Lagrangian Manifolds , Viscosity Solutions and Maslov Index ∗

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) ∈ M}. Recent work in the literature has shown that (i) V is a viscosity solution of H(x, ∂V/∂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 dimM ≤ 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 dimM. This analysis uses Arnold’s classification of Lagrangian singularities.