  We establish new, optimal gradient continuity estimates for solutions to a
class of 2nd order partial differential equations, $\mathscr{L}(X, \nabla u,
D^2 u) = f$, whose diffusion properties (ellipticity) degenerate along the
\textit{a priori} unknown singular set of an existing solution, $\mathscr{S}(u)
:= \{X : \nabla u(X) = 0 \}$. The innovative feature of our main result
concerns its optimality -- the sharp, encoded smoothness aftereffects of the
operator. Such a quantitative information usually plays a decisive role in the
analysis of a number of analytic and geometric problems. Our result is new even
for the classical equation $|\nabla u | \cdot \Delta u = 1$. We further apply
these new estimates in the study of some well known problems in the theory of
elliptic PDEs.
