Graph selectors and viscosity solutions on Lagrangian manifolds

University of Sheffield, Dept. of Automatic Control and Systems Engineering, Mappin Street, Sheffield, S1 3JD, UK; david@mccaffrey275.fsnet.co.uk

Received: 24 August 2005

Abstract

Let be a Lagrangian submanifold of for some closed manifold X. Let be a generating function for which is quadratic at infinity, and let W(x) be the corresponding graph selector for in the sense of Chaperon-Sikorav-Viterbo, so that there exists a subset of measure zero such that W is Lipschitz continuous on X, smooth on and for Let H(x,p)=0 for . Then W is a classical solution to on and extends to a Lipschitz function on the whole of X. Viterbo refers to W as a variational solution. We prove that W is also a viscosity solution under some simple and natural conditions. We also prove that these conditions are satisfied in many cases, including certain commonly occuring cases where H(x,p) is not convex in p.