Tangent measures of elliptic measure and applications
Tangent measures of elliptic measure and applications
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椭圆测度的切线测度及其应用
作者：
Jonas Azzam;Mihalis Mourgoglou
切测度和blow-up方法是理解区域边界的无穷小结构与调和测度行为之间关系的有力工具。我们介绍了一种方法来研究椭圆测度在任意域相关的（可能是非对称的）椭圆算子的发散形式，其系数有消失的平均振荡在边界上的正切措施。在此设置中，我们为域$ Omega子集mathbb{R}^{n+1}$显示以下内容：
1.我们推广了Kenig，Preiss和Toro [KPT 09]的结果，证明了{it any}域的内外椭圆测度的相互绝对连续性意味着正切测度是a.e.平坦的，椭圆测度的维数为n。
2.本文推广了Kenig和Toro [KT 06]的工作，证明了一般域上的二重内外椭圆测度的VMO等价性，并给出了其正切测度总是椭圆多项式的结论.
3.在满足容量密度条件、边界局部有限且具有a.e.正的下$n$-Hausdorff密度，证明了如果椭圆测度关于$n$-Hausdorff测度是绝对连续的，则边界是可求长的.这概括了阿克曼、巴杰、霍夫曼和马特尔的工作[ABHM 17]。
最后，我们推广了[Bad 11]的一个主要结果，证明了如果$omega$是一个Radon测度，且在一点上的所有切线测度都是在原点消失的调和多项式，则它们都是齐次调和多项式.
Tangent measure and blow-up methods, are powerful tools for understanding the relationship between the infinitesimal structure of the boundary of a domain and the behavior of its harmonic measure. We introduce a method for studying tangent measures of elliptic measures in arbitrary domains associated with (possibly non-symmetric) elliptic operators in divergence form whose coefficients have vanishing mean oscillation at the boundary. In this setting, we show the following for domains $ Omega subset mathbb{R}^{n+1}$:
1. We extend the results of Kenig, Preiss, and Toro [KPT09] by showing mutual absolute continuity of interior and exterior elliptic measures for {it any} domains implies the tangent measures are a.e. flat and the elliptic measures have dimension $n$.
2. We generalize the work of Kenig and Toro [KT06] and show that VMO equivalence of doubling interior and exterior elliptic measures for general domains implies the tangent measures are always elliptic polynomials.
3. In a uniform domain that satisfies the capacity density condition and whose boundary is locally finite and has a.e. positive lower $n$-Hausdorff density, we show that if the elliptic measure is absolutely continuous with respect to $n$-Hausdorff measure then the boundary is rectifiable. This generalizes the work of Akman, Badger, Hofmann, and Martell [ABHM17].
Finally, we generalize one of the main results of [Bad11] by showing that if $omega$ is a Radon measure for which all tangent measures at a point are harmonic polynomials vanishing at the origin, then they are all homogeneous harmonic polynomials.
具有 Ahlfors-David 正则边界的单边 NTA 域上的可修正性和椭圆测度
影响因子：
1.3
作者：
Akman, Murat;Badger, Matthew;Hofmann, Steve;Martell, Jose Maria
通讯作者：
Martell, Jose Maria