  This is in the sequel of authors' paper \cite{LPW} in which we had set up a
program to verify rigorously some formal statements associated with the
multiple component phase transitions with higher dimensional wells. The main
goal here is to establish a regularity theory for minimizing maps with a rather
non-standard boundary condition at the sharp interface of the transition. We
also present a proof, under simplified geometric assumptions, of existence of
local smooth gradient flows under such constraints on interfaces which are in
the motion by the mean-curvature. In a forthcoming paper, a general theory for
such gradient flows and its relation to Keller-Rubinstein-Sternberg's work
\cite{KRS1, KRS2} on the fast reaction, slow diffusion and motion by the mean
curvature would be addressed.
