In non-variational two-phase free boundary problems for harmonic measure, we examine how the relationship between the interior and exterior harmonic measures of a domain $\Omega \subset \mathbb R^n$ influences the geometry of its boundary. This type of free boundary problem was initially studied by Kenig and Toro in 2006, and was further examined in a series of separate and joint investigations by several authors. The focus of the present paper is on the singular set in the free boundary, where the boundary looks infinitesimally like zero sets of homogeneous harmonic polynomials of degree at least 2. We prove that if the Radon–Nikodym derivative of the exterior harmonic measure with respect to the interior harmonic measure has a Hölder continuous logarithm, then the free boundary admits unique geometric blowups at every singular point and the singular set can be covered by countably many $C^{1, \beta}$ submanifolds of dimension at most $n−3$. This result is partly obtained by adapting tools such as Garofalo and Petrosyan’s Weiss type monotonicity formula and an epiperimetric inequality for harmonic functions from the variational to the non-variational setting.

中文翻译：

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具有Hölder数据的谐波测量的两相问题中奇异集的正则性

在用于谐波测度的非变量两相自由边界问题中，我们研究了域$ \ Omega \ subset \ mathbb R ^ n $的内部和外部谐波测度之间的关系如何影响其边界的几何形状。这种类型的自由边界问题最初是由Kenig和Toro在2006年研究的，并由几位作者在一系列单独的联合研究中进行了进一步研究。本文的重点是在自由边界上的奇异集，其中边界看起来像无限次的齐次谐波多项式的零集至少为2。我们证明，如果外部谐波测度的Radon-Nikodym导数相对于内部谐波测度具有Hölder连续对数，那么自由边界在每个奇异点处都允许独特的几何爆炸，并且奇异集可以被数量众多的$ C ^ {1，\ beta} $维子流形最多覆盖$ n-3 $。通过改编Garofalo和Petrosyan的Weiss型单调性公式之类的工具以及从变分设置到非变分设置的谐波函数的表观不等式，可以部分获得此结果。