Singular limits of nonlinear elliptic and parabolic PDE

强竞争极限被证明是一种有效的方法，可以获得竞争参数较大的竞争系统的信息。由于缺乏变分结构，理解这类系统的动力学通常非常困难。另一方面，强相互作用极限是一个标量方程，它具有李雅普诺夫函数，更容易研究。这种奇异极限在数学上是有用的，并且与空间分离和相分离等重要的生物和物理现象相对应。然而，对于含有两个以上组分的竞争体系，在两组分情况下有效的技术失效，需要新的思路。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极限问题的解。这个建议的中心目标是证明多组分系统的所谓匡威问题的结果;也就是说，极限问题的解是否真的出现在原系统的解的极限上。这样的匡威问题是理解任何奇异极限的基本和重要的部分，并且对于多组分系统的强竞争极限还没有得到解决。我们还旨在证明当相互作用项仅在子域上大时两组分竞争系统的匡威问题的结果，以及对于具有不同立方型竞争耦合的系统，这发生在模拟某些量子力学系统时。

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.