1. 研究目的与意义
在描述生物竞争的Lotka-volterra模型以及与多组分Bose-Einstein 凝聚态相关的Gross-Pitaevskii方程组中，都涉及到一类奇异扰动非线性偏微分方程组. 在竞争参数趋于正无穷的奇异极限问题中, 解的支集相互分离, 这样就产生了一个自由边界问题. 许多数学工作者, 例如Caffarelli, 林芳华, Dancer, 和S. Terracini 等对此类问题作了大量的研究, 对它们的研究涉及偏微分方程、几何测度论、泛函分析等多个数学分支.本文拟研究一类强耦合椭圆系统解的渐近行为，拟将经典扩散问题的部分结果推广到交错扩散模型。
2. 研究内容和预期目标
课题拟研究一类具强竞争作用的强耦合椭圆方程组解的空间结构。竞争参数有限时系统解会有一定程度的混合，当竞争参数趋于无穷时，系统解的支集相互分离，产生一个自由边界问题。课题拟在这一结果的基础上继续研究自由边界的性质。拟解决如下问题：
1.自由边界的维数估计；
2.自由边界点的分类，奇异部分的部分正则性。
3. 国内外研究现状
在描述生物竞争的Lotka-volterra模型以及与多组分Bose-Einstein 凝聚态相关的Gross-Pitaevskii方程组中，都涉及到一类奇异扰动非线性偏微分方程组. 这类方程组其解的空间结构——包括共存或消亡，一直都是微分方程界研究的热点问题。特别是最近几年，人们对强相互作用导致的解的相分离现象表现出极大的兴趣，这是由于它们都涉及到一类奇异扰动的椭圆型或者抛物型方程组。在参数有限时方程组的解有一定程度的混合，但在参数趋于正无穷的奇异极限问题中，解的支集相互分离，从而产生一个自由边界问题。这引起了偏微分方程领域国内外学者极大的关注，美国德克萨斯大学的L.A.Caffarelli，Courrant数学所的林芳华，澳大利亚悉尼大学的E.N.Dancer，意大利米兰可比卡大学的S.Terracini，中国科学院的张志涛，扬州大学的刘祖汉等在相关方面取得了重要结果([11-23])。
Caffarelli, Karakhanyan和Lin [11，12]，Conti,Terracini 和 Verzini [13，14],Dancer, Wang和Zhang[16]等证明了第一类情形下（Lotka-Voltera型生物竞争模型）系统解关于参数的一致Holder界估计，以及趋向正无穷时的极限方程解的正则性和自由边界的正则性；Dancer,Wang和Zhang [16] 证明了第二类情形下（描述Bose-Einstein凝聚态的Gross-Pitaevskii方程组）系统解的一致Holder界估计；Caffarelli和Lin [12], Tavares和Terracini [23] 证明了自由边界的正则性；Dancer, Wang和Zhang[18]证明了两类模型的极限方程有着相同的结构（这是Terracini等的一个猜想）； Liu[19-21] 研究了第二类情形下系统复值解的空间分离行为，并证明了在某种特殊情形下，系统解将会发生熄灭（extinction）现象。
4. 计划与进度安排
2022年11月20日至30日 完成毕业论文文献搜索以及开题报告.
2022年02月26日至2022年03月019日 进入实习单位实习并系统学习泛函分析、偏微分方程基础课程,掌握基础知识及相关理念。
2022年03月19日至25日 系统阅读一些相关文献,了解国内外的研究现状.开始撰写论文初稿,并同时撰写指导周记以及完成中期检查表的书写.
5. 参考文献
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