论文标题
在没有对称性假设的平面上的三连接问题上
On the triple junction problem on the plane without symmetry hypotheses
论文作者
论文摘要
我们调查了艾伦 - 卡恩系统 \ begin {equation*} Δu-w_u（u）= 0，\ quad u：\ mathbb {r}^2 \ rightarrow \ mathbb {r}^2， \ end {equation*} 其中$ w \ in c^2（\ mathbb {r}^2，[0，+\ infty））$具有三个全局最小值的潜力。我们建立了具有三重连接结构的整个解决方案$ u $的存在。主要策略是研究变异问题的全球最小化$ u_ \ varepsilon $ \ begin {equation*} \ min \ int_ {b_1} \ left（\ frac {\ varepsilon} {2} | \ nabla u |^2+\ frac {1} {\ varepsilon} w（u）\ right） \ end {equation*} 出发点是一个能量下限，在估计弥散界面的位置和大小中起着至关重要的作用。我们不对溶液施加任何对称性假设。
We investigate the Allen-Cahn system \begin{equation*} Δu-W_u(u)=0,\quad u:\mathbb{R}^2\rightarrow\mathbb{R}^2, \end{equation*} where $W\in C^2(\mathbb{R}^2,[0,+\infty))$ is a potential with three global minima. We establish the existence of an entire solution $u$ which possesses a triple junction structure. The main strategy is to study the global minimizer $u_\varepsilon$ of the variational problem \begin{equation*} \min\int_{B_1} \left( \frac{\varepsilon}{2}|\nabla u|^2+\frac{1}{\varepsilon}W(u) \right)\,dz,\ \ u=g_\varepsilon \text{ on }\partial B_1. \end{equation*} The point of departure is an energy lower bound that plays a crucial role in estimating the location and size of the diffuse interface. We do not impose any symmetry hypothesis on the solution.
