所汇报的论文中主要证明了当系数满足一定条件时, 区域上二阶线性齐次椭圆方程的解的结点集在有限阶零点附近的 n-1 维Hausdorff测度是有界的, 并给出了一个具体的上界. 此外, 还对该结点集做了进一步的研究, 将其分解为两个具有不同类型的良好性质的子集的不交并. 为了证明主要定理, 文中给出了一定条件下多项式零点集测度的一个上界, 并对调和多项式做了进一步的估计. 在完成证明之后将所得主要定理应用到了几何分析的研究中, 给出了带有 C^(1,1 )度量的紧致黎曼流形上拉普拉斯算子的属于非零特征值的特征函数结点集的一个上界.

In this article for reporting, it is proved that on a domain, when coefficients satisfy certain conditions, the (n-1)-dimensional Hausdorff measure of the nodal set of a solution of a linear homogeneous elliptic equation of second order is finite in a neighborhood of any point of the domain at which the solution has finite order of vanishing and a specific upper bound is given. Besides, the nodal set is decomposed into a union of two disjoint subsets with good properties of different kinds. In order to prove the main theorem, an upper bound of Hausdorff measure of the nodal set of a polynomial is obtained and further research is done on harmonic polynomial in this article. After completing the proof for the main result, the principal theorem is applied to geometric analysis. An upper bound of the nodal set of any eigenfunction corresponding to any nonzero eigenvalue of Laplacian on a compact Riemannian manifold with C^(1,1 )metric is given.