Lagrangian Manifolds, Viscosity Solutions and Maslov Index

Let

be a Lagrangian manifold, let the 1-form

pdx p d x

be globally exact on

M M

and let

S(x,p) S ( x , p )

be defined by

dS=pdx d S = p d x

on

M. M .

Let

H(x,p) H ( x , p )

be convex in

p p

for all

x x

and vanish on

M M

. Let

V(x)=\inf \{S(x,p):p V ( x ) = inf ⁡ { S ( x , p ) : p

such that

(x,p)\in M\} ( x , p ) ∈ M }

. Recent work in the literature has shown that (i)

V V

is a viscosity solution of

H(x,\partial V/\partial x)=0 H ( x , ∂ V / ∂ x ) = 0

provided

V V

is locally Lipschitz, and (ii)

V V

is locally Lipschitz outside the set of caustic points for

M M

. It is well known that this construction gives a viscosity solution for finite time variational problems – the Lipschitz continuity of

V 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 V

is not obvious. We show that for dim

\,M\leq   M ≤

5, the local Lipschitz property follows from some geometrical assumptions on

M M

– in particular that the Maslov index vanishes on closed curves on

M. M .

We obtain a local Lipschitz constant for

V V

which is some uniform power of a local bound on

M M

, the power being determined by dim

M. M .

This analysis uses Arnold's classification of Lagrangian singularities