Sharp Domains of Determinacy and Hamilton-Jacobi Equations

Joly, Jean-Luc¹ ; Métivier, Guy² ; Rauch, Jeffrey³

¹ MAB 3 51 cours de la Libération Talence 33405, FRANCE
² MAB 351 cours de la Libération Talence 33405, FRANCE
³ Department of Mathematics University of Michigan 525 East University Ann Arbor MI 48109, USA

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2004-2005), Exposé no. 5, 9 p.

Résumé

If $L(t,x,{\partial }_t,{\partial }_x)$ 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 ${\Omega }_0\subset \{t=0\}$. The frozen constant coefficient operators $L(\underline{t},\underline{x},{\partial }_t,{\partial }_x)$ determine local convex propagation cones, ${\Gamma }^{+}(\underline{t},\underline{x})$. 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 space like 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.