Systems of nonlinear elliptic equations and free boundary problems on manifolds

本项目主要从三个方面进行研究。第一组问题是关于黎曼流形上的自由边界问题。一个主要任务是证明自由边界或FBP解的正则性结果。为此，建立一些单调公式来描述解在自由边界附近的渐近行为是很重要的。对于欧氏空间中的FBP，Alt-Caffarelli-Friedman和Caffarelli-Jerison-Kenig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，首席研究员和E.Teixeira试图为黎曼流形上的Laplace-Beltrami算子建立这些公式的类似。第二组问题是关于寻找具有常q-曲率的四-流形上的度量。这个几何问题可以转化为一个四阶偏微分方程解的存在性问题。主要研究人员和M.Ould Ahemedou试图通过使用林伟、Weinstein-Zhang、Burri-Coron等人的论点来处理这个方程所带来的各种主要困难，从而彻底解决这个存在性问题。第三组问题涉及某些二维椭圆型方程组，即Liouville组和Toda组的爆破解。与标量Liouvile型方程相比，对这些系统的爆破现象了解得很少。在第一组问题中，所谓的单调性公式应该为定义在黎曼流形上的自由边界问题的正则性理论的研究提供一个重要的新工具，从应用的角度来看，这类问题具有更一般和更有意义的背景。此外，这些新公式将为人们提供动力来推广到黎曼背景下的许多其他重要结果，这些结果目前仅在欧几里德空间领域中已知。第二组问题回顾了分析和几何之间的强烈相互作用。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，这些问题深刻而丰富的几何意义是人们试图理解和克服这种分析困难的巨大灵感来源。因此，解决第二组问题不仅将为研究其他具有类似复杂性的偏微分方程提供新的工具，而且还将有助于更好地理解几何中的许多相关开放问题。第三部分中的问题根植于物理、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是由类Liouville和Toda系统描述的。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到上述领域，并暴露出它们之间的深刻联系。

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.