Γ-type estimates for the one-dimensional Allen–Cahn’s action

Article type: Research Article

Authors: Bellettini, Giovanni^a | Nayam, Al-Hassem^b | Novaga, Matteo^{c; *}

Affiliations: [a] Dipartimento di Matematica, Università di Roma Tor Vergata, Roma, Italy. E-mail: belletti@mat.uniroma2.it | [b] Mathematics Section, The Abdus Salam International Center of Theoretical Physics – ICTP, Trieste, Italy. E-mail: anayam@ictp.it | [c] Dipartimento di Matematica, Università di Pisa, Pisa, Italy. E-mail: novaga@dm.unipi.it

Correspondence: [*] Corresponding author: Matteo Novaga, Dipartimento di Matematica, Università di Pisa, largo Pontecorvo 5, 56127 Pisa, Italy. E-mail: novaga@dm.unipi.it.

Abstract: In this paper we prove an asymptotic estimate, up to the second-order included, on the behaviour of the one-dimensional Allen–Cahn’s action functionals, around a periodic function with bounded variation and taking values in {±1}. The leading term of this estimate justifies and confirms, from a variational point of view, the results of Fusco–Hale [Dyn. Diff. Equation 1 (1989), 75–94] and Carr–Pego [Comm. Pure Appl. Math. 42 (1989), 523–576] on the exponentially slow motion of metastable patterns coexisting at the transition temperature.

Keywords: singular perturbations, Γ-expansion, Allen–Cahn’s action

DOI: 10.3233/ASY-151308

Journal: Asymptotic Analysis, vol. 94, no. 1-2, pp. 161-185, 2015