Asymptotic Behavior of Allen-Cahn Type Energies and Neumann Eigenvalues via Inner Variations

TL;DR

This paper investigates the asymptotic behavior of Allen-Cahn type energies and Neumann eigenvalues, establishing stability transfer to sharp interfaces and providing bounds relating eigenvalues of different operators.

Contribution

It introduces a method using inner variations to pass stability results to the sharp interface limit and derives asymptotic bounds for Neumann eigenvalues of the Allen-Cahn operator.

Findings

Stability passes to the sharp interface limit including boundary effects.

Provides asymptotic upper bounds for Neumann eigenvalues related to minimal surfaces.

Extends previous results to Cahn-Hilliard and Ohta-Kawasaki energies.

Abstract

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.