Multiple Brake Orbits and Homoclinics in Riemannian Manifolds
Tóm tắt
Let (M, g) be a complete Riemannian manifold,
$${\Omega\subset M}$$
an open subset whose closure is homeomorphic to an annulus. We prove that if ∂Ω is smooth and it satisfies a strong concavity assumption, then there are at least two distinct geodesics in
$${\overline\Omega=\Omega\cup\partial\Omega}$$
starting orthogonally to one connected component of ∂Ω and arriving orthogonally onto the other one. Using the results given in Giambò et al. (Adv Differ Equ 10:931–960, 2005), we then obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct brake orbits for a class of Hamiltonian systems. Under a further symmetry assumption, the result is improved by showing the existence of at least dim(M) pairs of geometrically distinct geodesics as above, brake orbits and homoclinic orbits. In our proof we shall use recent deformation results proved in Giambò et al. (Nonlinear Anal Ser A: Theory Methods Appl 73:290–337, 2010).
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