An Epiperimetric Inequality for the Regularity of Some Free Boundary Problems: The 2‐Dimensional Case
An Epiperimetric Inequality for the Regularity of Some Free Boundary Problems: The 2‐Dimensional Case
复制标题
一些自由边界问题正则性的周长不等式：二维情况
DOI：
10.1002/cpa.21785
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发表时间：
2016
影响因子：
3
通讯作者：
B. Velichkov
中科院分区：
文献类型：
--
作者：
L. Spolaor;B. Velichkov
使用直接的方法，我们证明了一个二维上围不等式的单阶段问题的标量和向量的情况下，双阶段问题。由此我们推导出在2维空间中，纯量单相和双相问题中自由边界的C1，α正则性，以及矢量情况下约化自由边界的C1，α正则性，而对分量函数的符号没有任何限制。此外，我们证明了在向量的情况下，{|u| =0}可能有尖点，但它们必须是有限数。© 2018 Wiley Periodicals，Inc.
Using a direct approach, we prove a two‐dimensional epiperimetric inequality for the one‐phase problem in the scalar and vectorial cases and for the double‐phase problem. From this we deduce, in dimension 2, the C1,α regularity of the free boundary in the scalar one‐phase and double‐phase problems, and of the reduced free boundary in the vectorial case, without any restriction on the sign of the component functions. Furthermore, we show that in the vectorial case each connected component of {|u|=0} might have cusps, but they must be a finite number. © 2018 Wiley Periodicals, Inc.