Previous |  Up |  Next


prolongation; partial differential equations; filtered manifolds; contact manifolds; weighted jet bundles
The aim of this article is to show that systems of linear partial differential equations on filtered manifolds, which are of weighted finite type, can be canonically rewritten as first order systems of a certain type. This leads immediately to obstructions to the existence of solutions. Moreover, we will deduce that the solution space of such equations is always finite dimensional.
[1] Beals, R., Greiner, P. C.: Calculus on Heisenberg manifolds. Ann. of Math. Stud. 119 (1988), x+194 pp. MR 0953082 | Zbl 0654.58033
[2] Goldschmidt, H.: Existence theorems for analytic linear partial differential equations. Ann. Math. 86 (1967), 246–270. DOI 10.2307/1970689 | MR 0219859 | Zbl 0154.35103
[3] Goldschmidt, H.: Prolongations of linear partial differential equations: A conjecture of Élie Cartan. Ann. Sci. École Norm. Sup. (4) 1 (1968), 417–444. MR 0235584
[4] Morimoto, T.: HASH(0x1cef7a8).
[5] Morimoto, T.: Théorème de Cartan-Kähler dans une classe de fonctions formelles Gevrey. C. R. Acad. Sci. Paris Sér. A Math. 311 (1990), 443–436. MR 1075665 | Zbl 0714.58060
[6] Morimoto, T.: Théorème d’existence de solutions analytiques pour des systèmes d’équations aux dérivées partielles non-linéaires avec singularités. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 1491–1496. MR 1366107 | Zbl 0842.22009
[7] Morimoto, T.: Lie algebras, geometric structures and differential equations on filtered manifolds. In “Lie Groups Geometric Structures and Differential Equations - One Hundred Years after Sophus Lie”, Adv. Stud. Pure Math., Math. Soc. of Japan, Tokyo, 2002, pp. 205–252. MR 1980903 | Zbl 1048.58015
[8] Spencer, D. C.: Overdetermined systems of linear partial differential equations. Bull. Amer. Math. Soc. 75 (1969), 179–239. DOI 10.1090/S0002-9904-1969-12129-4 | MR 0242200 | Zbl 0185.33801
[9] Taylor, M. E.: Noncommutative microlocal analysis I. Mem. Amer. Math. Soc. 52 (313) (1984), iv+182 pp. MR 0764508 | Zbl 0554.35025
[10] van Erp, E., : The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part 1. To appear in Ann. of Math. preprint arXiv: 0804.2490.
Partner of
EuDML logo