Hyperbolic domains of determinacy and Hamilton-Jacobi equations

If L(t, x, partial derivative(t), partial derivative(x)) is a linear hyperbolic system of partial differential operators for which local uniqueness in the Cauchy problem at spacelike hypersurfaces is known, we find nearly optimal domains of determinacy of open sets Omega(0) subset of {t= 0). The frozen constant coefficient operators L((t) under bar, (x) under bar, partial derivative(t),partial derivative(x)) determine local convex propagation cones, Gamma(+) ((t) under bar, (x) under bar). Influence curves are curves whose tangent always lies in these cones. We prove that the set of points Omega which cannot be reached by influence curves beginning in the exterior of Omega(0) is a domain of determinacy in the sense that solutions of Lu = 0 whose Cauchy data vanish in Omega(0) must vanish in Omega. We prove that Omega is swept out by continuous spacelike deformations of Omega(0) and is also the set described by maximal solutions of a natural Hamilton-Jacobi equation (HJE). The HJE provides a method for computing approximate domains and is also the bridge from the raylike description using influence curves to that depending on spacelike deformations. The deformations are obtained from level surfaces of mollified solutions of HJEs.