Stability analysis in N-dimensional gravitational collapse with an equation of state

We study the stability of occurrence of black holes and naked singularities that arise as the final states of a complete gravitational collapse of type I matter field in a spherically symmetric N-dimensional spacetime, with the equation of state p = k rho, 0 a parts per thousand currency sign k a parts per thousand currency sign 1. We prove that for a regular initial data comprising pressure (or density) profiles at an initial surface t = ti, from which the collapse evolves, there exists a large class of velocity functions and classes of solutions of Einstein equations such that the spacetime evolution goes to a final state which is either a black hole or a naked singularity. We use suitable function spaces for regular initial data leading the collapse to a black hole or a naked singularity and show that these data form an open subset of the set of all regular initial data. In this sense, both outcomes of collapse are stable. These results are discussed and analyzed in the light of the cosmic censorship hypothesis in black hole physics.