  Grazing set singularity leads to a surprising counter-example and breakdown
of the classical mathematical theory for $L^{\infty}$ diffusive expansion of
neutron transport equation with in-flow boundary condition in term of the
Knudsen number $\varepsilon$, one of the most classical problems in the kinetic
theory. Even though a satisfactory new theory has been established by
constructing new boundary layers with favorable $\varepsilon$-geometric
correction for convex domains, the severe grazing singularity from non-convex
domains has prevented any positive mathematical progress. We develop a novel
and optimal $L^2$ expansion theory for general domain (including non-convex
domain) by discovering a surprising $\varepsilon^{\frac{1}{2}}$ gain for the
average of remainder.
