On the second inner variation of the Allen-Cahn Functional and its applications

2010/09/28 by Nam Q. Le, Le, Nam Q. · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1009.5649

to appear in Indiana Univ. Math. J

arxiv created 2010/09/28 · arxiv updated 2010/09/29

Abstract

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.

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