一般区域上Minkowsky空间中平均曲率方程研究

The proposal will aim to study the mean curvature equations in Minkowsky space. It includes that: the existence and multiplicity of solutions of Minkowsky-curvature equations in the convex domain; the global bifurcation structure of radial solutions and the conditions to guarantee positive solution sets are of 1-dimension manifolds; the spectrum theorem of singular differential operators and the criteria for regular Sturm-Liouville problems to approximate the singular Sturm-Liouville problems; the approximate symmetry theorems for positive solutions of the finite difference problems associated to the mean curvature equations in a ball; the existence and multiplicity of radial decreasing ground.state solutions and the solutions of breather-shape; prove the Bonheuer-Noris-Weth conjecture and establish the similar results for mean curvature equations in Minkowsky space. The research is expected not only to solve mean curvature equations in Minkowsky space, but also to provide theoretical implications for computing the numerical solutions of mean curvature equations.

本项目拟研究Minkowsky空间中给定平均曲率方程的下列问题：凸形区域和一般有界区域上Minkowsky空间中给定平均曲率方程Dirichlet问题正解的存在性和多解性；正径向解的全局分歧结构及正解的连通分支是一维流形的判据；奇异微分算子的谱结构；奇异Sturm-Liouville问题的正则逼近；球形区域上Minkowsky空间中给定平均曲率方程Dirichlet问题的相应有限差分问题正解的逼近对称性；Minkowsky空间中给定平均曲率方程的径向递减基态解和多个呼吸子形式古典径向解的存在性和多解性；证明Bonheuer-Noris-Weth猜想并对Minkowsky空间中给定平均曲率方程建立类似于Bonheuer-Noris-Weth猜想的结果. 本研究不仅会促进平均曲率方程的理论发展，而且对平均曲率方程的数值求解具有重要的理论指导意义.

本项目拟研究Minkowski空间中给定平均曲率方程的下列问题：凸形区域和一般有界区域上Minkowski空间中给定平均曲率方程Dirichlet问题正解的存在性和多解性；正径向解的全局分歧结构及正解的连通分支是一维流形的判据；奇异微分算子的谱结构；奇异Sturm-Liouville问题的正则逼近；球形区域上Minkowski空间中给定平均曲率方程Dirichlet问题的相应有限差分问题正解的逼近对称性；Minkowski空间中给定平均曲率方程的径向递减基态解和多个呼吸子形式古典径向解的存在性和多解性；证明Bonheuer-Noris-Weth猜想并对Minkowski空间中给定平均曲率方程建立类似于Bonheuer-Noris-Weth猜想的结果. 本研究不仅会促进平均曲率方程的理论发展，而且对平均曲率方程的数值求解具有重要的理论指导意义.