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.
