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Minimizers for the Thin One-Phase Free Boundary Problem
Communications on Pure and Applied Mathematics ( IF 3 ) Pub Date : 2021-07-16 , DOI: 10.1002/cpa.22011 Max Engelstein 1 , Aapo Kauranen 2 , Martí Prats 3 , Georgios Sakellaris 3 , Yannick Sire 4
Communications on Pure and Applied Mathematics ( IF 3 ) Pub Date : 2021-07-16 , DOI: 10.1002/cpa.22011 Max Engelstein 1 , Aapo Kauranen 2 , Martí Prats 3 , Georgios Sakellaris 3 , Yannick Sire 4
Affiliation
We consider the “thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in 
plus the area of the positivity set of that function in 
. We establish full regularity of the free boundary for dimensions 
, prove almost everywhere regularity of the free boundary in arbitrary dimension, and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight.
中文翻译：
薄单相自由边界问题的最小化器
我们考虑“薄单相”自由边界问题，与最小化函数 in 的加权狄利克雷能量加上该函数的正集面积 in 相关。我们建立维度的自由边界的完全正则性，证明几乎无处不在自由边界在任意维度上的正则性，并在自由边界存在时提供对自由边界奇异集的内容和结构估计。所有这些结果都适用于相关权重的全范围。
更新日期：2021-07-16
plus the area of the positivity set of that function in . We establish full regularity of the free boundary for dimensions , prove almost everywhere regularity of the free boundary in arbitrary dimension, and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight.
中文翻译：
薄单相自由边界问题的最小化器
我们考虑“薄单相”自由边界问题，与最小化函数 in 的加权狄利克雷能量
加上该函数的正集面积 in 相关。我们建立维度的自由边界的完全正则性，证明几乎无处不在自由边界在任意维度上的正则性，并在自由边界存在时提供对自由边界奇异集的内容和结构估计。所有这些结果都适用于相关权重的全范围。
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