On Lagrange Manifolds and Viscosity Solutions

We consider the use of Lagrange manifolds to construct viscosity solutions of first order Hamiltonian-Jacobi equations. Recent work of several authors is indicated in which the essential underlying structure consists of a Lagrange manifold on which 1) the desired Hamiltonian function vanishes and 2) the canonical 1-form p·dx of classical mechanics has an integral S(x, p). We explore the proposition that a viscosity solution W (x) of the Hamiltonian-Jacobi equation is obtained by minimizing the function S over points in the Lagrange manifold that project to the state x. We prove that the function W (x) produced by this construction is necessarily a viscosity supersolution, and if Lipschitz is also a subsolution. Elementary examples illustrate the construction, including situations in which the subsolution property fails. Connections with Riccati PDEs, L2-gain in nonlinear systems, small-noise quasipotentials, and simple variational examples are all described.