Let (M,g) be a (complete) Riemannian surface, and let Ω⊂M be an open subset whose closure is homeomorphic to a disk. We prove that if ∂Ω is smooth and it satisfies a strong concavity assumption, then there are at least two distinct orthogonal geodesics in Ω⋃∂Ω. Using the results given in Giambò et al. (Adv Differ Eq 10:931–960, 2005), we then obtain a proof of the existence of two distinct brake orbits for a class of Hamiltonian systems. 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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