广义平均曲率方程中的非线性分析研究

Mean curvature problems come from differential geometry and physics. They are widely used in the study of problems related with surface. They attract the attention of scholars at home and abroad and become hot issues of international research now. The project intends to study prescribed mean curvature equations systematically by using the theory and the method of nonlinear analysis. We will develop the Time-map analysis method, explore the relations between the concavity-convexity of nonlinearities and the concavity-convexity of Time-map, and investigate the nonexistence of radial solutions for prescribed mean curvature equations with concave-convex nonlinearities. Meanwhile, we will discuss the exact number of solutions for the one dimensional prescribed mean curvature equations with concave-convex nonlinearities. We will study the existence of multiple solutions for prescribed mean curvature equations by constructing new invariant sets of descending flow and combining with the method of upper and lower solutions, and investigate the existence of periodic solutions for functional differential equations with one dimensional mean curvature operator by giving an extension of Mawhin's continuation theorem. The research work of this project can not only enrich the theories and methods of nonlinear analysis, but also provide new theory to solve the application problems related to the mean curvature. Therefore, the research has important theoretical value and broad application prospects.

平均曲率问题来源于微分几何与物理学，在研究与曲面相关的问题方面有着广泛的应用，受到国内外学者的关注，是国际上的研究热点问题。本项目拟采用非线性分析的理论方法对广义平均曲率方程进行系统的研究：发展Time-map分析法，探讨广义平均曲率方程中非线性项的凹凸性与Time-map凹凸性的关系，并用来研究带凹凸非线性项的广义平均曲率方程径向解的非存在性问题及带凹凸非线性项的一维广义平均曲率方程解的确切个数问题；构造新的下降流不变集，并与上下解方法结合，对广义平均曲率方程多解的存在性问题进行研究；推广Mawhin连续性定理，并用其对含一维平均曲率算子的泛函微分方程周期解的存在性问题进行研究。对本项目的研究不仅可以丰富非线性分析的理论和方法，还可以为解决与平均曲率相关的应用问题提供新理论。因此,该研究具有重要的理论价值和广阔的应用前景。

本项目通过发展time-map 分析法，研究了两类含平均曲率算子的非线性方程，通过解的分岔图和time-map的分析性质给出问题恰好存在一个解、恰好存在两个解及恰好不存在解的充分条件；通过发展拓扑度方法，得到了几类线性和拟线性算子方程参数的确切存在区间以及正解对参数的依赖关系等方面的结果。这些研究成果既丰富了非线性分析的理论和方法，又为解决与平均曲率算子等拟线性相关的应用问题提供了理论支持。