Symplectic Homogenization

Viterbo, Claude

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 67-140

Let $H(q,p)$ be a Hamiltonian on $T^{*}{T}^n$. Under suitable assumptions on $H$, we show that the sequence ${\left(H_k\right)}_{k\ge 1}$ defined by $H_k(q,p)=H(kq,p)$ converges in the $\gamma$-topology—defined in [Vit92]—to an integrable continuous Hamiltonian $\overline{H}\left(p\right)$. This is extended to the case of non-autonomous Hamiltonians, and the more general setting in which only some of the variables are homogenized: we consider the sequence $H(kx,y,q,p)$ and prove it has a $\gamma$-limit $\overline{H}(y,q,p)$, thus yielding an “effective Hamiltonian”. The goal of this paper is to prove convergence of the above sequences, state the first properties of the homogenization operator, and give some applications to solutions of Hamilton-Jacobi equations, construction of quasi-states, etc. We also prove that when $H$ is convex in $p$, the function $\overline{H}$ coincides with Mather’s $\alpha$ function defined in [Mat91] and associated to the Legendre dual of $H$. This gives a new proof—in the torus case—of its symplectic invariance first discovered by P. Bernard in [Ber07].

Soit $H(q,p)$ un hamiltonien défini sur $T^{*}{T}^n$. Sous des hypothèses convenables, on montre que la suite ${\left(H_k\right)}_{k\ge 1}$ définie par $H_k(q,p)=H(kq,p)$ converge pour la topologie $\gamma$, définie dans [Vit92], vers un hamiltonien intégrable $\overline{H}\left(p\right)$. Ceci s’étend au cas de hamiltoniens non-autonomes, et au cas où seulement certaines variables sont homogénéisées : par exemple la suite $H_k(kx,y,p_x,p_y)$ qui dans ce cas aura une limite $\overline{H}(y,p_x,p_y)$, qui est un « hamiltonien effectif ». Le but de cet article est de démontrer la convergence de ces suites, ainsi que les premières propriétés de l’opérateur d’homogénéisation et d’en donner des applications aux solutions d’équations de Hamilton-Jacobi, aux quasi-états symplectiques, etc. On démontre aussi que lorsque $H$ est convexe en $p$, la fonction $\overline{H}$ coïncide avec la fonction $\alpha$ de Mather (cf. [Mat91]) associée au dual de Legendre de $H$. Cela redémontre, dans le cas du tore, que cette fonction est symplectiquement invariante, comme l’avait démontré P. Bernard ([Ber07]) dans le cas général.

Classification : 37J05, 53D35, 35F20, 49L25, 37J40, 37J50
Keywords: Homogenization, symplectic topology, Hamiltonian flow, Hamilton-Jacobi equation, variational solutions
Mots-clés : Flot hamiltonien, homogénéisation, Hamilton-Jacobi, symplectique

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