Abstract:Let $\Omega \subset \mathbb {R}^{n+1}$, $n \geq 1$, be an open and connected set. Set $\mathcal {T}_n$ to be the set of points $\xi \in \partial \Omega$ so that there exists an approximate tangent $n$-plane for $\partial \Omega$ at $\xi$ and $\partial \Omega$ satisfies the weak lower Ahlfors-David $n$-regularity condition at $\xi$. We first show that $\mathcal {T}_n$ can be covered by a countable union of boundaries of bounded Lipschitz domains. Then, letting $\partial ^\star \Omega$ be a subset of $\mathcal {T}_n$ where $\Omega$ satisfies an appropriate thickness condition, we prove that $\partial ^\star \Omega$ can be covered by a countable union of boundaries of bounded Lipschitz domains contained in $\Omega$. As a corollary we obtain that if $\Omega$ has locally finite perimeter, $\partial \Omega$ is weakly lower Ahlfors-David $n$-regular, and the measure-theoretic boundary coincides with the topological boundary of $\Omega$ up to a set of $\mathcal {H}^n$-measure zero, then $\partial \Omega$ can be covered, up to a set of $\mathcal {H}^n$-measure zero, by a countable union of boundaries of bounded Lipschitz domains that are contained in $\Omega$. This implies that in such domains, $\mathcal {H}^n|_{\partial \Omega }$ is absolutely continuous with respect to harmonic measure.

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中文翻译：

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近似切线、谐波测量和具有可校正边界的域

摘要：令$\Omega \subset \mathbb {R}^{n+1}$, $n \geq 1$ 是一个开连通集。将 $\mathcal {T}_n$ 设置为点集 $\xi \in \partial \Omega$ 以便在 $\xi$ 处存在 $\partial \Omega$ 的近似切线 $n$-plane 和$\partial \Omega$ 满足 $\xi$ 处的弱下 Ahlfors-David $n$-正则性条件。我们首先证明 $\mathcal {T}_n$ 可以被有界 Lipschitz 域的边界的可数并集覆盖。然后，令 $\partial ^\star \Omega$ 是 $\mathcal {T}_n$ 的子集，其中 $\Omega$ 满足合适的厚度条件，我们证明 $\partial ^\star \Omega$ 可以被覆盖由包含在 $\Omega$ 中的有界 Lipschitz 域的边界的可数联合组成。作为推论，我们得到如果 $\Omega$ 具有局部有限周长，$\partial \Omega$ 是弱低的Ahlfors-David $n$-regular，测度理论边界与$\Omega$ 的拓扑边界重合直到一组$\mathcal {H}^n$-measure零，那么 $\partial \Omega$ 可以被包含在 $\Omega$ 中的有界 Lipschitz 域的边界的可数并集覆盖，直到一组 $\mathcal {H}^n$-measure 0。这意味着在这些域中，$\mathcal {H}^n|_{\partial \Omega }$ 相对于谐波测度是绝对连续的。

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