椭圆方程解的临界点集的几何结构研究

椭圆方程解的临界点集几何结构的研究是偏微分方程领域的重要研究内容之一，也是当前偏微分方程和几何分析领域关注的热点和难点问题。零点集和临界点集反映了解的重要几何特性，又与解的增长性、渐近性等相关，是研究偏微分方程解的一些深刻性态的重要工具之一。本课题拟深入研究椭圆型方程解的临界点集的几何结构。借助偏微分方程、几何分析、变分学和几何测度论等的理论、思想与方法，(1)在平面环形区域上研究具有零Dirichlet边值条件的半线性椭圆方程解的临界约当曲线的存在性与不存在性问题；(2)在高维空间区域上研究具有零Dirichlet边值条件的常数非齐次Poisson方程解的临界点集的几何结构。本课题的研究将进一步丰富偏微分方程、变分学和几何分析等的相关理论。

The geometric structure of critical point set of solutions to elliptic equations is one of the important research contents in the field of partial differential equations. It is also the hot and difficult problem in the field of partial differential equation and geometric analysis. The zero point set and critical point set reflect the important geometric properties of solution. They are also related to the growth and asymptotic properties of the solution. They are one of the important tools for studying some profound properties of the solutions of PDEs. This topic intends to study the geometric structure of the critical point set of the solutions of elliptic equations. We will use the theories, ideas and methods of partial differential equations, geometric analysis, variational calculus and geometric measure theory. (1) We study the existence and nonexistence problem of critical Jordan curves for solutions of semilinear elliptic equations with zero Dirichlet boundary value conditions on a planar annular domain. (2) We study the geometric structure of critical point set of the solutions to the constant inhomogeneous Poisson equation with zero Dirichlet boundary value conditions on high-dimensional space domains. The research of this topic will further enrich the related theories of partial differential equations, variational calculus and geometric analysis.