AI 中文总结
研究二维艾伦 - 卡恩系统有界整体极小化解，通过特殊代数结构产生的校准恒等式，得到其完整分类，明确无穷远处有三叉结构时解的显式轮廓，并证明\(D_3\)等变类中极小化解与该轮廓一致。
AI 中文摘要
我们研究使艾伦 - 卡恩泛函\(J(u,\Omega)=\int_\Omega \left(\frac12 |u|^2+W(u)\mathrm{d}\mathbf{x}\right)\)【其中\(W(u_1,u_2)=|u|^4+2u_1u_2^2-\frac23 u_1^3-|u|^2+\frac23\)】极小化的有界整体解\(u:\mathbb{R}^2\to \mathbb{R}^2\)。我们得到了整体极小化解的完整分类。特别地，当\(u\)在无穷远处具有三叉结构时，经过平移和坐标正交变换，\(u\)具有显式轮廓\(u_*(\mathbf{x})=\sum_{i=1}^3 \frac{e^{\sqrt2 a_i\cdot \mathbf{x}}}{\sum_{j=1}^3 e^{\sqrt2 a_j\cdot \mathbf{x}}}a_i\)。我们还证明了在\(D_3\)等变类中通过极小化得到的解与\(u_*\)一致。关键要素是由\(W\)的特殊代数结构产生的校准恒等式。
英文摘要
We study bounded entire solutions $u:\mathbb{R}^2\to \mathbb{R}^2$ that minimize the Allen-Cahn functional \begin{equation*} J(u,Ω)=\int_Ω\left(\frac12 |\nabla u|^2+W(u)\right)\,d\mathbf{x}, \end{equation*} with the $D_3$-invariant triple-well potential \begin{equation*} W(u_1,u_2)=|u|^4+2u_1u_2^2-\frac23 u_1^3-|u|^2+\frac23. \end{equation*} We obtain a complete classification of entire minimizing solutions. In particular, when $u$ has a triple-junction structure at infinity, up to translation and orthogonal change of coordinates, $u$ has the explicit profile \begin{equation*} u_*(\mathbf{x})=\sum_{i=1}^3 \frac{e^{\sqrt2 a_i\cdot \mathbf{x}}}{\sum_{j=1}^3 e^{\sqrt2 a_j\cdot \mathbf{x}}}a_i. \end{equation*} We also demonstrate that the solutions obtained by minimizing within the $D_3$-equivariant class coincide with $u_*$. The key ingredient is a calibration identity arising from the special algebraic structure of $W$.
Comments18 pages
