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 {displaystyle I} . It is named for Élie Cartan and Erich Kähler.

Contents 1 Meaning 2 Statement 3 Proof and assumptions 4 References 5 External links Meaning It is not true that merely having {displaystyle dI} contained in {displaystyle 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.

Statement Let {displaystyle (M,I)} be a real analytic EDS. Assume that {displaystyle Psubseteq M} is a connected, {displaystyle k} -dimensional, real analytic, regular integral manifold of {displaystyle I} with {displaystyle r(P)geq 0} (i.e., the tangent spaces {displaystyle T_{p}P} are "extendable" to higher dimensional integral elements).

Moreover, assume there is a real analytic submanifold {displaystyle Rsubseteq M} of codimension {displaystyle r(P)} containing {displaystyle P} and such that {displaystyle T_{p}Rcap H(T_{p}P)} has dimension {displaystyle k+1} for all {displaystyle pin P} .

Then there exists a (locally) unique connected, {displaystyle (k+1)} -dimensional, real analytic integral manifold {displaystyle Xsubseteq M} of {displaystyle 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, vol. 4, (1977) Chapt. XVIII.13 R. Bryant, S. S. Chern, R. Gardner, H. Goldschmidt, P. Griffiths, Exterior Differential Systems, Springer Verlag, New York, 1991. External links Alekseevskii, D.V. (2001) [1994], "Pfaffian problem", Encyclopedia of Mathematics, 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

Si quieres conocer otros artículos parecidos a Cartan–Kähler theorem puedes visitar la categoría Partial differential equations.

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