A Free Boundary Problem for the Parabolic Poisson Kernel
A Free Boundary Problem for the Parabolic Poisson Kernel
复制标题
抛物线泊松核的自由边界问题
DOI：
10.1016/j.aim.2017.04.032
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Max Engelstein
中科院分区：
文献类型：
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作者：
Max Engelstein
我们研究了 Hofmann、Lewis 和 Nyström [14] 提出的抛物线弦弧域，并证明了低于连续阈值的自由边界正则性结果。更准确地说，我们表明，泊松核在 VMO 中具有对数的 Reifenberg 平坦抛物线弦弧域实际上必定是消失弦弧域（即满足消失卡尔森测度条件）。这将 Kenig 和 Toro [26] 的结果推广到抛物线设置，并肯定地回答了 Hofmann 等人上述论文中未解决的问题。该证明的关键一步是对抛物线问题的“平坦”爆炸进行分类。
We study parabolic chord arc domains, introduced by Hofmann, Lewis and Nyström [14], and prove a free boundary regularity result below the continuous threshold. More precisely, we show that a Reifenberg flat, parabolic chord arc domain whose Poisson kernel has logarithm in VMO must in fact be a vanishing chord arc domain (i.e. satisfies a vanishing Carleson measure condition). This generalizes, to the parabolic setting, a result of Kenig and Toro [26] and answers in the affirmative a question left open in the aforementioned paper of Hofmann et al. A key step in this proof is a classification of “flat” blowups for the parabolic problem.