end{document}-body problem, with the existence of action minimizing half entire expansive...

We investigate the local properties , including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type: { ∂ t u ¯ − y − a ∇ ⋅ ( y a ∇ u ¯ ) = 0 a m p ; in B 1 + × ( − 1 , 0 ) − ∂ y a u ¯ = q ( x , t ) u a m p ; on B 1 × { 0 } × ( − 1 , 0 ) , \begin{equation*} \begin {cases} \partial _t \ov...

As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type for exponents \(a>-1\), where the weight \(\rho \) vanishes with non zero gradient on a regular hypersurface \(\Gamma \), which can be either a part of the boundary of \(\Omega \) or mostly contained in its interior. As an appl...

We prove the presence of topological chaos at high internal energies for a new class of mechanical refraction billiards coming from Celestial Mechanics. Given an open and bounded domain D ∈ R 2 with smooth boundary, a central mass generates a Keplerian potential in it, while, in R 2 ∖ D ‾ , a harmonic oscillator-type potential acts. At the interfac...

As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type: \[ -\mathrm{div}\left(\rho^aA\nabla w\right)=\rho^af+\mathrm{div}\left(\rho^aF\right) \quad\textrm{in}\; \Omega \] for exponents $a>-1$, where the weight $\rho$ vanishes in a non degenerate manner on a regular hypersurface $\G...

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 o...

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...

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 an application, we derive Liouville-type theorems for subsolutions of some elliptic systems, and we analyze segregation phenomena for systems of equations where the diffusion...

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 a 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 re...

We give a complete characterization of the boundary traces $\varphi_i$ ($i=1,\dots,K$) supporting spiraling waves, rotating with a given angular speed $\omega$, which appear as singular limits of competition-diffusion systems of the type \[ \frac{\partial}{\partial t} u_i -\Delta u_i = \mu u_i -\beta u_i \sum_{j \neq i} a_{ij} u_j \text{ in } \Omeg...

Drawing on her experience on the Governing Board of Italy’s National Evaluation Agency (ANVUR), the author asks if strategic initiatives involving merit-based fund allocation threaten academic freedom. While admitting such threats exist, she argues that the alternatives to qualitative assessment are “delusional” and thus not preferrable. The risk t...

The aim of this work is to continue the analysis, started in [10], of the dynamics of a point-mass particle \begin{document}$ P $\end{document} moving in a galaxy with an harmonic biaxial core, in whose center sits a Keplerian attractive center (e.g. a Black Hole). Accordingly, the plane \begin{document}$ \mathbb{R}^2 $\end{document} is divided int...

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: x¨(t)=∑j=1N∇Vj(x(t)-cj).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgree...

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...

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 $\Sigma$, where a Snell's law o...

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, anisotropi...

We consider a class of equations in divergence form with a singular/degenerate weight − div ( | y | a A ( x , y ) ∇ u ) = | y | a f ( x , y ) or div ( | y | a F ( 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 ∈ R , and possibly of their derivatives up t...

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 pos...

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...

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 H\"older continuity of solutions which are odd in $y\in\mathbb{R}$, and p...

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 includingLa=div(|y|a∇), with a∈(−1,1) and their perturbations. As they belong to the Muckenhoupt class A2, these operators appear in the seminal works of Fabes, Kenig, Jerison and Serapioni [1], [...

We are concerned with the analysis of finite time collision trajectories for a class of singular anisotropic homogeneous potentials of degree $-\alpha$, with $\alpha\in(0,2)$ and their lower order perturbations. It is well known that, under reasonable generic assumptions, the asymptotic normalized configuration converges to a central configuration....

We study the local structure and the regularity of free boundaries of segregated minimal 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 extendi...

We consider a class of equations in divergence form with a singular/degenerate weight −div(|y| a A(x, y)∇u) = |y| a f (x, y) or div(|y| a F (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 ∈ R, and possibly of their derivatives up to order two or more (Schaud...

We prove the existence of parabolic arcs with prescribed asymptotic direction for the equation \begin{equation*} \ddot{x} = - \dfrac{x}{\lvert x \rvert^{3}} + \nabla W(t,x), \qquad x \in \mathbb{R}^{d}, \end{equation*} where $d \geq 2$ and $W$ is a (possibly time-dependent) lower order term, for $\vert x \vert \to +\infty$, with respect to the Kepl...

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 extend...

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=div(|y|^a∇), with a ∈ (−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...

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,...

