Abstract

Let MM be a Lagrangian manifold, let the 1-form pdxpdx be globally exact on MM and let S(x,p)S(x,p) be defined by dS=pdxdS=pdx on M.M. Let H(x,p)H(x,p) be convex in pp for all xx and vanish on MM. Let V(x)=inf⁡{S(x,p):pV(x)=\inf \{S(x,p):p such that (x,p)∈M}(x,p)\in M\}. Recent work in the literature has shown that (i) VV is a viscosity solution of H(x,∂V/∂x)=0H(x,\partial V/\partial x)=0 provided VV is locally Lipschitz, and (ii) VV is locally Lipschitz outside the set of caustic points for MM. It is well known that this construction gives a viscosity solution for finite time variational problems – the Lipschitz continuity of VV 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 VV is not obvious. We show that for dim M≤\,M\leq 5, the local Lipschitz property follows from some geometrical assumptions on MM – in particular that the Maslov index vanishes on closed curves on M.M. We obtain a local Lipschitz constant for VV which is some uniform power of a local bound on MM, the power being determined by dimM.M. This analysis uses Arnold's classification of Lagrangian singularities

Author information

Contact details are reproduced from the original publication and may be historical.

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

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Suggested citation

D. McCaffrey, S. P. Banks. “Lagrangian Manifolds, Viscosity Solutions and Maslov Index.” Journal of Convex Analysis 9 (2002), No. 1, 185–224. https://doi.org/10.68381/jca09009

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Published by Heldermann Verlag, 2002. Rights now held by Banach Press LLC.