# Sharp Domains of Determinacy and Hamilton-Jacobi Equations

Jean-Luc Joly^{[1]}; Guy Métivier^{[2]}; Jeffrey Rauch^{[3]}

- [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 (2004-2005)

- Volume: 2004-2005, page 1-9

## Access Full Article

top## Abstract

top## How to cite

topJoly, 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 -

## References

top- M.D. Bronshtein, Smoothness of polynomials depending on parameters, Siberian Mat. Zh. 20(1979)493-501 (Russian). English transl. Siberian Math. J. 20(1980)347-352. Zbl0429.30007MR537355
- R. Courant, Methods of Mathematical Physics vol II, Interscience Publ., 1962. Zbl0099.29504
- S. Fomel and J. Sethian, Fast phase space computation of multiple arrivals, Proc. Nat. Acad. Sci. 99(2002), 7329-7334. Zbl1002.65113MR1907838
- K. O. Friedrichs and P. D. Lax, Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci. USA 68(1971), 1686-1688. Zbl0229.35061MR285799
- L. Gårding, Linear hyperbolic partial differential operators with constant coefficients, Acta Math. 85(1951), 1-62. Zbl0045.20202MR41336
- S. K. Godunov , An interesting class of quasilinear systems. Dokl. Akad. Nauk SSSR 139 (1972), p. 521. Zbl0125.06002MR131653
- S. K. Godunov, Symmetric form of the equations of magnetohydrodynamics. Numerical Methods for Mechanics of Continuum Medium 1 (1972), p. 26.
- A. Haar, Über eindeutigkeit und analytizität der lösungen partieller differenzialgleichungen, Atti del Congr. Intern. dei Mat., Bologna 3(1928), 5-10.
- L. Hörmander, Uniqueness theorems and estimates for normally hyperbolic partial differential equations of the second order, Tolfte Skandinaviska Matematikerkongressen, Lunds Universitets Matematiska Institution, Lund 1954, pp 105-115. Zbl0057.32501MR65783
- L. Hörmander, The Analysis of Linear Partial Differential Operators vol. I (§9.6), Grundlehren der mathematischen Wissenschaften #256, Springer-Verlag 1983. Zbl0521.35002
- L. Hörmander, The Analysis of Linear Partial Differential Operators vol. II, Grundlehren der mathematischen Wissenschaften #257, Springer-Verlag 1983. Zbl0521.35002
- J.-L. Joly, G. Métivier, and J.Rauch, Hyperbolic Domains of Determination and Hamilton- Jacobi Theory, preprint, (http://www.math.lsa.umich.edu/~rauch). MR1069957
- F. John, On linear partial differential equations with analytic coefficients, Unique continuation of data, C.P.A.M. 2(1949), 209-253. Zbl0035.34601MR36930
- P. Lax, Lectures on Hyperbolic Partial Differential Equations, Stanford, 1963.
- P. Lax, Shock waves amd entropy, in Contributions to Nonlinear Functional Analysis ed. E. Zarantonello, Academic Press NY 1971, 603-634. Zbl0268.35014MR393870
- J. Leray, Hyperbolic Differential Equations, Institute for Advanced Study, 1953. MR80849
- P.-L. Lions, Generalized Solutions of Hamilton-Jacobi Equations Pittman Lecture Notes, 1982. Zbl0497.35001MR667669
- A. Marchaud, Sur les champs continus de demi cônes convexes et leurs intégrales, Compositio Math. 2(1936)89-127. Zbl62.0803.02MR1556934
- C. B. Morrey Jr., Multiple Integrals in the Calculus of Variations, Springer-Verlag, 1966. Zbl0142.38701MR202511
- J. Rauch, Partial Differential Equations, Graduate Texts in Mathematics #128, Springer-Verlag, 1991. Zbl0742.35001MR1223093
- S. Ruuth, B. Merriman, and S. Osher, A fixed grid method for capturing the motion of self-intersecting wavefronts and related PDEs, J.Comp.Phys. 163(2000), 1-21. Zbl0963.65097MR1777719
- J. Steinfhoff, M. Fan, and L. Wang, A new Eulerian method for the computation of propagating short acoustic and electromagnetic pulses, J. Comp.Phys. 157(2000), 683-706. Zbl1043.78556MR1739115
- S. Wakabayashi, Remarks on hyperbolic polynomials, Tsukuba J. Math. 10(1986), 17-28. Zbl0612.35005MR846411

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.