On the second inner variation of the Allen-Cahn functional and its applications Nam Le 49J4535B2549S05Allen-Cahn functionalsecond variationMorse indexstability 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. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4505 10.1512/iumj.2011.60.4505 en Indiana Univ. Math. J. 60 (2011) 1843 - 1856 state-of-the-art mathematics http://iumj.org/access/