Abstract

Consider a compact manifold with boundary, homeomorphic to the N-dimensional disk, and a Tonelli Lagrangian function defined on the tangent bundle. In this paper, we study the multiplicity problem for Euler-Lagrange orbits that satisfy the conormal boundary conditions and that lay on the boundary only in their extreme points. In particular, for suitable values of the energy function and under mild hypotheses, if the Tonelli Lagrangian is reversible then the minimal number of Euler-Lagrange orbits with prescribed energy that satisfies the conormal boundary conditions is N. If L is not reversible, then this number is two.

Highlights

  • Let Ω be a compact and connected N-manifold of class C3 with boundary ∂Ω ∈ C2, homeomorphic to an N-dimensional disk D N ⊂ R N

  • When the Lagrangian is the energy function of a Riemannian or Finsler metric, a solution of the Euler-Lagrange equations is a geodesic and the conormal boundary conditions are nothing but the orthogonality condition of the geodesic with the boundary

  • We reduce our study on a fixed-time problem, so we can avoid to take care of the possible sources of non-compactness of the time variable in the free-time Lagrangian action functional

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Summary

Introduction

Let Ω be a compact and connected N-manifold of class C3 with boundary ∂Ω ∈ C2 , homeomorphic to an N-dimensional disk D N ⊂ R N. When the Lagrangian is the energy function of a Riemannian or Finsler metric, a solution of the Euler-Lagrange equations is a geodesic and the conormal boundary conditions are nothing but the orthogonality condition of the geodesic with the boundary. The Riemannian and Finsler geodesic chords on a manifold with boundary are strictly related with the brake orbits in a potential well for a Hamiltonian system of classical type, namely when the hamiltonian function is fiberwise even and convex (cf [1]). Seifert conjectured in [2] that there are at least N brake orbits in an N-dimensional potential well of a natural Hamiltonian system, where the brake-orbits correspond to the geodesics of a Riemannian metric This conjecture has been recently proved in [3], exploiting some partial results achieved by the same authors in different previous works (cf [4,5,6,7,8,9]), while a preliminary result for the Finsler case is presented in [10].

Framework Setup and Notation
Sobolev and Functional Spaces
The Free-Time Action Functional
The Energy Critical Values
The Functional Fκ
Proof of the Main Theorem
Conclusions
Methods

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