On upper semicontinuity of the Allen–Cahn twisted eigenvalues

Article type: Research Article

Authors: Tawri, Krutika^{; *}

Affiliations: Department of Mathematics, Indiana University, Bloomington, 831 E 3rd St, Bloomington, IN 47405, USA. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We give an asymptotic upper bound for the kth twisted eigenvalue of the linearized Allen–Cahn operator in terms of the kth eigenvalue of the Jacobi operator, taken with respect to the minimal surface arising as the asymptotic limit of the zero sets of the Allen–Cahn critical points. We use an argument based on the notion of second inner variation developed in Le (On the second inner variations of Allen–Cahn type energies and applications to local minimizers. J. Math. Pures Appl. (9) 103 (2015) 1317–1345).

Keywords: Allen-Cahn functional, local minimizer, twisted eigenvalues, inner variations

DOI: 10.3233/ASY-211753

Journal: Asymptotic Analysis, vol. 130, no. 3-4, pp. 323-334, 2022