Sharp Domains of Determinacy and Hamilton-Jacobi Equations

Joly, Jean-Luc, Métivier, Guy, and Rauch, Jeffrey. "Sharp Domains of Determinacy and Hamilton-Jacobi Equations." Séminaire Équations aux dérivées partielles 2004-2005 (2004-2005): 1-9. <http://eudml.org/doc/11117>.

@article{Joly2004-2005,
abstract = {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 \lbrace t=0\rbrace $. 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 $L\,u=0$ 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.},
affiliation = {MAB 3 51 cours de la Libération Talence 33405, FRANCE; MAB 351 cours de la Libération Talence 33405, FRANCE; Department of Mathematics University of Michigan 525 East University Ann Arbor MI 48109, USA},
author = {Joly, Jean-Luc, Métivier, Guy, Rauch, Jeffrey},
journal = {Séminaire Équations aux dérivées partielles},
language = {eng},
pages = {1-9},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Sharp Domains of Determinacy and Hamilton-Jacobi Equations},
url = {http://eudml.org/doc/11117},
volume = {2004-2005},
year = {2004-2005},
}

TY - JOUR
AU - Joly, Jean-Luc
AU - Métivier, Guy
AU - Rauch, Jeffrey
TI - Sharp Domains of Determinacy and Hamilton-Jacobi Equations
JO - Séminaire Équations aux dérivées partielles
PY - 2004-2005
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2004-2005
SP - 1
EP - 9
AB - 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 \lbrace t=0\rbrace $. 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 $L\,u=0$ 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.
LA - eng
UR - http://eudml.org/doc/11117
ER -