  An explicit output-feedback boundary feedback law is introduced that
stabilizes an unstable linear constant-coefficient reaction-diffusion equation
on an $n$-ball (which in 2-D reduces to a disk and in 3-D reduces to a sphere)
using only measurements from the boundary. The backstepping method is used to
design both the control law and a boundary observer. To apply backstepping the
system is reduced to an infinite sequence of 1-D systems using spherical
harmonics. Well-posedness and stability are proved in the $H^1$ space. The
resulting control and output injection gain kernels are the product of the
backstepping kernel used in control of one-dimensional reaction-diffusion
equations and a function closely related to the Poisson kernel in the $n$-ball.
