偏微分方程解的临界点集的几何分布及测度估计的研究是偏微分方程领域的重要研究内容，也是当前偏微分方程领域关注的热点和难点问题之一。它们既是解的重要几何特性，又与解的渐近性和增长性等相关，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和抛物方程解的临界点集的几何分布及Hausdorff测度估计。应用偏微分方程理论、几何分析和几何测度论的思想方法，(1)在多连通区域上研究一类具有Dirichlet边值条件的拟线性椭圆方程解的临界点集的几何分布；(2)在外区域上分别研究具有三类边值条件的调和方程解的临界点集的几何结构；(3)在n维光滑紧黎曼流形上研究Laplace特征方程解的临界点集的Hausdorff测度估计；(4)研究抛物方程初边值问题解的空间临界点的Klamkin猜想相关问题。开展本课题的研究，可进一步丰富偏微分方程、几何分析和几何测度论的理论。

The study of geometric distribution and measure estimation of the critical set is an important research content for the solutions of partial differential equations, and it is also one of the hot and difficult problems in the field of partial differential equations. They are not only important geometric properties of solutions, but also related to the asymptotic behavior and growth of solutions. They are one of the important tools to study some profound properties of solutions of partial differential equations. In this subject, we mainly investigate the geometric distribution and Hausdorff measure estimation of the critical set of solutions to elliptic and parabolic equations. Based on the method of partial differential equations, geometric analysis and geometric measure theory, (1) we study the geometric distribution of the critical set of solutions to a quasilinear elliptic equations with Dirichlet boundary conditions in a multiply connected domain; (2) we investigate the geometric structure of the critical set of solutions for harmonic equation with three kinds of boundary conditions in an exterior domain respectively; (3) we consider the Hausdorff measure estimation of the critical set of eigenfunctions to the Laplacian on the n-dimensional smooth Riemannian manifold; (4) and we also discuss related problem of the Klamkin conjecture of the spatial critical point of the solution to the initial boundary value problem of parabolic equation. Therefore the research on this subject can further enrich the theory of partial differential equation, geometry analysis and geometric measure theory.