Isolation of scalar Allen-Cahn local minimizers

We show that scalar local minimizers of the Allen-Cahn functional with the double-well potential and homogeneous Neumann boundary conditions on a bounded Lipschitz domain are isolated in the $L^1$-topology. This affirmatively answers a question raised by Kohn and Sternberg (Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989)). The proof implements an analytic Lyapunov-Schmidt reduction inspired by the theory of monotone dynamical systems in works of Jiang-Yu and Hirsch-Smith in an elliptic setting.