Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Asymptotic Behavior of Allen-Cahn Type Energies and Neumann Eigenvalues
  via Inner Variations   We use the notion of first and second inner variations as a bridge allowing
one to pass to the limit of first and second Gateaux variations for the
Allen-Cahn, Cahn-Hilliard and Ohta-Kawasaki energies. Under suitable
assumptions, this allows us to show that stability passes to the sharp
interface limit, including boundary terms, by considering non-compactly
supported velocity and acceleration fields in our variations. This complements
the results of Tonegawa, and Tonegawa and Wickramasekera, where interior
stability is shown to pass to the limit. As a further application, we prove an
asymptotic upper bound on the $k^{th}$ Neumann eigenvalue of the linearization
of the Allen-Cahn operator, relating it to the $k^{th}$ Robin eigenvalue of the
Jacobi operator, taken with respect to the minimal surface arising as the
asymptotic location of the zero set of the Allen-Cahn critical points. We also
prove analogous results for eigenvalues of the linearized operators arising in
the Cahn-Hilliard and Ohta-Kawasaki settings. These complement the earlier
result of the first author where such an asymptotic upper bound is achieved for
Dirichlet eigenvalues for the linearized Allen-Cahn operator. Our asymptotic
upper bound on Allen-Cahn Neumann eigenvalues extends, in one direction, the
asymptotic equivalence of these eigenvalues established in the work of
Kowalczyk in the two-dimensional case where the minimal surface is a line
segment and specific Allen-Cahn critical points are suitably constructed.

9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
24
The categories of this paper are: dg-ga
17
The categories of this paper are: q-fin
26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an
18
The categories of this paper are: econ
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
22
The categories of this paper are: chao-dyn