Seminario Geometría
Existence of expansive solutions and Hamilton-Jacobi equations for the N-body problem
Ponente: Davide Polimeni (U. Torino)Fecha: miércoles 13 de mayo de 2026 - 11:30Lugar: Aula Gris 1, ICMAT
Resumen:
We prove, for the classical \(N\)-body problem, the existence of action minimizing half-entire expansive solutions with prescribed asymptotic direction and initial configuration of the bodies, tackling the cases of hyperbolic, parabolic and hyperbolic-parabolic arcs in a unified 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. In addition, we also manage to give a precise description of the growth of parabolic and hyperbolic-parabolic solutions. These results, first proved for \((-1)\)-homogeneous Newtonian potentials, can also be estended to the case of \((-\alpha)\)-homogeneous Newtonian potentials, with \(\alpha>0\).
Building on these results, we introduce a renormalized value function defined as a linear correction of a minimum of the renormalized Lagrangian action. We prove that these value functions, which are associated with the \(d\)-dimensional (\(d \ge 2\)) N-body problem, 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 \(H^{d(N−1)−1}\)-rectifiability, and we provide an upper bound on the Hausdorff dimension of the set of regular conjugate points.
This is a joint work with Diego Berti and Susanna Terracini.
