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A quadratic Bolza-type problem in a non-complete Riemannian manifold

Pages: 173 - 181, Issue Special, July 2003

 Abstract        Full Text (197.4K)              

Anna Maria Candela - Dipartimento Di Matematica, Universita' degli Studi di Bari "Aldo Moro", via E. Orabona 4, 70125 Bari, Italy (email)
J.L. Flores - Departamento de Álgebra, Geometría y Topología, Facultdad de Ciencias, Universidad de Granada, Avenida Fuentenueva s/n, 18071 Granada, Spain (email)
M. Sánchez - Departamento de Álgebra, Geometría y Topología, Facultdad de Ciencias, Universidad de Granada, Avenida Fuentenueva s/n, 18071 Granada, Spain (email)

Abstract: Let the nonlinear equation $D_s(dotx) + \lambda \nabla_x V (x, s) = 0$ be defined in a non–complete Riemannian manifold $M$ and consider those ones of its solutions which join any couple of fixed points in $M$ in a fixed arrival time $T > 0$. If $V$ has a quadratic growth with respect to $x$ and if $M$ has a convex boundary, then a "best constant" $bar(\lambda)(T)>$ 0 exists such that if $0 \<= \lambda \< bar(\lambda)(T)$ the problem admits at least one solution while infinitely many ones exist if the topology of $M$ is not trivial.

Keywords:  Bolza¨Ctype problem, quadratic growth, convex boundary, variational approach, penalization arguments, Ljusternik¨CSchnirelman Theory.
Mathematics Subject Classification:  70H03, 58E05, 49J40.

Received: September 2002;      Revised: March 2003;      Published: April 2003.