The main objective of this paper is to prove
new necessary conditions to the existence of
KAM tori.
To do so, we develop a
set of
explicit a-priori estimates for smooth
solutions of Hamilton-Jacobi equations,
using a combination of methods from
viscosity solutions,
KAM and Aubry-Mather theories.
These estimates
are valid
in any
space dimension, and can be checked numerically
to detect gaps between KAM tori and Aubry-Mather sets.
We apply these results to detect non-integrable regions in
several
examples such as
a forced pendulum, two coupled penduli, and
the double pendulum.