We investigate the nodal properties of solutions $u = u(x,t)$ to a nonlocal parabolic reaction-diffusion equation. We characterise the possible blow-ups and we examine the structure of the nodal set of such solutions. More precisely, we prove that their nodal set has at least parabolic Hausdorff codimension one in $\mathbb{R}^N\times\mathbb{R}$, an...

We consider a class of variational problems for densities that repel each other at a distance. Typical examples are given by the Dirichlet functional and the Rayleigh functional $$\begin{aligned} D(\mathbf{u}) = \sum_{i=1}^k \int_{\Omega} |\nabla u_i|^2 \quad \text{or} \quad R(\mathbf{u}) = \sum_{i=1}^k \frac{\int_{\Omega} |\nabla u_i|^2}{\int_{\Om...

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 $D\subset \mathbb{R}^d$, $\Lambda>0$ and $\varphi_i\in H^{1/2}(\partial D)$, we deal with \[ \min{\left\{\sum_{i=1}^k\int_D|\nabla v_i|^2+\Lambda...

This paper deals with solutions to the equation \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- > 0$, $q \in (0,1)$, $B_1=B_1(0)$ is the unit ball in $\mathbb{R}^N$, $N \ge 2$, and $u^+:= \max\{u,0\}$, $u^-:= \max\{-u,0\}$ are the posi...

We are concerned with the nodal set of solutions to equations of the form \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- > 0$, $q \in [1,2)$, $B_1=B_1(0)$ is the unit ball in $\mathbb{R}^N$, $N \ge 2$, and $u^+:= \max\{u,0\}$, $u^-:=...

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. Unlik...

We deal with non negative functions satisfying (-\Delta)^s u_s=0 & \mathrm{in}\quad C, u_s=0 & \mathrm{in}\quad \mathbb{R}^n\setminus C, 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 sam...

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 b...

In this paper we study the regularity of the optimal sets for the shape optimization problem $$\min\Big\{\lambda_{1}(\Omega)+\dots+\lambda_{k}(\Omega) : \Omega \subset {\mathbb {R}}^{d} {\rm open},\quad |\Omega| = 1\Big\},$$where \({\lambda_{1}(\cdot),\ldots,\lambda_{k}(\cdot)}\) denote the eigenvalues of the Dirichlet Laplacian and \({|\cdot|}\) t...

For the N-centre problem in the three dimensional space, $${\ddot{x}} = -\sum_{i=1}^{N} \frac{m_i \,(x-c_i)}{\vert x - c_i \vert^{\alpha+2}}, \qquad x \in {\mathbb{R}}^3 {\setminus} \{c_1,\ldots,c_N\},$$where \({N \geqq 2}\), \({m_i > 0}\) and \({\alpha \in [1,2)}\), we prove the existence of entire parabolic trajectories having prescribed asymptot...

For a competition-diffusion system involving the fractional Laplacian of the form −(−\mathrm{\Delta })^{s}u = uv^{2},\:−(−\mathrm{\Delta })^{s}v = vu^{2},\:u,v > 0\:\text{in}\:\mathbb{R}^{N}, with s \in (0,1) , we prove that the maximal asymptotic growth rate for its entire solutions is 2s . Moreover, since we are able to construct symmetric soluti...

Let \({\Omega \subset \mathbb{R}^N}\) be an open bounded domain and \({m \in \mathbb{N}}\). Given \({k_1,\ldots,k_m \in \mathbb{N}}\), we consider a wide class of optimal partition problems involving Dirichlet eigenvalues of elliptic operators, of the following form $${\rm inf}\left\{F({\lambda_{k_{1}}}(\omega_1),\ldots,\lambda_{k_m}(\omega_m)):\ (...

We consider a magnetic Schrödinger operator with magnetic field concentrated at one point (the pole) of a domain and half integer circulation, and we focus on the behavior of Dirichlet eigenvalues as functions of the pole. Although the magnetic field vanishes almost everywhere, it is well known that it affects the operator at the spectral level (th...

In this paper we study the Neumann problem\begin{equation*}\begin{cases}-\Delta u+u=u^p \& \text{ in }B\_1 \\u \textgreater{} 0, \& \text{ in }B\_1 \\\partial\_\nu u=0 \& \text{ on } \partial B\_1,\end{cases}\end{equation*}and we show the existence of multiple-layer radial solutions as $p\rightarrow+\infty$.

