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 noncompactly 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 kth Neumann eigenvalue of the linearization of the Allen-Cahn operator, relating it to the kth 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.