  In this paper we study 1-equivariant wave maps of finite energy from
1+3-dimensional Minkowski space exterior to the unit ball at the origin into
the 3-sphere. We impose a Dirichlet boundary condition at r=1, meaning that the
unit sphere in R^3 gets mapped to the north pole. Finite energy implies that
spacial infinity gets mapped to either the north or south pole. In particular,
each such equivariant wave map has a well-defined topological degree which is
an integer. We establish relaxation of such a map of arbitrary energy and
degree to the unique stationary harmonic map in its degree class. This settles
a recent conjecture of Bizon, Chmaj, Maliborski who observed this asymptotic
behavior numerically.
