Calculus of Variations and Geometric Measure Theory

A. Davini - A. Siconolfi

A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations

created by davini on 07 Apr 2006
modified on 10 Nov 2006


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


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.