  This paper extends the Carleman estimates to high dimensional parabolic
equations with highly degenerate symmetric coefficients on a bounded domain of
Lipschitz boundary and use these estimates to study the controlla?bility the
corresponding equations. Due to the nonsmoothness and degeneracy of boundary,
the partial integration by parts in Carleman estimates have no meaning on the
degenerate and nonsmooth parts of the boundary. To get around of this
difficulty, we construct special weight function, and transform some integral
terms in degenerate regions into a non-degenerate ones carefully so that the
obtained Carleman estimates can still be used to the controllability problem.
Our results includes some well-known works as some special cases as well as
some interesting new examples.
