Cartan–Kähler theorem

Cartan–Kähler theorem In mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals {estilo de exibição I} . It is named for Élie Cartan and Erich Kähler.

Conteúdo 1 Meaning 2 Declaração 3 Proof and assumptions 4 Referências 5 External links Meaning It is not true that merely having {displaystyle dI} contained in {estilo de exibição I} is sufficient for integrability. There is a problem caused by singular solutions. The theorem computes certain constants that must satisfy an inequality in order that there be a solution.

Letra de declaração {estilo de exibição (M,EU)} be a real analytic EDS. Assuma isso {displaystyle Psubseteq M} is a connected, {estilo de exibição k} -dimensional, real analytic, regular integral manifold of {estilo de exibição I} com {estilo de exibição r(P)geq 0} (ou seja, the tangent spaces {estilo de exibição T_{p}P} são "extendable" to higher dimensional integral elements).

Além disso, assume there is a real analytic submanifold {displaystyle Rsubseteq M} of codimension {estilo de exibição r(P)} contendo {estilo de exibição P} e tal que {estilo de exibição T_{p}Rcap H(T_{p}P)} has dimension {displaystyle k+1} para todos {displaystyle pin P} .

Then there exists a (localmente) unique connected, {estilo de exibição (k+1)} -dimensional, real analytic integral manifold {displaystyle Xsubseteq M} do {estilo de exibição I} that satisfies {displaystyle Psubseteq Xsubseteq R} .

Proof and assumptions The Cauchy-Kovalevskaya theorem is used in the proof, so the analyticity is necessary.

References Jean Dieudonné, Eléments d'analyse, volume. 4, (1977) Chapt. XVIII.13 R. Bryant, S. S. Chern, R. Gardner, H. Goldschmidt, P. Griffiths, Exterior Differential Systems, Editora Springer, Nova york, 1991. External links Alekseevskii, D.V. (2001) [1994], "Pfaffian problem", Enciclopédia de Matemática, EMS Press R. Bryant, "Nine Lectures on Exterior Differential Systems", 1999 E. Cartan, "On the integration of systems of total differential equations," transl. by D. H. Delphenich E. Kähler, "Introduction to the theory of systems of differential equations," transl. by D. H. Delphenich Categories: Partial differential equationsTheorems in analysis