Analytic dependence on parameters for Evans' approximated Weak KAM solutions

Abstract

We consider a variational principle for approximated Weak KAM solutions proposed by Evans. For Hamiltonians in quasi-integrable form $h(p)+\varepsilon f(\varphi,p)$, we prove that the map which takes the parameters $(\varepsilon,P,\varrho)$ to Evans' approximated solution $u_{\varepsilon,P,\varrho}$ is real analytic. In the mechanical case, we compute a recursive system of periodic partial differential equations identifying univocally the coefficients for the power series of the perturbative parameter $\varepsilon$.