分歧分析和三类平均曲率型方程解集的全局结构

自上世纪六十年代以来，算子方程的分歧理论受到众多数学家的关注，并取得了许多深刻的结果。但分歧点的存在性和分歧分支的全局结构问题还远没有解决。本项目的第一个目的是对一类完全非线性算子方程建立局部Dancer型的单边分歧定理。近年来，来源于广义相对论、黎曼几何和宇宙学的平均曲率型方程受到了广泛关注。我们的第二个目的是用分歧分析法研究三类平均曲率型方程空间型解集的全局结构。具体包括：FLRW时空中的平均曲率型方程、相对静态时空中的平均曲率型方程和欧式-洛伦兹型平均曲率方程。我们首先建立几类对应特征值问题的谱理论用以寻找分歧点。然后用分歧法研究这三类平均曲率型方程空间型解集的(单边)全局(区间)分歧理论及其全局结构，进而获得定号或变号解的梯度估计、正则性、对称性、存在性、非存在性、多解性及唯一性。我们的方法和结果很多都是原始创新，推广和改进了之前的相关结果。该项目既是基础性的研究也是多学科的交叉。

The study on bifurcation phenomena of operator equations has attracted many famous mathematician's attentions since 1960s. Many celebrated and profound results have been obtained on this topic. However, the existence of bifurcation points and the global structure of bifurcation branch are still far from being solved. The first aim of this project is to establish the local Dancer type unilateral bifurcation theorems for a class of fully nonlinear operator equations. In recent years, equations of mean curvature type have been widely concerned which is derived from the general relativity, Riemannian geometry and cosmology. Our second goal is to obtain the global structure of the set of spacelike solutions of three classes of mean curvature equations via bifurcation method. It includes the mean curvature equation in FLRW spacetime, the mean curvature equation in the relative static spacetime and the Euclidean-Lorenz type mean curvature equation. To find bifurcation points, we establish the spectral theory of several classes of corresponding eigenvalue problems. Then, we study the the (unilateral) global (interval) bifurcation theory and the global structure of the set of spacelike solutions of the above three types mean curvature equations. And obtain the gradient estimate, regularity, symmetry, existence, nonexistence, multiplicity and uniqueness of one-sign or sign-changing solutions. Many of our methods and results are original innovations, some of which generalize and improve the previous results. Our project is not only a basic research but also a cross disciplinary subject.