A quadratic Bolza-type problem in a Riemannian manifold

Abstract

A classical nonlinear equation D s x ̇ + ∇ xV(x,s)=0 on a complete Riemannian manifold M is considered. The existence of solutions connecting any two points x 0,x 1∈ M is studied, i.e., for T>0 the critical points of the functional J T(x)= 1 2 ∫ 0 T 〈 x ̇ , x ̇ 〉 ds− ∫ 0 T V(x,s)ds with x(0)= x 0, x( T)= x 1. When the potential V has a subquadratic growth with respect to x, J T admits a minimum critical point for any T>0 (infinitely many critical points if the topology of M is not trivial). When V has an at most quadratic growth, i.e., V(x,s)⩽λd 2(x, x ̄ )+k , this property does not hold, but an optimal arrival time T( λ)>0 exists such that, if 0< T< T( λ), any pair of points in M can be joined by a critical point of the corresponding functional. For the existence and multiplicity results, variational methods and Ljusternik–Schnirelman theory are used. The optimal value T(λ)=π/ 2λ is fulfilled by the harmonic oscillator. These ideas work for other related problems.