  We show that the Morse index of a properly embedded free boundary minimal
hypersurface in a strictly mean convex domain of the Euclidean space grows
linearly with the dimension of its first relative homology group (which is at
least as big as the number of its boundary components, minus one). In ambient
dimension three, this implies a lower bound for the index of a free boundary
minimal surface which is linear both with respect to the genus and the number
of boundary components. Thereby, the compactness theorem by Fraser and Li
implies a strong compactness theorem for the space of free boundary minimal
surfaces with uniformly bounded Morse index inside a convex domain. Our
estimates also imply that the examples constructed, in the unit ball, by
Fraser-Schoen and Folha-Pacard-Zolotareva have arbitrarily large index.
Extensions of our results to more general settings (including various classes
of positively curved Riemannian manifolds and other convexity assumptions) are
discussed.
