  In this paper, we study the relation between the second inner variations of
the Allen-Cahn functional and its Gamma-limit, the area functional. Our result
implies that the Allen-Cahn functional only approximates well the area
functional up to the first order. However, as an application of our result, we
prove, assuming the single-multiplicity property of the limiting energy, that
the Morse indices of critical points of the Allen-Cahn functional are bounded
from below by the Morse index of the limiting minimal hypersurface.
