Dynamical horizons in excised black hole evolutions

Starting from the 3+1 description of a dynamical horizon, i.e. a world-tube of marginally trapped surfaces of increasing area, we discuss a set of inner boundary conditions to be imposed on a black hole space-time in generic evolution. These boundary conditions are particularly useful when employing an excision technique in a constrained evolution scheme for the numerical resolution of Einstein equations, namely with the excised surface coinciding with the black hole (apparent) horizon. Special attention is devoted to the relation between the foliation uniqueness theorems of a dynamical horizon, and the optimal choice of the lapse function on the horizon.