On weak KAM theory for \(N\)-body problems (Q2908169)

The author considers an \(N\)-body problem with masses interacting though a potential \(V = 1/r^{2 k}\) for \(k \in (0,1)\); this generalizes the standard (Newtonian) case, obtained for \(k = 1/2\).NEWLINENEWLINEThe paper provides an upper bound for the minimal action of the set of paths which bind, in time \(T\), any two configurations which are contained in some ball of radius \(R\); this for any \(T >0\) and \(R>0\). These estimates allows to obtain the Hölder regularity of the critical action potential.NEWLINENEWLINEThe weak KAM theorem for these \(N\)-body problems follows as an application of the above results; the proof goes through a setting in terms of Hamilton-Jacobi equations and a study of (global) viscosity solutions of these. It also turns out that there exist solutions which are invariant under the action of isometries on the configuration space.