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

Let $(M,g)$ be a (complete) Riemannian surface, and let $\mathrm{\Omega }\subset M$ be an open subset whose closure is homeomorphic to a disk. We prove that if $\mathrm{\partial }\mathrm{\Omega }$ is smooth and it satisfies a strong concavity assumption, then there are at least two distinct orthogonal geodesics in $\overline\Omega=\Omega \bigcup\partial\Omega$. 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].