Nonexistence of stable solutions to quasilinear elliptic equations on Riemannian manifolds

We prove nonexistence of nontrivial, possibly sign changing, stable solutions to a class of quasilinear elliptic equations with a potential on Riemannian manifolds, under suitable weighted volume growth conditions on geodesic balls

The secular evolution of the Kuiper belt after a close stellar encounter

We show the effects of the perturbation caused by a passing by star on the Kuiper belt objects (KBOs) of our Solar System. The dynamics of the Kuiper belt (KB) is followed by direct $N$-body simulations. The sampling of the KB has been done with $N$ up to $131,062$, setting the KBOs on initially nearly circular orbits distributed in a ring of surface density $\Sigma \sim r^{-2}$. This modelization allowed us to investigate the secular evolution of the KB upon the encounter with the perturbing star. Actually, the encounter itself usually leads toward eccentricity and inclination distributions similar to observed ones, but tends also to excite the low-eccentricity population ($e < 0.1$ around $a\sim 40$\,$\mathrm{AU}$ from the Sun), depleting this region of low eccentricities. The following long-term evolution shows a "cooling" of the eccentricities repopulating the low-eccentricity area. In dependence on the assumed KBO mass spectrum and sampled number of bodies, this repopulation takes place in a time that goes from 0.5 Myr to 100 Myr. Due to the unavoidable limitation in the number of objects in our long-term simulations ($N \leq 16384$), we could not consider a detailed KBO mass spectrum, accounting for low mass objects, thus our present simulations are not reliable in constraining correlations among inclination distribution of the KBOs and other properties, such as their size distribution. However, our high precision long term simulations are a starting point for future larger studies on massively parallel computational platforms which will provide a deeper investigation of the secular evolution ($\sim 100\,$Myr) of the KB over its whole mass spectrum.Comment: 13 pages, 12 figures, 5 table