A logarithmic epiperimetric inequality for the obstacle problem
A logarithmic epiperimetric inequality for the obstacle problem
复制标题
障碍问题的对数周长不等式
DOI：
10.1007/s00039-018-0451-1
复制
发表时间：
2017
影响因子：
2.2
通讯作者：
B. Velichkov
中科院分区：
文献类型：
--
作者：
Maria Colombo;L. Spolaor;B. Velichkov
研究了任意维障碍问题的正则集和奇异集的正则性。我们的方法与韦斯（Invent Math 138：23-50，Wei 99 a）的上围不等式有关，该不等式适用于正则点，并提供了Caffarelli（Acta Math 139：155-184，Caf 77）以前介绍的方法的替代方法。在他的论文中，韦斯使用了一个矛盾的论点，经常设置和他问的问题，如果这样的epiperimetric不等式可以证明在一个直接的方式（即，展示明确的竞争对手），这将有重大影响的正则性的自由边界在D > 2。我们肯定地回答了韦斯的问题，在正则点上直接证明了上围不等式，更重要的是我们引入了一个新的工具，我们称之为对数上围不等式。它允许研究整个奇异集的正则性，并产生一个明确的对数模连续的C1正则性，从而改善以前的结果Caffarelli和Monneau，并提供了一个完全替代的方法。这是第一个实例在文献中（即使在最小曲面的背景下）的一个上围不等式的对数型和第一个实例中的上围不等式的奇点有一个直接的证明。我们的对数上围不等式在奇点有一个相当普遍的性质，并将适用于提供类似的结果在不同的情况下，例如薄障碍问题。
We study the regularity of the regular and of the singular set of the obstacle problem in any dimension. Our approach is related to the epiperimetric inequality of Weiss (Invent Math 138:23–50, Wei99a), which works at regular points and provides an alternative to the methods previously introduced by Caffarelli (Acta Math 139:155–184, Caf77). In his paper, Weiss uses a contradiction argument for the regular set and he asks the question if such epiperimetric inequality can be proved in a direct way (namely, exhibiting explicit competitors), which would have significant implications on the regularity of the free boundary in dimension d > 2. We answer positively the question of Weiss, proving at regular points the epiperimetric inequality in a direct way, and more significantly we introduce a new tool, which we call logarithmic epiperimetric inequality. It allows to study the regularity of the whole singular set and yields an explicit logarithmic modulus of continuity on the C1 regularity, thus improving previous results of Caffarelli and Monneau and providing a fully alternative method. It is the first instance in the literature (even in the context of minimal surfaces) of an epiperimetric inequality of logarithmic type and the first instance in which the epiperimetric inequality for singular points has a direct proof. Our logarithmic epiperimetric inequality at singular points has a quite general nature and will be applied to provide similar results in different contexts, for instance for the thin obstacle problem.
几乎面积最小化电流的光滑锥体上的（对数）外周不等式和规律性
影响因子：
2
作者：
Engelstein, Max;Spolaor, Luca;Velichkov, Bozhidar
通讯作者：
Velichkov, Bozhidar
DOI：
10.1515/crelle-2019-0041
发表时间：
2020
期刊：
Journal für die reine und angewandte Mathematik
影响因子：
--
作者：
Spolaor, Luca;Colombo, Maria;Velichkov, Bozhidar
通讯作者：
Velichkov, Bozhidar