论文标题
二维矢量Allen-Cahn系统的渐近学
Asymptotics for two-dimensional vectorial Allen-Cahn systems
论文作者
论文摘要
在标量案例中众所周知并确定了用于多孔梯度驱动问题的编码 - 一个接口的形成，在标量情况下，该方程通常称为allen-cahn方程。证明依赖于对能量密度的单调性公式的大范围，这本身与所谓的差异函数消失有关。相比之下，矢量案例非常开放。缺乏结果和洞察力是与缺乏已知合适的单调性公式有关的很大扩展。在本文中，我们将重点放在\ emph {椭圆形的二维情况下}，并依靠偏微分方程的分析引入方法，这使得能够避免缺乏能量密度的单调性公式。在本文的最后一部分中，我们恢复了一个\ emph {新的单调公式}，该公式}依赖于\ emph {新的差异关系}。这些工具允许在二维中扩展到矢量案例，大多数为标量案例获得的结果。我们还强调了矢量案例的一些\ emph {特定功能}。
The formation of codimension-one interfaces for multi-well gradient-driven problems is well-known and established in the scalar case, where the equation is often referred to as the Allen-Cahn equation. The proofs rely for a large on a monotonicity formula for the energy density, which is itself related to the vanishing of the so-called discrepancy function. The vectorial case in contrast is quite open. This lack of results and insight is to a large extend related to the absence of known appropriate monotonicity formula. In this paper, we focus on the \emph{elliptic case in two dimensions}, and introduce methods, relying on the analysis of the partial differential equation, which allow to circumvent the lack of monotonicity formula for the energy density. In the last part of the paper, we recover a \emph{new monotonicity formula} which relies on a \emph{new discrepancy relation}. These tools allow to extend to the vectorial case in two dimensions most of the results obtained for the scalar case. We emphasize also some \emph{specific features} of the vectorial case.
