Susanna Terracini

On the existence of collisionless equivariant minimizers for the classical n-body problem

We show that the minimization of the Lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n-body problem, provided that some simple conditions on the symmetry group are satisfied. More precisely, we give a fairly general condition on symmetry groups G of the loop space Λ for the n-body problem (with potential of homogeneous degree -α, with α>0) which ensures that the restriction of the Lagrangian action

An optimal partition problem related to nonlinear eigenvalues

\mathcal{A}

Elliptic Equations with Multi-Singular Inverse-Square Potentials and Critical Nonlinearity

to the space ΛG of G-equivariant loops is coercive and its minimizers are collisionless, without any strong force assumption. In proving that local minima of ΛG are free of collisions we develop an averaging technique based on Marchal’s idea of replacing some of the point masses with suitable shapes (see [10]). As an application, several new orbits can be found with some appropriate choice of G. Furthermore, the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits.

Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential

We extend to the case of many competing densities the results of the paper (Ann. Inst. H. Poincare 6 (2002)). More precisely, we are concerned with an optimal partition problem in N-dimensional domains related to the method of nonlinear eigenvalues introduced by Nehari, (Acta Math. 105 (1961)). We prove existence of the minimal partition and some extremality conditions. Moreover, in the case of two-dimensional domains we give an asymptotic formula near the multiple intersection points. Finally, we show some connections between the variational problem and the behavior of competing species systems with large interaction.

On spectral minimal partitions: the case of the sphere

ABSTRACT This article deals with a class of nonlinear elliptic equations involving a critical power-nonlinearity as well as a potential featuring multiple inverse square singularities. We show that existence of solutions heavily depends on the strength and the location of the singularities. We associate to the problem the corresponding Rayleigh quotient and give both sufficient and necessary conditions on masses and location of singularities for the minimum to be achieved. Both the cases of whole ℝ N and bounded domains are taken into account.

Action minimizing orbits in the n-body problem with simple choreography constraint

Asymptotics of solutions to Schroedinger equations with singular magnetic and electric potentials is investigated. By using a 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) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order -1.

Collisionless periodic solutions to some three-body problems

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 \(\mathbb{S}^2\) is up to a rotation the so-called Y-partition. This question is connected to a celebrated conjecture of Bishop in harmonic analysis.

Periodic Solutions to Some Problems of n-Body Type

In 1999 Chenciner and Montgomery found a remarkably simple choreographic motion for the planar three-body problem (see [11]). In this solution, three equal masses travel on an figure-of-eight shaped planar curve; this orbit is obtained by minimizing the action integral on the set of simple planar choreographies with some special symmetry constraints. In this work our aim is to study the problem of n masses moving in under an attractive force generated by a potential of the kind 1/rα, α > 0, with the only constraint to be a simple choreography: if q1(t),...,qn(t) are the n orbits then we impose the existence of such that where τ = 2π/n. In this setting, we first prove that for every and α > 0, the Lagrangian action attains its absolute minimum on the planar regular n-gon relative equilibrium. Next, we deal with the problem in a rotating frame and show a richer phenomenology: indeed, while for some values of the angular velocity, the minimizers are still relative equilibria, for others, the minima of the action are no longer rigid motions.

SIGN-CHANGING SOLUTIONS OF COMPETITION-DIFFUSION ELLIPTIC SYSTEMS AND OPTIMAL PARTITION PROBLEMS

We provide sufficient conditions for the existence of periodic solutions to some three-body problems. Periodic solutions are found as minima of the associated action integral and are shown to be free of double and triple collisions.

Oscillating solutions to second-order ODEs with indefinite superlinear nonlinearities

AbstractWe prove the existence of at least one T-periodic solution to a dynamical system of the type