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

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2003, 2003(Special): 173-181. doi: 10.3934/proc.2003.2003.173

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

Anna Maria Candela ^1, , J.L. Flores ^2, and M. Sánchez ^2,

1.	Dipartimento Di Matematica, Universita' degli Studi di Bari "Aldo Moro", via E. Orabona 4, 70125 Bari, Italy
2.	Departamento de Álgebra, Geometría y Topología, Facultdad de Ciencias, Universidad de Granada, Avenida Fuentenueva s/n, 18071 Granada, Spain, Spain

Received September 2002 Revised March 2003 Published April 2003

Abstract
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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: penalization arguments, quadratic growth, BolzaÂ¨Ctype problem, variational approach, convex boundary, LjusternikÂ¨CSchnirelman Theory..

Mathematics Subject Classification: 70H03, 58E05, 49J4.

Citation: Anna Maria Candela, J.L. Flores, M. Sánchez. A quadratic Bolza-type problem in a non-complete Riemannian manifold. Conference Publications, 2003, 2003 (Special) : 173-181. doi: 10.3934/proc.2003.2003.173

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