一维空间中的分数阶Allen--Cahn层的能量
The energy of fractional Allen--Cahn layers in dimension one
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中文总结 AI 辅助
本文研究s∈(1/2,1]时一维分数阶Allen--Cahn层的能量，证明其连续且严格递减，得到两端点的显式渐近展开，严格递减性的证明结合理论与计算机辅助方法。
中文摘要 AI 辅助
我们研究s∈(1/2,1]时一维分数阶Allen--Cahn层解Φ_s的能量E(s):=E_s[Φ_s]，其中Φ_s是满足(-Δ)^s Φ_s = Φ_s - Φ_s³、Φ_s(±∞)=±1且Φ_s(0)=0的唯一奇函数、递增函数。主要结果是对该区间上的能量E给出精确的定性与定量描述：证明能量连续且严格递减，并得到两端点处的显式渐近展开；在上端点处，证得E(s)=2√2/3 + κ₁(1-s) + o(1-s)，其中κ₁有显式公式；在下端点处，证得能量随s趋近1/2时趋于无穷，即E(s)=1/[π(s-1/2)] + O(1)；严格递减性的证明借助计算机辅助，在内部子区间上简化为有限个不等式，通过区间算术计算验证；证明结合了Cabré--Sire的层构造、Palatucci--Savin--Valdinoci的极小性定理、能量的s导数恒等式以及显式近似层的计算机辅助 coercivity估计。
英文摘要
We study the energy $\mathcal{E}(s) := E_{s}[Φ_s]$ of the one-dimensional fractional Allen--Cahn layer solution $Φ_s$, defined as the unique odd, increasing solution of $(-Δ)^{s} Φ_s = Φ_s - Φ_s^{3}$ with $Φ_s(\pm\infty)=\pm 1$ and $Φ_s(0)=0$, 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 $\mathcal{E}(s) = \frac{2\sqrt{2}}{3} + κ_1 (1-s) + o(1-s)$ with an explicit formula for $κ_1$. At the lower endpoint we prove that the energy goes to infinity as $\mathcal{E}(s)=\frac{1}{π(s-1/2)}+O(1)$. 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 Cabré--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.
