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几何中的退化椭圆型方程

批准号11571019 学科分类几何分析 ( A010303 )
项目负责人韩青 负责人职称教授 依托单位北京大学
资助金额45.00
万元
项目类别面上项目 研究期限2016 年 01 月 01 日 至
2019 年 12 月 31 日
中文主题词退化椭圆型方程;Monge-Ampere 方程;极小曲面方程;极值 Kahler 度量;边界正则性
英文主题词degenerate elliptic equations;Monge-Ampere equations;minimal surface equations;extremal Kahler metrics;boundary regularity

摘要

中文摘要 几何分析在几何学的研究中起了重要的作用,它的兴起是在上世纪七十年代中后期和八十年代初期。在此期间,一系列与非线性椭圆型方程相关的重要几何问题得到了解决,这包括Yamabe问题,正质量猜想以及Calabi猜想。椭圆型方程在这些问题的解决中起了关键性的作用。随着(严格)椭圆型方程理论的完善,退化椭圆型方程已成为偏微分方程中重要的和活跃的课题之一,特别是与几何和物理有关的退化椭圆型方程更是引起了广泛的关注。由于退化性的多样化,到目前为止退化椭圆型方程并没有完整的理论,与之相关的一些重要问题也没有得到完全解决。本项目着重研究与退化椭圆型方程相关的几类重要几何问题。这些问题的共同特点如下:方程本身定义在紧带边流形上,方程在内部为严格椭圆, 退化只发生在边界。本项目试图通过对这几类退化椭圆型方程的研究,发现退化性对方程解的存在性和正则性的影响,从而建立与之相关的理论。
英文摘要 Geometric analysis plays an important role in the geometry. It has evolved into a field in mathematics by the late 1970s and early 1980s. During this period, a series of important geometric problems related to nonlinear elliptic equations were solved, including the Yamabe problem, the positive mass conjecture and the Calabi conjecture. Elliptic equations play a crucial role in the solutions of these problems. With the perfection of the (strict) elliptic equations, degenerate elliptic equations have become one of the most important and most active areas in differential equations, especially those equations related to the geometry and physics. Due to the diversity of degeneracy, there has not been a complete theory of degenerate elliptic equations and many important problems were not solved. This project studies several of these problems in geometry related to degenerate elliptic equations. These problems have the following same features: the equations are defined in compact manifolds with boundary; they are elliptic in the interior of the manifolds and the degeneracy occurs on the boundary. Through the study of these degenerate elliptic equations, it is attempted to find the impact of the degeneracy on the existence and the regularity of solutions and therefore to establish a relevant theory.
结题摘要

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