The adjoint method, recently introduced by Evans, is used to study obstacle problems, weakly coupled systems, cell problems for weakly coupled systems of Hamilton − Jacobi equations, and weakly coupled systems of obstacle type. In particular, new results about the speed of convergence of some approximation procedures are derived.

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 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...

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