Global comparison of finite-dimensional reduction schemes in smooth variational problems.

*(English. Russian original)*Zbl 1036.49019
Math. Notes 67, No. 5, 631-638 (2000); translation from Mat. Zametki 67, No. 5, 745-754 (2000).

The usage of various finite-dimensional reduction schemes in problems of variational calculus makes it necessary to compare results obtained by different schemes and, in particular, to compare analytic and topological properties of the pair of functions corresponding to one another of the action functions in a specified pair of schemes. In variational boundary problems reduction schemes are related to the interpretation of the problem as a functional-operator equation \( f(x)=0\) where \(f\) is a potential operator of the zero index acting in a given Banach space. From this equation it is passed to the equivalent extremal problem about minimization of a smooth functional \(V(x)\) which is potential of \(f\) on a Banach space or a Banach manifold \(X\). This variational problem can fairly often be reduced to a similar minimization problem in a finite-dimensional space \(\mathbb{R}^n\)

In this paper a criterion for global smooth equivalence of a pair of key functions corresponding to a smooth functional in the calculus of variations for a given pair of finite-dimensional reduction schemes is established. Here the main condition is the possibility to deform the reduction schemes into each other preserving the coactivity of the key functions. The theorem proved concerns global smooth equivalence of the key functions calculated by the Lyapunov-Schmidt and Morse-Bott reduction schemes in the two-point boundary value problem for a mechanical system.

In this paper a criterion for global smooth equivalence of a pair of key functions corresponding to a smooth functional in the calculus of variations for a given pair of finite-dimensional reduction schemes is established. Here the main condition is the possibility to deform the reduction schemes into each other preserving the coactivity of the key functions. The theorem proved concerns global smooth equivalence of the key functions calculated by the Lyapunov-Schmidt and Morse-Bott reduction schemes in the two-point boundary value problem for a mechanical system.

Reviewer: Asaf D. Iskenderov (Baku)

##### MSC:

49J40 | Variational inequalities |

47J30 | Variational methods involving nonlinear operators |

58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |

49J53 | Set-valued and variational analysis |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

34C25 | Periodic solutions to ordinary differential equations |

37J45 | Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) |

##### Keywords:

variational problem; finite-dimensional reduction; Lyapunov-Schmidt and Morse-Bott reduction schemes; functional-operator equation; potential operator; Banach space; Banach manifold; key functions
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\textit{Yu. I. Sapronov} and \textit{S. L. Tsarev}, Math. Notes 67, No. 5, 631--638 (2000; Zbl 1036.49019); translation from Mat. Zametki 67, No. 5, 745--754 (2000)

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##### References:

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