发表机构
Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究环形域中两种群长程分离的椭圆系统，利用解的唯一性和旋转不变性证明自由边界为同心球面，并唯一确定，同时分析半径随参数变化及收敛性，推导出半径的常微分方程。
AI 中文摘要
我们考虑一个依赖于小参数 $\epsilon > 0$ 的椭圆方程组，该方程组模拟种群的长程分离。该系统先前在文献 \cite{CL2} 中针对二维自由边界的正则性进行了研究，并在 \cite{ChPaTo26_2} 和 \cite{ChPaTo26_3} 中针对高维自由边界的部分正则性进行了研究。在本文中，我们考虑任意维度环形域中 $K = 2$ 个种群的特殊情形。利用解的唯一性和旋转不变性，我们证明自由边界由同心球面组成。此外，通过应用 \cite{CL2} 中导出的自由边界条件，我们证明自由边界是唯一确定的。我们还考察了自由边界半径如何根据域、相互作用距离和边界数据而变化。特别地，我们证明当相互作用距离趋于零时，自由边界收敛到相邻模型的自由边界。我们还研究了椭圆系统的单参数族，其中域、相互作用距离和边界数据依赖于 $t$。我们分别在二维和维度 $n \geq 3$ 的情况下推导了自由边界半径的常微分方程。文中给出并讨论了若干例子。
英文摘要
We consider a system of elliptic equations, depending on a small parameter $ε> 0$, which models the long range segregation of populations. The system has been previously studied in \cite{CL2} for the regularity of the free boundary in dimension $2$ and in \cite{ChPaTo26_2} and \cite{ChPaTo26_3} for the partial regularity of the free boundary in higher dimensions. In this paper, we consider the special case with $K = 2$ populations in an annular domain in arbitrary dimensions. Using the uniqueness and the rotational invariance of the solution, we show that the free boundary consist of concentric spheres. Moreover, by an application of the free boundary condition derived in \cite{CL2}, we show that the free boundary is uniquely determined. We also examine how the radius of the free boundary change according to the domain, the interaction distance and the boundary data. In particular, we show that the free boundary converges to that of the adjacent model as the interaction distance tends to zero. We also study the one parameter family of the elliptic system, in which the domain, the interaction distance, and the boundary data depends on $t$. We derive an ODE for the radius of the free boundary in dimension $2$ and dimension $n \geq 3$ separately. Several examples are given and discussed.
Comments23 pages
