Multiple brake orbits in $\mathbf m$-dimensional disks

Let $(M,g)$ be a (complete) Riemannian surface, and let $Ω\subset M$ be an open subset whose closure is homeomorphic to a disk. We prove that if $\partialΩ$ is smooth and it satisfies a strong concavity assumption, then there are at least two distinct orthogonal geodesics in $\overlineΩ=Ω\bigcup\partialΩ$. Using the results given in [6], 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 [7].

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