On the regularization of the collision solutions of the one-center problem with weak forces
Roberto Castelli Susanna Terracini
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 transmission trajectories, is continuous, though not differentiable, with respect to the initial data.
keywords: regularization technique Two-body problem weak singular potential. logarithmic potential
Addendum to: Symbolic dynamics for the $N$-centre problem at negative energies
Nicola Soave Susanna Terracini
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.
keywords: symbolic dynamics Levi-Civita regularization. $N$-centre problem chaotic motions
DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface
Susanna Terracini Juncheng Wei
The field of nonlinear elliptic equations/systems has experienced a new burst of activities in recent years. This includes the resolution of De Giorgi's conjecture for Allen-Cahn equation, the classification of stable/finite Morse index solutions for Lane-Emden equation, the regularity of interfaces of elliptic systems with large repelling parameter, Caffarelli-Silvestre extension of fractional laplace equations, the analysis of Toda type systems, etc. This special volume touches several aspects of these new activities.

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Harmonic functions in union of chambers
Laura Abatangelo Susanna Terracini
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 frequency at the infinite ends of the domain; finally we sketch the case of several different chambers.
keywords: Harmonic functions unbounded domains asymptotic estimates.
Symbolic dynamics for the $N$-centre problem at negative energies
Nicola Soave Susanna Terracini
We consider the planar $N$-centre problem, with homogeneous potentials of degree $-\alpha < 0$, $\alpha \in [1,2)$. 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 existence result allows to characterize the associated dynamical system with a symbolic dynamics, where the symbols are the partitions of the $N$ centres in two non-empty sets.
keywords: chaotic motions $N$-centre problem symbolic dynamics Levi-Civita regularization.
Uniform Hölder regularity with small exponent in competition-fractional diffusion systems
Susanna Terracini Gianmaria Verzini Alessandro Zilio
For a class of competition-diffusion nonlinear systems involving the $s$-power of the laplacian, $s\in(0,1)$, of the form \[ (-\Delta)^{s} u_i=f_i(u_i) - \beta u_i\sum_{j\neq i}a_{ij}u_j^2,\qquad i=1,\dots,k, \] we prove that $L^\infty$ boundedness implies $\mathcal{C}^{0,\alpha}$ boundedness for $\alpha>0$ sufficiently small, uniformly as $\beta\to +\infty$. This extends to the case $s\neq1/2$ part of the results obtained by the authors in the previous paper [arXiv: 1211.6087v1].
keywords: strongly competing systems optimal regularity of limiting profiles Fractional laplacian singular perturbations. spatial segregation
On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials
Veronica Felli Alberto Ferrero Susanna Terracini
The asymptotic behavior of solutions to Schrödinger 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 singular potentials. Furthermore, by an iterative Brezis-Kato procedure, pointwise upper estimate are derived.
keywords: Hardy's inequality singular cylindrical potentials Quantum $N$-body problem Schrödinger operators.
Nodal minimal partitions in dimension $3$
Bernard Helffer Thomas Hoffmann-Ostenhof Susanna Terracini
In continuation of [20], we analyze the properties of spectral minimal $k$-partitions of an open set $\Omega$ in $\mathbb R^3$ which are nodal, i.e. produced by the nodal domains of an eigenfunction of the Dirichlet Laplacian in $\Omega$. We show that such a partition is necessarily a nodal partition associated with a $k$-th eigenfunction. Hence we have in this case equality in Courant's nodal theorem.
keywords: Optimal partitions; Eigenvalues; Nodal domains; Courant nodal Theorem; Spectral minimal partitions.
On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity
Veronica Felli Elsa M. Marchini Susanna Terracini
Asymptotics of solutions to Schrödinger equations with singular dipole-type potentials are 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 are considered.
keywords: Hardy's inequality dipole moment Singular potentials Schrödinger operators.

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