  The electrostatic potential of any test charge distribution in Schwarzschild
space with boundary values is derived. We calculate the Green's function,
generalize the second Green's identity for p-forms and find the general
solution. Boundary value problems are solved. With a multipole expansion the
asymptotic property for the field of any charge distribution is derived. It is
shown that one produces a Reissner--Nordstrom black hole if one lowers a test
charge distribution slowly toward the horizon. The symmetry of the distribution
is not important. All the multipole moments fade away except the monopole. A
calculation of the gravitationally induced electrostatic self-force on a
pointlike test charge distribution held stationary outside the black hole is
presented.
