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

Abstract

We study the energy $\mathcal{E}(s) := E_{s}[\Phi_s]$ of the one-dimensional fractional Allen--Cahn layer solution $\Phi_s$, defined as the unique odd, increasing solution of $(-\Delta)^{s} \Phi_s = \Phi_s - \Phi_s^{3}$ with $\Phi_s(\pm\infty)=\pm 1$ and $\Phi_s(0)=0$, for $s\in(1/2,1]$. Our main results are a sharp qualitative and quantitative description of the energy $\mathcal{E}$ on this interval. We show that the energy is continuous and strictly decreasing, and we obtain explicit asymptotic expansions at both endpoints. At the upper endpoint we prove $\mathcal{E}(s) = \frac{2\sqrt{2}}{3} + \kappa_1 (1-s) + o(1-s)$ with an explicit formula for $\kappa_1$. At the lower endpoint we prove that the energy goes to infinity as $\mathcal{E}(s)=\frac{1}{\pi(s-1/2)}+O(1)$. The strict decrease is proved with computer assistance. On an interior subinterval it is reduced to finitely many inequalities verified by interval-arithmetic computations. The proofs combine the Cabr\'e--Sire construction of the layer, the minimality theorem of Palatucci--Savin--Valdinoci, an identity for the $s$ derivative of the energy, and a computer-assisted coercivity estimate at explicit approximate layers.