We study regularity issues for systems of elliptic equations of the type \[ -\Delta u_i=f_{i,\beta}(x)-\beta \sum_{j\neq i} a_{ij} u_i |u_i|^{p-1}|u_j|^{p+1} \] set in domains $\Omega \subset \mathbb{R}^N$, for $N \geq 1$. The paper is devoted to the derivation of $\mathcal{C}^{0,\alpha}$ estimates that are uniform in the competition parameter $\be...

We continue the analysis started in [Noris,Terracini,Indiana Univ Math J,2010] and [Bonnaillie-No\"el,Noris,Nys,Terracini,Analysis & PDE,2014], concerning the behavior of the eigenvalues of a magnetic Schr\"odinger operator of Aharonov-Bohm type with half-integer circulation. We consider a planar domain with Dirichlet boundary conditions and we con...

For a class of competition-diusion nonlinear systems involving the s-power of the laplacian, s 2 (0; 1), of the form (-δ)s/ui/ = fi/ (ui/) δ ∑uiXj6=iaiju2j ; i = 1; k;we prove that L1 boundedness implies C0; boundedness for 0 sufficiently small, uniformly as ! +1. This extends to the case s 6= 1=2 part of the results obtained by the authors in the...

We study the nature of the nonlinear Schrödinger equation ground states on the product spaces ℝ n ×M k , where M k is a compact Riemannian manifold. We prove that for small L 2 masses the ground states coincide with the corresponding ℝ n ground states. We also prove that above a critical mass the ground states have nontrivial M k dependence. Finall...

We consider solutions of the competitive elliptic system \[ \left\{ \begin{array}{ll} -\Delta u_i = - \sum_{j \neq i} u_i u_j^2 & \text{in $\mathbb{R}^N$} \\ u_i >0 & \text{in $\mathbb{R}^N$} \end{array}\right. \qquad i=1,\dots,k. \] We are concerned with the classification of entire solutions, according with their growth rate. The prototype of our...

In a singular potential setting, we generalize a method which allows to show that minimizers under topological constraints of the action functional (or of the Maupertuis') are collision-free. This methods applies to $3$-dimensional problems of celestial mechanics exhibiting a particular cylindrical symmetry, as well as to planar problems of $N$-cen...

Let $\Omega\subset \mathbb{R}^N$ be an open bounded domain and $m\in \mathbb{N}$. Given $k_1,\ldots,k_m\in \mathbb{N}$, we consider a wide class of optimal partition problems involving Dirichlet eigenvalues of elliptic operators, including the following \[ \inf\left\{\Phi(\omega_1,\ldots,\omega_m):=\sum_{i=1}^m \lambda_{k_i}(\omega_i):\ (\omega_1,\...

We consider a magnetic operator of Aharonov-Bohm type with Dirichlet boundary conditions in a planar domain. We analyse the behavior of its eigenvalues as the singular pole moves in the domain. For any value of the circulation we prove that the k-th magnetic eigenvalue converges to the k-th eigenvalue of the Laplacian as the pole approaches the bou...

This paper aims at completing and clarifying a delicate step in the proof of Theorem 5.3 of our paper [1], where it was used the differentiability of a function F, which a priori can appear not necessarily differentiable.

Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichlet Laplacian is continuous with respect to the channel section parameter. This in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation. Under an explicit nondegeneracy c...

In continuation with the paper arXiv:1202.4414, we investigate the asymptotic behavior of weighted eigenfunctions in two half-spaces connected by a thin tube. We provide several improvements about some convergences stated in arXiv:1202.4414; most of all, we provide the exact asymptotic behavior of the implicit normalization for solutions given in a...

L’anno vede la continuazione dell’esperienza dei fascicoli monografici, accolti con favore dai lettori. Il numero 57–58 èinteramente dedicato alla figura diTullio Levi-Civita che I’editoriale introduce come «uno dei più importanti matematici italiani, riconosciuto per le sue qualità a livello internazionale». Continua I’editoriale: Pietro Nastasi e...

For a class of competition-diffusion nonlinear systems involving the square root of the Laplacian, including the fractional Gross-Pitaevskii system, we prove that uniform boundedness implies Holder boundedness for every exponent less than 1/2, uniformly as the interspecific competition parameter diverges. Moreover we prove that the limiting profile...

Asymptotics of solutions to Schrodinger equations with singular dipole-type poten- tials is investigated. We evaluate the exact behavior near the singularity of solutions to elliptic equations with potentials which are purely angular multiples of radial inverse-square functions. Both the linear and the semilinear (critical and subcritical) cases ar...

