Singular limits of nonlinear elliptic and parabolic PDE

双组分竞争种群系统的强竞争极限被证明是获取竞争参数较大时竞争系统信息的一种有效方法。一般来说，由于缺乏变分结构，理解这种系统的动力学是非常困难的。另一方面，强相互作用极限是一个带有Lyapunov函数的标量方程，更容易研究。这样的奇异极限在数学上是有用的，并且与空间分离和相分离的重要生物和物理现象相对应。然而，对于两个以上组成部分的竞争系统，在两个组成部分的情况下有效的技术失败了，需要新的想法。最近，利用自由边界问题的工具，已经取得了一些成功，证明了竞争系统的解收敛于一个极限问题的解。本提案的中心目标是证明多组分系统的所谓逆问题的结果；也就是说，取决于极限问题的解是否作为原系统解的极限出现。这种反向问题是理解任何单一极限的必要和重要部分，对于多组分系统的强竞争极限还有待解决。我们还旨在证明当相互作用项仅在子域上较大时两个组件竞争系统的逆向问题的结果，以及具有不同的三次竞争耦合类型的系统，这发生在某些量子力学系统的建模中。

Strong-competition limits of competing-population systems with two componentshave proved an effective way of obtaining information about the competition system for large values of the competition parameter. Understanding the dynamics of such systems is very difficult in general, due to their lack of variational structure. On the other hand, the strong-interaction limit is a scalar equation which has a Lyapunov function and is much more easily studied. Such singular limits are both mathematically useful, and correspond to the important biological and physical phenomena of spatial segregation and phase separation.For competition systems with more than two components, however, the techniquesthat are effective in the two-component case fail and new ideas are needed. Some success has been had been recently, using tools from free-boundary problems, to show convergence of solutions of the competition system to a solution of a limitproblem. The central goal of this proposal is to prove results on the so-called converse problem for multi-component systems; that is, on whether or not a solution of the limit problem does in fact arise as the limit of solutions of the original system. Such converse problemsare an essential and important part of the understanding of any singular limit, and have yet to be addressed at all for strong-competition limits of multi-component systems.We also aim to prove results on the converse problem for competition systems oftwo components when the interaction terms are large only on a subdomain, and forsystems with a different, cubic type of competitive coupling, which occurs in modelling certain quantum mechanical systems.