Structure of sets which are well approximated by zero sets of harmonic polynomials
Structure of sets which are well approximated by zero sets of harmonic polynomials
复制标题
由调和多项式的零集很好地近似的集合的结构
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
T. Toro
中科院分区：
文献类型：
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作者：
Matthew Badger;Max Engelstein;T. Toro
调和多项式的零点集在调和测度自由边界正则性问题的研究中起着至关重要的作用。为了理解这些自由边界的精细结构，需要详细研究这些零点集的奇点。在本文中，我们研究如何“度$k$点”坐在零套调和多项式在$mathbb R^n$的度$d$（所有的$ngeq 2$和$1leq kleq d$）和内部集，允许任意好的局部逼近的零套调和多项式。我们得到了一般的结构定理后一种类型的集，包括尖锐的Hausdorff和Minkowski维数估计的奇异集的“度$k$点”（$kgeq 2$）没有证明唯一性的爆破或援助的PDE方法，如单调性公式。此外，我们表明，在存在一定的拓扑分离条件下，尖锐的尺寸估计改善，并依赖于宇称的$k$。本文应用于Kenig和Toro提出的连续阈值下调和测度的两相自由边界正则性问题。
The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries a detailed study of the singular points of these zero sets is required. In this paper we study how "degree $k$ points" sit inside zero sets of harmonic polynomials in $mathbb R^n$ of degree $d$ (for all $ngeq 2$ and $1leq kleq d$) and inside sets that admit arbitrarily good local approximations by zero sets of harmonic polynomials. We obtain a general structure theorem for the latter type of sets, including sharp Hausdorff and Minkowski dimension estimates on the singular set of "degree $k$ points" ($kgeq 2$) without proving uniqueness of blowups or aid of PDE methods such as monotonicity formulas. In addition, we show that in the presence of a certain topological separation condition, the sharp dimension estimates improve and depend on the parity of $k$. An application is given to the two-phase free boundary regularity problem for harmonic measure below the continuous threshold introduced by Kenig and Toro.