We deal with a class of Lipschitz vector functions $U=(u_1,...,u_h)$ whose components are non negative, disjointly supported and verify an elliptic equation on each support. Under a weak formulation of a reflection law, related to the Poho\u{z}aev identity, we prove that the nodal set is a collection of $C^{1,\alpha}$ hyper-surfaces (for every $0<\...

For a class of quasilinear elliptic equations involving the p-Laplace operator, we develop an abstract critical point theory in the presence of sub-supersolutions. Our approach is based upon the proof of the invariance under the gradient flow of enlarged cones in the $W^{1,p}_0$ topology. With this, we prove abstract existence and multiplicity theo...

We characterize the set of harmonic functions with Dirichlet boundary conditions in unbounded domains which are union of two different chambers. We analyse the asymptotic behavior of the solutions in connection with the changes in the domain's geometry; we classify all (possibly sign-changing) infinite energy solutions having given asymptotic frequ...

This paper aims at completing and clarifying a delicate step in the proof of Theorem 5.3 of our paper \cite{ST}, where it was used the differentiability of a function $F$, which a priori can appear not necessarily differentiable.

In the recent literature, the phenomenon of phase separation for binary mixtures of Bose–Einstein condensates can be understood, from a mathematical point of view, as governed by the asymptotic limit of the stationary Gross–Pitaevskii system \left\{ \begin{array}{l} -\Delta u + u^3+\beta uv^2=\lambda u,\\ -\Delta v + v^3+\beta u^2v=\mu v,\\ u,v\in...

We consider a semilinear elliptic equation on a smooth bounded domain $\Om$ in $\R^2$, assuming that both the domain and the equation are invariant under reflections about one of the coordinate axes, say the y-axis. It is known that nonnegative solutions of the Dirichlet problem for such equations are symmetric about the axis, and, if strictly posi...

We study the nature of the Nonlinear Schr\"odinger equation ground states on the product spaces $\R^n\times M^k$, where $M^k$ is a compact Riemannian manifold. We prove that for small $L^2$ masses the ground states coincide with the corresponding $\R^n$ ground states. We also prove that above a critical mass the ground states have nontrivial $M^k$...

In this paper we prove the existence of infinitely many sign-changing solutions for the system of $m$ Schr\"odinger equations with competition interactions $$ -\Delta u_i+a_i u_i^3+\beta u_i \sum_{j\neq i} u_j^2 =\lambda_{i,\beta} u_i \quad u_i\in H^1_0(\Omega), \quad i=1,...,m $$ where $\Omega$ is a bounded domain, $\beta>0$ and $a_i\geq 0\ \foral...

We present a global variational approach to the search for multiple nodal solutions u∈H 1(ℝ N ) to a class of elliptic equations of type¶−Δu(x)=f(|x|,u(x)), x∈ℝ N ,¶where N≧ 2, f is superlinear and subcritical, and f(|x|≡0.

We study the effect of a forcing term in the context of the search of multiple nodal solutions u∈h 1(ℝ N ) to a class of elliptic equations of type¶−Δu(x)=f(|x|,u(x))+h(|x|), x∈ℝ N ,¶where f(|x|≡0 and f is superlinear but subcritical at infinity.

We study the qualitative properties of a limiting elliptic system arising in phase separation for Bose-Einstein condensates with multiple states: \Delta u=u v^2 in R^n, \Delta v= v u^2 in R^n, u, v>0\quad in R^n. When n=1, we prove uniqueness of the one-dimensional profile. In dimension 2, we prove that stable solutions with linear growth must be o...

Consider two domains connected by a thin tube: it can be shown that, generically, the mass of a given eigenfunction of the Dirichlet Laplacian concentrates in only one of them. The restriction to the other domain, when suitably normalized, develops a singularity at the junction of the tube, as the channel section tends to zero. Our main result stat...

Glossary Definition of the Subject Introduction Simple Choreographies and Relative Equilibria Symmetry Groups and Equivariant Orbits The 3-Body Problem Minimizing Properties of Simple Choreographies Generalized Orbits and Singularities Asymptotic Estimates at Collisions Absence of Collision for Locally Minimal Paths Future Directions Bibliography

We consider the planar $N$-centre problem, with homogeneous potentials of degree $-\a<0$, $\a \in [1,2)$. We prove the existence of infinitely many collisions-free periodic solutions with negative and small energy, for any distribution of the centres inside a compact set. The proof is based upon topological, variational and geometric arguments. The...

