Journal of Convex Analysis 09 (2002), No. 1, 185--224
Copyright Heldermann Verlag 2002
Lagrangian Manifolds, Viscosity Solutions and Maslov Index
David McCaffrey
Dept. of Automatic Control and Systems Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, Great Britain
david.mccaffrey@opc.shell.com
S. P. Banks
Dept. of Automatic Control and Systems Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, Great Britain
s.banks@sheffield.ac.uk
[Abstract-pdf]
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.
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