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We consider geometric properties of 3-jet non-degenerate functions in connection with Nekhoroshev theory. In particular, after showing that 3-jet non-degenerate functions are "almost quasi-convex", we prove that they are steep and compute explicitly the steepness indices (which do not exceed 2) and the steepness coefficients. [ABSTRACT FROM AUTHOR]

We study the nonlinear Klein–Gordon (NLKG) equation on a manifold M in the nonrelativistic limit, namely as the speed of light c tends to infinity. In particular, we consider a higher-order normalized approximation of NLKG (which corresponds to the NLS at order r = 1 ) and prove that when M is a smooth compact manifold or R d , the solution of the approximating equation approximates the solution of the NLKG locally uniformly in time. When M = R d , d ≥ 2 , we also prove that for r ≥ 2 small radiation solutions of the order-r normalized equation approximate solutions of the nonlinear NLKG up to times of order O (c 2 (r - 1)) . We also prove a global existence result uniform with respect to c for the NLKG equation on R 3 with cubic nonlinearity for small initial data and Strichartz estimates for the Klein–Gordon equation with potential on R 3 . [ABSTRACT FROM AUTHOR]

From KAM theory it follows that the measure of phase points which do not lie on Diophantine, Lagrangian, 'primary' tori in a nearly integrable, real-analytic Hamiltonian system is $$O(\sqrt{\varepsilon })$$ , if $$\varepsilon $$ is the size of the perturbation. In this paper we discuss how the constant in front of $$\sqrt{\varepsilon }$$ depends on the unperturbed system and in particular on the phase-space domain. [ABSTRACT FROM AUTHOR]