Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the second inner variation of the Allen-Cahn Functional and its
  applications   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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The categories of this paper are: hep-ph
24
The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
7
The categories of this paper are: hep-th
21
The categories of this paper are: adap-org