论文标题
具有正alt-caffarelli-friedman限制点的爆炸的重构和唯一性
Rectifiability and uniqueness of blow-ups for points with positive Alt-Caffarelli-Friedman limit
论文作者
论文摘要
我们研究了一对非阴性亚谐波函数的不相交支撑之间的界面的规律性。 Alt-caffarelli-friedman（ACF）单调公式的接口部分是渐近的阳性，形成了$ \ MATHCAL {H}^{n-1} $ - 可矫正设置。此外，对于$ \ Mathcal {h}^{n-1} $ - a.e。这样的点，这两个功能具有独特的爆炸，即它们的Lipschitz重新列为$ W^{1,2} $汇聚到一对非等级截断的线性函数，其支撑物在近似切线平面相交。所使用的主要工具包括Naber-Valtorta框架和我们最近的结果，在ACF单调性公式中建立了急剧的定量剩余项。我们还将结果应用于自由边界问题。
We study the regularity of the interface between the disjoint supports of a pair of nonnegative subharmonic functions. The portion of the interface where the Alt-Caffarelli-Friedman (ACF) monotonicity formula is asymptotically positive forms an $\mathcal{H}^{n-1}$-rectifiable set. Moreover, for $\mathcal{H}^{n-1}$-a.e. such point, the two functions have unique blowups, i.e. their Lipschitz rescalings converge in $W^{1,2}$ to a pair of nondegenerate truncated linear functions whose supports meet at the approximate tangent plane. The main tools used include the Naber-Valtorta framework and our recent result establishing a sharp quantitative remainder term in the ACF monotonicity formula. We also give applications of our results to free boundary problems.
