Extension to arbitrary smooth double-well potentials

Determine whether the finiteness and isolation conclusions of Theorem 1.3 for scalar stable critical points and local minimizers of the Allen–Cahn functional remain valid when the analytic double-well potential W(u)=(u^2-1)^2/2 is replaced by an arbitrary smooth function.

Background

The paper proves finiteness of stable critical points and isolation of scalar local minimizers for the Allen–Cahn functional with the analytic quartic double-well potential W(u)=(u^2-1)^2/2. A central ingredient is the identity theorem for real analytic functions of one variable, which forces the reduced scalar function arising from the Lyapunov–Schmidt reduction either to have isolated zeros or to vanish identically along an analytic curve.

The authors explicitly note that the identity-theorem argument has no counterpart for arbitrary smooth functions. Consequently, it remains unresolved whether the finiteness and isolation results persist when analyticity is replaced by mere smoothness.