  The gravitational Dirichlet problem -- in which the induced metric is fixed
on boundaries at finite distance from the bulk -- is related to simple notions
of UV cutoffs in gauge/gravity duality and appears in discussions relating the
low-energy behavior of gravity to fluid dynamics. We study the Einstein-Maxwell
version of this problem, in which the induced Maxwell potential on the wall is
also fixed. For flat walls in otherwise-asymptotically-flat spacetimes, we
identify a moduli space of Majumdar-Papapetrou-like static solutions
parametrized by the location of an extreme black hole relative to the wall.
Such solutions may be described as balancing gravitational repulsion from a
negative-mass image-source against electrostatic attraction to an
oppositely-signed image charge. Standard techniques for handling divergences
yield a moduli space metric with an eigenvalue that becomes negative near the
wall, indicating a region of negative kinetic energy and suggesting that the
Hamiltonian may be unbounded below. One may also surround the black hole with
an additional (roughly spherical) Dirichlet wall to impose a regulator whose
physics is more clear. Negative kinetic energies remain, though new terms do
appear in the moduli-space metric. The regulator-dependence indicates that the
adiabatic approximation may be ill-defined for classical extreme black holes
with Dirichlet walls.
