欧氏空间中加倍测度的限制与延拓

This project mainly concerns with some hot issues in the doubling measure theory. A thorough study on these issues is of great theoretical importance to the understanding of the geometric measure theory and harmonic analysis as well. The following topics will be addressed in this project. The first is concerned with the restrictions and extensions of doubling measures on Jordan domains with piecewise smooth boundary. The second is concerned with the restrictions and extensions of doubling measures on the middle interval Cantor sets. The third is concerned with doubling properties of Moran measures on uniform Cantor sets and their extensions. The last topic is concerned with the restrictions and extensions of doubling measures on fat sets and thin sets for doubling measures. In the study of these issues we shall employ knowledge and techniques from various fields such as fractal geometry, real analysis, geometric measure theory, harmonic analysis，etc..

本项目主要研究加倍测度理论中的几个热点问题，这些问题的深入研究对于丰富和发展几何测度论以及调和分析都具有十分重要的理论意义。本项目拟研究如下几个问题：(1)具有分片光滑边界的Jordan域上的加倍测度限制与延拓；(2)中间区间Cantor集上加倍测度的限制与延拓；(3)均匀Cantor集上Moran测度的加倍性质的刻画及其延拓；（4）加倍测度肥集、瘦集上加倍测度的限制与延拓。项目成员将综合运用分形几何、实分析、几何测度论、调和分析等方面的技巧和知识，坚持独立思考，努力发掘一些新思想和新方法，圆满解决这些问题。

本项目成员经过三年努力，主要解决了以下几类问题。1、我们研究了[0,1]区间上的广义三分Cantor集上的二项式测度的加倍性质。我们将它分成三种类型，第一种类型我们得到了测度具有加倍性质的等价条件；其它类型我们证明了该条件是充分但不是必要条件。2,我们确定了由两类滑动平均描述的水平集的Hausdorff维数。不类似的结论补充了Pfaffelhuber & del Junco等人的研究工作，同时揭示了这两类滑动平均的收敛的差异性。3、研究了非退化椭圆方程的Liouville型定理，将Laplace 方程的结果推广到含Grushin 算子的方程。4，研究了某类具有Robin边界条件的非线性椭圆的解得非存在性问题。在某些情况下，该问题没有Morse指数有限的解。