  To make a Born-Infeld (BI) black hole thermally stable, we consider two types
of boundary conditions, i.e., the asymptotically anti-de Sitter (AdS) space and
a Dirichlet wall placed in the asymptotically flat space. The phase structures
and transitions of these two types of BI black holes, namely BI-AdS black holes
and BI black holes in a cavity, are investigated in a grand canonical ensemble,
where the temperature and the potential are fixed. For BI-AdS black holes, the
globally stable phases can be the thermal AdS space. For small values of the
potential, there is a Hawking-Page-like first order phase transition between
the BI-AdS black holes and the thermal-AdS space. However, the phase transition
becomes zeroth order when the values of the potential are large enough. For BI
black holes in a cavity, the globally stable phases can be a naked singularity
or an extremal black hole with the horizon merging with the wall, which both
are on the boundaries of the physical parameter region. The thermal flat space
is never globally preferred. Besides a first order phase transition, there is a
second order phase transition between the globally stable phases. Thus, it
shows that the phase structures and transitions of BI black holes with these
two different boundary conditions have several dissimilarities.
