自由边界问题相关的Hausdorff测度估计

发布时间：2018-07-28 08:23

【摘要】：'Quantitative stratification for some free-boundary problems' 这篇文章利用最近非常流行的量化分层(Quantitative stratification)技巧对单相自由边界问题中的自由边界进行量化分层,并且给出相应'层'的Hausdorff测度估计。我论文的主要目的是理解该文章的证明思路,想法及相关技巧,补充文章证明中略去的细节并将该文章的结论推广到双相自由边界问题中。文章的主要技巧是量化分层,离散Reifenberg定理的运用以及相应的树构造:量化分层将处理对象由k对称函数推广到近k对称函数,从而适用性更强,同时近k对称函数也具有类似于k对称函数好的性质;离散Reifenberg定理则在满足球Weiss密度足够大的前提下,给出了自由边界非常好的测度估计;而若离散Reifenberg定理所需要的条件不成立时,我们则利用巧妙的树构造来推出最终结论。最后,我证明了单相自由边界问题的Weiss单调性公式可以推广到双相自由边界问题中,从而单相自由边界的Hausdorff测度估计对双相问题也成立。
[Abstract]:This paper makes use of the recently popular quantized hierarchical (Quantitative stratification) technique to quantize the free boundary in single-phase free boundary problems, and gives the estimate of the Hausdorff measure of the corresponding 'layer'. The main purpose of my thesis is to understand the proof ideas, ideas and related techniques of this paper, to supplement the details of the proof and to extend the conclusion of this paper to the biphasic free boundary problem. The main skills of this paper are quantization stratification, the application of discrete Reifenberg theorem and the corresponding tree construction. Quantization stratification extends the processing object from k-symmetric function to near-k-symmetric function, so it is more applicable. At the same time, the near k-symmetric function is similar to the k-symmetric function, and the discrete Reifenberg theorem gives a very good estimate of the free boundary on the condition that the spherical Weiss density is large enough. If the condition required by discrete Reifenberg theorem is not valid, we use the ingenious tree structure to deduce the final conclusion. Finally, I prove that the Weiss monotonicity formula for the single-phase free boundary problem can be extended to the two-phase free boundary problem, so that the Hausdorff measure estimate of the single-phase free boundary problem is also valid for the biphasic problem.
【学位授予单位】：中国科学技术大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8

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