论文标题
在平滑界面上的不均匀艾伦-CAHN方程的边界界面聚类
Clustering of Boundary Interfaces for an inhomogeneous Allen-Cahn equation on a smooth bounded domain
论文作者
论文摘要
我们考虑不均匀的allen-cahn方程$$ε^2Δu\，+\，v（y）（1-u^2）\，u \，= \，0 \ quad \ quad \ mbox {in} \Ω \ mbox {on} \ \partialΩ，$ $ $ω$是$ {\ Mathbb r}^2 $中的一个有界域，带有光滑边界$ \partialΩ$和$ v（x）$是一个积极的光滑函数，$ε> 0 $是一个小参数，$ n是一个小参数，$ν$代表$ n eft $ $ $ $ $ $ \ p part $ \ p pot a poptial $。对于任何固定的整数$ n \ geq 2 $，我们将展示群集解决方案的存在$u_ε$，$ n $ - 过渡层接近$ \ partial topial goart gart距离$ o（ε| \lnε|）$，前提是普遍的平均值curvature $ \ natial $ \ sutial us $ \ s $ \ set $ \ sut $ \ set $ \ reptial $ \ rectial $发生共鸣。我们的结果是A. Malchiodi，W.-M。的延伸（具有尺寸二）。 Ni，J。Wei在Pacific J. Math中。 （第229卷，2007年，第2、447-468号）和A. Malchiodi，J。Wei，J。IxidointPoint理论应用。 （第1卷，2007年，第2、305-336号）
We consider the inhomogeneous Allen-Cahn equation $$ ε^2Δu\,+\,V(y)(1-u^2)\,u\,=\,0\quad \mbox{in}\ Ω, \qquad \frac {\partial u}{\partial ν}\,=\,0\quad \mbox{on}\ \partial Ω, $$ where $Ω$ is a bounded domain in ${\mathbb R}^2$ with smooth boundary $\partialΩ$ and $V(x)$ is a positive smooth function, $ε>0$ is a small parameter, $ν$ denotes the unit outward normal of $\partialΩ$. For any fixed integer $N\geq 2$, we will show the existence of a clustered solution $u_ε$ with $N$-transition layers near $\partial Ω$ with mutual distance $O(ε|\ln ε|)$, provided that the generalized mean curvature $\mathcal{H} $ of $\partialΩ$ is positive and $ε$ stays away from a discrete set of values at which resonance occurs. Our result is an extension of those (with dimension two) by A. Malchiodi, W.-M. Ni, J. Wei in Pacific J. Math. (Vol. 229, 2007, no. 2, 447-468) and A. Malchiodi, J. Wei in J. Fixed Point Theory Appl. (Vol. 1, 2007, no. 2, 305-336)
