Boris A. Percino-Figueroa

arXiv2024

A geometric reduction method for some fully nonlinear first-order PDEs on semi-Riemannian manifolds

Given a semi-Riemannian manifold $(M,\langle \cdot,\cdot\rangle_g),$ we use the transnormal functions defined on $M$ to reduce fully nonlinear first order PDEs of the form \[ F(x,u,\langle \nabla_g u, \nabla_g u \rangle_g) = 0,\qquad \text{on }M \] into ODEs and obtain local existence results of solutions which are constant along the level sets of the transnormal functions. In particular, we apply this reduction method to obtain new solutions to eikonal equations with a prescribed geometry.

math.AP↗

arXiv2017

Null Angular Momentum and Weak KAM Solutions of the Newtonian $N$-Body Problem

In [Arch. Ration. Mech. Anal. 213 (2014), 981-991] it has been proved that in the Newtonian $N$-body problem, given a minimal central configuration $a$ and an arbitrary configuration $x$, there exists a completely parabolic orbit starting on $x$ and asymptotic to the homothetic parabolic motion of $a$, furthermore such an orbit is a free time minimizer of the action functional. In this article we extend this result in abundance of completely parabolic motions by proving that under the same hypothesis it is possible to get that the completely parabolic motion starting at $x$ has zero angular momentum. We achieve this by characterizing the rotation invariant weak KAM solutions as those defining a lamination on the configuration space by free time minimizers with zero angular momentum.

math.DS↗