Systems of nonlinear elliptic equations and free boundary problems on manifolds

非线性椭圆方程组和流形上的自由边界问题

• 批准号: 1027628
• 负责人: Lei Zhang
• 金额: $ 7.41万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1027628&HistoricalAwards=false
• 关键词: Systems; nonlinear; elliptic; equations; free

This project pursues research in three major directions. The first set of problems is about free boundary problems (FBP) on Riemannian manifolds. One major task is to prove regularity results for the free boundary or the solution of the FBP. For this purpose it is important to establish some monotonicity formulas to describe the asymptotic behavior of the solutions near the free boundary. For FBP in Euclidean spaces Alt-Caffarelli-Friedman and Caffarelli-Jerison-Kenig have established some celebrated monotonicity formulas, which play a central role in the regularity theory. As their first goal, the principal investigator and E. Teixeira seek to establish the analogues of these formulas for the Laplace-Beltrami operator on Riemannian manifolds. The second set of problems is related to finding a metric on four-manifolds with constant Q-curvature. This geometric problem can be translated to an existence problem for a certain fourth-order partial differential equation. The principal investigator and M. Ould Ahemedou seek to solve this existence problem completely by using arguments of Lin-Wei, Weinstein-Zhang, Bahri-Coron, and others to handle various major difficulties presented by this equation. The third set of problems concerns the blow-up solutions for certain systems of two-dimensional elliptic equations, namely, the Liouville and Toda systems. In comparison with scalar Liouville-type equations, the blow-up phenomenon for these systems is very poorly understood. The principal investigator and C.S. Lin seek to develop the necessary tools for obtaining a thorough understanding of blow-up for these two systems.In the first set of problems, the so-called monotonicity formulas should provide a major new tool for the study of regularity theory for free boundary problems defined on Riemannian manifolds, the more general and meaningful context for such problems from the viewpoint of applications. Moreover, these new formulas will provide motivation for people to extend to the Riemannian setting many other important results that are known currently only in the realm of Euclidean spaces. The second set of problems reviews strong interplay between analysis and geometry. On one hand, these problems exhibit some major analytical difficulties, the overcoming of which will require new ideas and methods. On the other hand, the deep and rich geometric meaning of these problems is a great source of inspiration for people to try to understand and surmount such analytical difficulties. Thus, solving the second set of problems will not only provide new tools for investigating other partial differential equations with similar complications, but also lead to a better understanding of many related open problems in geometry. The problems in part three are rooted in various fields of physics, chemistry, and ecology, as some important models in these fields are described by the Liouville- and Toda-like systems. Solving the challenging mathematical questions related to these systems will likely impact the aforementioned fields and expose the deep connections between them.

该项目在三个主要方向进行研究。第一组问题是关于黎曼流形上的自由边界问题。一个主要的任务是证明自由边界或FBP的解决方案的正则性结果。为此，建立一些单调性公式来描述解在自由边界附近的渐近行为是很重要的。Alt-Caffarelli-Friedman和Caffarelli-Jerison-Kenig在Euclidean空间中对FBP建立了一些著名的单调性公式，这些公式在正则性理论中起着核心作用。作为他们的第一个目标，主要研究者和E。特谢拉试图建立类似的这些公式的拉普拉斯-贝尔特拉米算子黎曼流形。第二组问题是关于寻找一个度量上的四个流形与常数Q曲率。这个几何问题可以转化为某个四阶偏微分方程的存在性问题。主要研究者和M. Ould Ahemedou试图通过使用Lin-Wei，Weinstein-Zhang，Bahri-Coron等人的论点来解决这个方程的存在性问题。第三组问题涉及某些二维椭圆方程组的爆破解，即刘维和户田系统。与标量Liouville型方程相比，对这些系统的爆破现象知之甚少。主要研究者和C.S.在第一组问题中，所谓的单调性公式应该为研究定义在黎曼流形上的自由边界问题的正则性理论提供一个重要的新工具，从应用的角度来看，这类问题的背景更一般和有意义。此外，这些新的公式将提供动力，人们将许多其他重要的结果，目前已知的只有在欧氏空间领域的黎曼设置。第二组问题回顾了分析和几何之间的相互作用。一方面，这些问题表现出一些重大的分析困难，克服这些困难需要新的思路和方法。另一方面，这些问题所蕴含的深刻而丰富的几何意义，也是人们试图理解和克服这些分析困难的巨大灵感来源。因此，解决第二组问题将不仅为研究其他具有类似复杂性的偏微分方程提供新的工具，而且还导致更好地理解许多相关的几何开放问题。第三部分中的问题根植于物理学、化学和生态学的各个领域，因为这些领域中的一些重要模型由Liouville和Toda类系统描述。解决与这些系统相关的具有挑战性的数学问题可能会影响上述领域，并揭示它们之间的深层联系。