The wire ballast resistor (BR) is one of the simplest physical systems that exhibit bistability and pattern formation. An annular BR is suggested as a simple two-dimensional extension of the wire BR. The nonuniformity of the electric current density in the annular BR leads to translational symmetry breaking in the temperature domain dynamics. As a result, the steady-state position of the domain wall is “pinned” and the system exhibits coarsening. The two-phase steady-state relaxation towards it and coarsening in the annular BR are investigated analytically and numerically.