BMO Solvability and $$A_{\infty }$$A∞ Condition of the Elliptic Measures in Uniform Domains
BMO Solvability and $$A_{\infty }$$A∞ Condition of the Elliptic Measures in Uniform Domains
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均匀域中椭圆测度的 BMO 可解性和 $$A_{infty }$$A∞ 条件
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发表时间：
2016
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通讯作者：
Zihui Zhao
中科院分区：
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作者：
Zihui Zhao
本文研究了具有有界可测系数的散度型椭圆算子的Dirichlet边值问题.证明了对于具有Ahlfors正则边界的一致区域，这类问题的BMO可解性等价于椭圆测度关于表面测度的定量绝对连续性，即，$$\omega _L\in A_{\infty }（\sigma）$$ωL∈A∞（σ）。这推广了Dindos、Kenig和Pipher关于Lipschitz域的先前结果（参见Dindos等人，J Geom Anal 21：78-95，2011）。
We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO solvability of such problems is equivalent to a quantitative absolute continuity of the elliptic measure with respect to the surface measure, i.e., $$\omega _L\in A_{\infty }(\sigma )$$ωL∈A∞(σ). This generalizes a previous result on Lipschitz domains by Dindos, Kenig, and Pipher (see Dindos et al. in J Geom Anal 21:78–95, 2011).