The energy of fractional Allen--Cahn layers in dimension one

Technical Summary: The Energy of Fractional Allen–Cahn Layers in Dimension One

Problem Statement
This paper investigates the energy $E(s)$ of the unique odd, monotone increasing layer solution $\Phi_s$ to the one-dimensional fractional Allen–Cahn equation:
$$(-\Delta)^s \Phi_s = \Phi_s - \Phi_s^3, \quad \Phi_s(\pm\infty) = \pm 1, \quad \Phi_s(0) = 0,$$
for the fractional exponent $s \in (1/2, 1]$. The energy functional is defined as:
$$E(s) := E_s[\Phi_s] = \frac{c_{1,s}}{4} \iint_{\mathbb{R}\times\mathbb{R}} \frac{(\Phi_s(x) - \Phi_s(y))^2}{|x-y|^{1+2s}} \, dx dy + \int_{\mathbb{R}} W(\Phi_s(x)) \, dx,$$
where $W(t) = \frac{1}{4}(1-t^2)^2$. While the existence, uniqueness, and decay properties of $\Phi_s$ were established by Cabr'e and Sire, and the minimality by Palatucci, Savin, and Valdinoci, sharp quantitative information regarding the dependence of the energy $E(s)$ on the parameter $s$ was absent from the literature beyond the explicit value at $s=1$.

Methodology
The authors employ a hybrid analytical and computer-assisted strategy to characterize $E(s)$ across the interval $(1/2, 1]$. The proof of strict monotonicity is divided into three regimes based on the cover $(1/2, 1] = (1/2, 53/100] \cup [53/100, 49/50] \cup [49/50, 1]$.

1. Analytical Asymptotics at Endpoints:

   • Near $s=1$: The authors perform a perturbation analysis around the explicit classical solution $\Phi_1(x) = \tanh(x/\sqrt{2})$. They construct a first-order corrector $\psi$ by inverting the linearized operator $J = -\partial_x^2 + (3\Phi_1^2 - 1)$ on the odd subspace. Using a fixed-point argument in $H^{2s}$, they prove the true layer $\Phi_s$ is close to the approximate profile $\Phi_1 + (1-s)\psi$. This allows for a rigorous expansion of the energy and its derivative, yielding explicit constants for the linear term.
   • Near $s=1/2$: The analysis relies on a Pohozaev-type identity derived from the minimality of $\Phi_s$, which relates the kinetic and potential energy parts. By using an arctan profile as a competitor, they establish an upper bound. A Fourier-side analysis of the Gagliardo seminorm isolates the singularity at low frequencies, proving the energy diverges as $(s-1/2)^{-1}$ with a specific residue.

2. Computer-Assisted Proof on the Interior:

   • On the interior interval $I_0 = [0.530, 0.980]$, where no closed-form solution exists, the authors construct numerical approximate layers $\Phi_s^{\text{ap}}$.
   • They utilize a fixed-point argument to rigorously bound the distance between the true layer $\Phi_s$ and the numerical approximation in the $H^{2s}$ norm.
   • To prove strict decrease ($E'(s) < 0$), they evaluate the derivative functional $J(s, \Phi_s)$ at grid points. Since the true layer is unknown, they compute $J(s, \Phi_s^{\text{ap}})$ and transfer the sign to the true layer using Lipschitz estimates derived from the distance bound.
   • Semiconcavity: A crucial component is establishing a profile-free upper bound on the second derivative of the energy with respect to $s$ (semiconcavity). This allows the strict negativity of the derivative at discrete grid points to propagate to the entire interval, provided the grid spacing is sufficiently small relative to the curvature bound.
   • Validation: All numerical bounds, including Schur inequalities for operator coercivity and the evaluation of functionals, are certified using interval arithmetic (Arb library) at 200-bit precision.

Key Contributions and Results

1. Asymptotic Expansions (Theorem 1.1):

   • At $s=1$: The energy admits the expansion $E(s) = \frac{2\sqrt{2}}{3} + \kappa_1(1-s) + o(1-s)$, where $\kappa_1$ is given by an explicit integral:
     $$\kappa_1 = -2\pi \int_0^\infty \frac{\xi^2 \log \xi}{\sinh^2(\pi\xi/\sqrt{2})} \, d\xi \approx 1.073.$$
   • At $s=1/2$: The energy exhibits a simple pole:
     $$E(s) = \frac{1/\pi}{s - 1/2} + O(1).$$
     The authors explicitly rule out logarithmic intermediate terms in the expansion.

2. Strict Monotonicity and Continuity (Theorem 1.2):

   • The map $s \mapsto E(s)$ is continuous and strictly decreasing on $(1/2, 1]$.
   • Consequently, $E$ extends to a homeomorphism from $[1/2, 1]$ onto $[2\sqrt{2}/3, +\infty)$.

3. Technical Innovations:

   • Derivation of an explicit identity for the derivative $E'(s)$ as a spectral integral involving the Fourier transform of the layer's derivative.
   • Development of a rigorous framework combining asymptotic analysis with computer-assisted proofs to handle nonlocal operators where explicit solutions are unavailable.
   • Establishment of a profile-independent semiconcavity bound for the energy, which is essential for bridging the gap between discrete numerical checks and continuous monotonicity.

Significance
The paper provides the first sharp quantitative description of the energy of fractional Allen–Cahn layers in one dimension. It resolves the behavior of the energy cost of a single phase transition as the order of nonlocality varies. The strict monotonicity result is significant because standard variational arguments fail to determine the sign of the derivative due to competing effects in the Fourier multiplier $|\xi|^{2s-2}$. The work demonstrates that combining rigorous asymptotic expansions with computer-assisted interval arithmetic can yield definitive results for nonlocal PDE problems where traditional analytical methods are insufficient. The authors note that the strict decrease is the most delicate result of the paper, relying on a delicate balance of spectral estimates and numerical verification.