Lagrangian Manifolds, Viscosity Solutions and Maslov Index

D. McCaffrey ; S. P. Banks

Journal of convex analysis, Tome 9 (2002) no. 1, pp. 185-224.

Résumé

Let $M$ be a Lagrangian manifold, let the 1-form $pdx$ be globally exact on $M$ and let $S(x,p)$ be defined by $dS=pdx$ on $M.$ Let $H(x,p)$ be convex in $p$ for all $x$ and vanish on $M$. Let $V(x)=\inf \{S(x,p):p$ such that $(x,p)\in M\}$. Recent work in the literature has shown that (i) $V$ is a viscosity solution of $H(x,\partial V/\partial x)=0$ provided $V$ is locally Lipschitz, and (ii) $V$ is locally Lipschitz outside the set of caustic points for $M$. It is well known that this construction gives a viscosity solution for finite time variational problems -- the Lipschitz continuity of $V$ follows from that of the initial condition for the variational problem. However, this construction also applies to infinite time variational problems and stationary Hamilton-Jacobi-Bellman equations where the regularity of $V$ is not obvious. We show that for dim$\,M\leq$ 5, the local Lipschitz property follows from some geometrical assumptions on $M$ -- in particular that the Maslov index vanishes on closed curves on $M.$ We obtain a local Lipschitz constant for $V$ which is some uniform power of a local bound on $M$, the power being determined by dim$M.$ This analysis uses Arnold's classification of Lagrangian singularities.