We study the possible regularization of collision solutions for one centre problems with a weak singularity. In the case of logarithmic singularities, we consider the method of regularization via smoothing of the potential. With this technique, we prove that the extended flow, where collision solutions are replaced with a transmission trajectory, i...

In this paper we deal with the cubic Schr\"odinger system $ -\Delta u_i = \sum_{j=1}^n \beta_{ij}u_j^2 u_i$, $u_1,\dots,u_n \geq 0$ in $\mathbb{R}^N (N\leq 3)$, where $\beta=(\beta_{i,j})_{ij}$ is a symmetric matrix with real coefficients and $\beta_{ii}\geq 0$ for every $i=1,\ldots,n$. We analyse the existence and nonexistence of nontrivial soluti...

We continue the variational approach to parabolic trajectories introduced in our previous paper [5], which sees parabolic orbits as minimal phase transitions. We deepen and complete the analysis in the planar case for homogeneous singular potentials. We characterize all parabolic orbits connecting two minimal central configurations as free-time Mor...

For the class of anisotropic Kepler problems in $\RR^d\setminus\{0\}$ with homogeneous potentials, we seek parabolic trajectories having prescribed asymptotic directions at infinity and which, in addition, are Morse minimizing geodesics for the Jacobi metric. Such trajectories correspond to saddle heteroclinics on the collision manifold, are struct...

Asymptotics of solutions to Schrödinger equations with singular magnetic and electric potentials is investigated. By using an Almgren type monotonicity formula, separation of variables, and an iterative Brezis-Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) el...

We investigate existence and qualitative behavior of solutions to nonlinear Schrödinger equations with critical exponent and singular electromagnetic potentials. We are concerned with magnetic vector potentials which are homogeneous of degree –1, including the Aharonov–Bohm class. In particular, by variational arguments we prove a result of multipl...

This paper completes and partially improves some of the results of [arXiv:0809.5002] about the asymptotic behavior of solutions of linear and nonlinear elliptic equations with singular coefficients via an Almgren type monotonicity formula

We investigate symmetry properties of solutions to equations of the form -Δu = + f(|x|, u) in ℝ N for N ≥ 4, with at most critical nonlinearities. By using geometric arguments, we prove that solutions with low Morse index (namely 0 or 1) and which are biradial (i.e. are invariant under the action of a toric group of rotations), are in fact complete...

The asymptotic behavior of solutions to Schr\"odinger equations with singular homogeneous potentials is investigated. Through an Almgren type monotonicity formula and separation of variables, we describe the exact asymptotics near the singularity of solutions to at most critical semilinear elliptic equations with cylindrical and quantum multi-body...

We prove the existence of infinitely many periodic solutions with prescribed period to a class of problems of n-body type.

In continuation of [20], we analyse the properties of spectral mini-mal k-partitions of an open set Ω in R 3 which are nodal, i.e. produced by the nodal domains of an eigenfunction of the Dirichlet Laplacian in Ω. We show that such a k-partition is necessarily the nodal one asso-ciated with a k-th eigenfunction. Hence we have in this case equality...

For the positive solutions of the Gross–Pitaevskii system we prove that L∞-boundedness implies C0,α-boundedness for every α ϵ (0,1), uniformly as β → +∞. Moreover, we prove that the limiting profile as β → +∞ is Lipschitz-continuous. The proof relies upon the blowup technique and the monotonicity formulae by Almgren and Alt, Caffarelli, and Friedma...

In continuation of previous work, we analyze the properties of spectral minimal partitions and focus in this paper our analysis on the case of the sphere. We prove that a minimal 3-partition for the sphere \mathbbS2\mathbb{S}^2 is up to a rotation the so-called Y-partition. This question is connected to a celebrated conjecture of Bishop in harmon...

We consider spectral minimal partitions. Continuing work of the the present authors about problems for planar domains, [23], we focus on the sphere and obtain a sharp result for 3-partitions which is related to questions from harmonic analysis, in particular to a conjecture of Bishop.

We consider two-dimensional Schrödinger operators in bounded domains. We analyze relations between the nodal domains of eigenfunctions, spectral minimal partitions and spectral properties of the corresponding operator. The main results concern the existence and regularity of the minimal partitions and the characterization of the minimal partitions...

In this paper we consider a stationary Schroedinger operator in the plane, in presence of a magnetic field of Aharonov-Bohm type with semi-integer circulation. We analyze the nodal regions for a class of solutions such that the nodal set consists of regular arcs, connecting the singular points with the boundary. In case of one magnetic pole, which...