调和测度与泊松核的自由边界正则性
Free boundary regularity for harmonic measures and Poisson kernels
摘要：
本文的基本目的之一是研究欧几里得空间的“类超曲面”子集的几何与它们所支持的测度的性质之间的关系。在这种情况下，我们表明，某些加倍的措施性质决定其支持的几何形状。如果C乘以以支撑为中心的半径为r的球的测度大于半径为2 r且中心相同的球的测度，则称Radon测度是常数C的两倍。证明了如果\R^{n+1}上的测度的加倍常数接近于n维勒贝格测度的加倍常数，那么它的支撑可以用n维仿射空间很好地逼近，只要支撑是相对平坦的.首先，我们考虑的集合是\R^{n+1}中的域的边界。在R^{n+1}中，n维Hausdorff测度可能不定义在域的边界上。因此，我们把我们的注意力转向调和测度，这是表现良好的小的假设。利用局部平坦域的调和测度的加倍性质，得到了局部平坦域的一个新的刻画。沿着这些路线，我们调查如何“弱”的正则性的泊松核的域确定其边界的几何形状。
英文摘要：
One of the basic aims of this paper is to study the relationship between the geometry of ``hypersurface like'' subsets of Euclidean space and the properties of the measures they support. In this context we show that certain doubling properties of a measure determine the geometry of its support. A Radon measure is said to be doubling with constant C if C times the measure of the ball of radius r centered on the support is greater than the measure of the ball of radius 2r and the same center. We prove that if the doubling constant of a measure on \R^{n+1} is close to the doubling constant of the n-dimensional Lebesgue measure then its support is well approximated by n-dimensional affine spaces, provided that the support is relatively flat to start with. Primarily we consider sets which are boundaries of domains in \R^{n+1}. The n-dimensional Hausdorff measure may not be defined on the boundary of a domain in R^{n+1}. Thus we turn our attention to the harmonic measure which is well behaved under minor assumptions. We obtain a new characterization of locally flat domains in terms of the doubling properties of their harmonic measure. Along these lines we investigate how the ``weak'' regularity of the Poisson kernel of a domain determines the geometry of its boundary.
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