*Published Paper*

**Inserted:** 7 apr 2006

**Last Updated:** 10 nov 2006

**Journal:** SIAM J. Math. Anal.

**Volume:** 38

**Number:** 2

**Pages:** 478-502

**Year:** 2006

**Abstract:**

We consider the Hamilton--Jacobi equation
\[
\partial_t u+H(x,Du)=0\qquad \hbox{in $(0,+\infty)\times{\mathbb T}^N$}
\]
where ${\mathbb T}^N$ is the flat $N$--dimensional torus, and the
Hamiltonian $H(x,p)$ is assumed continuous in $x$ and strictly
convex and coercive in $p$. We study the large time behavior of
solutions, and we identify the limit through a Lax--type
formula. Some convergence results are also given for $H$ solely
convex. Our qualitative method is based on the analysis of the
dynamical properties of the so called Aubry set, performed in the
spirit of a previous work by A. Fathi and A. Siconolfi (*Calc. Var.* **22**, (2005), no. 1, pp. 185--228). This can be viewed as a generalization
of the techniques used by A. Fathi (*C. R. Acad. Sci. Paris Sér. I Math.*, **327**, no. 3 (1998), pp. 267--270) and J.M. Roquejoffre (*J. Math. Pures Appl.* **80**, 1 (2001), pp. 85--104). Analogous
results have been obtained by G. Barles, P. E. Souganidis (*SIAM J. Math. Anal.* **31**, no. 4 (2000), pp. 925--939) using PDE methods.