The helically-reduced wave equation as a symmetric-positive system

Journal of Mathematical Physics

Volumn 44, Issue 12, 2003, Pages 6223-6232

  Torre, C G ^a  

^a UTAH STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0344287482     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1623930     Document Type: Article

References (18)

1. R. Courant and D. Hubert, Methods of Mathematical Physics, Vol. 2 (Interscience, New York, 1962).
2. M. Taylor, Partial Differential Equations (Springer, New York, 1996).
3. For a survey of results on differential equations of mixed type, see, e.g., M. Smirnov, Equations of Mixed Type (American Mathematical Society, Providence, RI, 1978); J. Stewart, Class. Quantum Grav. 18, 4983 (2001).
4. For a survey of results on differential equations of mixed type, see, e.g., M. Smirnov, Equations of Mixed Type (American Mathematical Society, Providence, RI, 1978); J. Stewart, Class. Quantum Grav. 18, 4983 (2001).
5. J. Blackburn and S. Detweiler, Phys. Rev. D 46, 2318 (1992).
6. J. Whelan, C. Beetle, W. Landry, and R. Price, Class. Quantum Grav. 19, 1285 (2002).
7. J. Friedman, K. Uryu, and M. Shibata, Phys. Rev. D 6, 064035 (2002).
8. J. Whelan, W. Krivan, and R. Price, Class. Quantum Grav. 17, 4895 (2000).
9. It should be noted that one of the goals in Refs. 7 and 5 is to replace Sommerfeld conditions with "minimum energy radiation balanced" boundary conditions, which are nonlocal. It would appear that, generally speaking, nothing is known about the use of such boundary conditions with symmetric-positive equations.
10. K. O. Friedrichs, Commun. Pure Appl. Math. 11, 333 (1958).
11. P. Lax and R. Phillips, Commun. Pure Appl. Math. 13, 427 (1960).
12. P. Olver, Applications of Lie Groups to Differential Equations (Springer-Verlag, New York, 1993).
13. I. Anderson, M. Fels, and C. Torre, Commun. Math. Phys. 212, 653 (2000).
14. The restriction to ε>0 is needed to cast the field equations (2.11) into symmetric positive form.
15. More general formulations are possible. For example, symmetric-positive equations can be defined for sections of a vector bundle. See Ref. 9.
16. a into M.
17. Recall that the equations (4.15) are symmetric-positive provided |a| is sufficiently large.
18. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol I (Academic, New York, 1980).