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Periodic solutions on non-compact riemannian manifolds

Periodic solutions on non-compact riemannian manifolds Ann. Univ. Ferrara - Sez. VII - Sc. Mat. Vol. XXXVIII, 65-75 (1992) Periodic Solutions on Non-Compact Riemannian Manifolds(*). ELVIRA MIRENGHI (**) - MARIA TUCCI (***) 1. - Introduction. Let (g~, (.,.)R) be a Riemannian manifold and V: g~ � R-~ R a T-peri- odic potential function of class C 1 ; in this paper we look for T-periodic curves x: [0, T]--) :~ which are solutions of the problem (1.1) Dt (&(t)) = - VR V(x(t), t) Dt (&(t)) being the covariant derivative of &(t) with respect to the Riemanni- an structure and VR the Riemannian gradient. If g~ is compact the existence of T-periodic solutions of (1.1) has been studied in [1]. If :~ is not compact the existence of infinitely many solutions of (1.1) sat- isfying the boundary conditions x(0)= x0, x(1)= Xl has been stated in[8] when the topology of ~ is not trivial. Without compactness assumptions the research of the T-periodic sol- utions of (1.1) is not so easy because in general the action functional related to that problem does not satisfy the Palais-Smale condition. In this paper we will prove that under suitable conditions on V and ~, there exist infinitely many T-periodic http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ANNALI DELL UNIVERSITA DI FERRARA Springer Journals

Periodic solutions on non-compact riemannian manifolds

Mirenghi, Elvira; Tucci, Maria
ANNALI DELL UNIVERSITA DI FERRARA , Volume 38 (1): 11 – Dec 1, 1992
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Publisher
Springer Journals
Copyright
Copyright © Università degli Studi di Ferrara 1992
ISSN
0430-3202
eISSN
1827-1510
DOI
10.1007/bf02827084
Publisher site
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Abstract

Ann. Univ. Ferrara - Sez. VII - Sc. Mat. Vol. XXXVIII, 65-75 (1992) Periodic Solutions on Non-Compact Riemannian Manifolds(*). ELVIRA MIRENGHI (**) - MARIA TUCCI (***) 1. - Introduction. Let (g~, (.,.)R) be a Riemannian manifold and V: g~ � R-~ R a T-peri- odic potential function of class C 1 ; in this paper we look for T-periodic curves x: [0, T]--) :~ which are solutions of the problem (1.1) Dt (&(t)) = - VR V(x(t), t) Dt (&(t)) being the covariant derivative of &(t) with respect to the Riemanni- an structure and VR the Riemannian gradient. If g~ is compact the existence of T-periodic solutions of (1.1) has been studied in [1]. If :~ is not compact the existence of infinitely many solutions of (1.1) sat- isfying the boundary conditions x(0)= x0, x(1)= Xl has been stated in[8] when the topology of ~ is not trivial. Without compactness assumptions the research of the T-periodic sol- utions of (1.1) is not so easy because in general the action functional related to that problem does not satisfy the Palais-Smale condition. In this paper we will prove that under suitable conditions on V and ~, there exist infinitely many T-periodic

Journal

ANNALI DELL UNIVERSITA DI FERRARA – Springer Journals

Published: Dec 1, 1992

Keywords: Periodic Solution; Riemannian Manifold; Loop Space; Close Geodesic; Riemannian Structure

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