HYPERBOLIC DOMAINS OF DETERMINACY AND HAMILTON–JACOBI EQUATIONS

Abstract

If L(t, x, ∂t, ∂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 Ω0⊂ {t = 0}. The frozen constant coefficient operators [Formula: see text] determine local convex propagation cones, [Formula: see text]. Influence curves are curves whose tangent always lies in these cones. We prove that the set of points Ω which cannot be reached by influence curves beginning in the exterior of Ω0is a domain of determinacy in the sense that solutions of Lu = 0 whose Cauchy data vanish in Ω0must vanish in Ω. We prove that Ω is swept out by continuous spacelike deformations of Ω0and 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.