Consider the three-body problem, in the regime where one body revolves far away around the other two, in space, the masses of the bodies being arbitrary but fixed; in this regime, there are no resonances in mean motions. The so-called secular dynamics governs the slow evolution of the Keplerian ellipses. We show that it contains a horseshoe and all the chaotic dynamics which goes along with it, corresponding to motions along which the eccentricity of the inner ellipse undergoes large, random excursions. The proof goes through the surprisingly explicit computation of the homoclinic solution of the first order secular system, its complex singularities and the Melnikov potential.
In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem (-Delta)(s) u = f(u) in Omega, u equivalent to 0 in R-n\Omega. Here, s is an element of (0, 1), (-Delta)(s) is the fractional Laplacian in R-n, and Omega is a bounded C-1,C-1 domain.
To establish the identity we use, among other things, that if u is a bounded solution then u/delta(s)vertical bar(Omega) is C-alpha up to the boundary partial derivative Omega, where delta(x) = dist(x, partial derivative Omega). In the fractional Pohozaev identity, the function u/delta(s)vertical bar(partial derivative Omega) plays the role that partial derivative u/partial derivative nu plays in the classical one. Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over partial derivative Omega) which is completely local.
As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities.
We prove some interior regularity results for potential functions of optimal transportation problems with power costs. The main point is that our problem is equivalent to a new optimal transportation problem whose cost function is a sufficiently small perturbation of the quadratic cost, but it does not satisfy the well known condition (A.3) guaranteeing regularity. The proof consists in a perturbation argument from the standard Monge–Ampère equation in order to obtain, first, interior C1,1 estimates for the potential and, second, interior Hölder estimates for
second derivatives. In particular, we take a close look at the geometry of optimal
transportation when the cost function is close to quadratic in order to understand
how the equation degenerates near the boundary.
Two mass points of equal masses m 1 = m 2 > 0 move under Newton's law of attraction in a non-collision hyperbolic orbit while their center of mass is at rest. We consider a third mass point, of mass m 3 = 0, moving on the straight line L perpendicular to the plane of motion of the first two mass points and passing through their center of mass. Since m 3 = 0, the motion of m 1 and m 2 is not affected by the third, and from the symmetry of the motion it is clear that m 3 remains on the line L. The hyperbolic restricted 3-body problem is to describe the moton of m 3. Our main result is the characterization of the global flow of this problem.