Optimal Regularity for the Thin Obstacle Problem with $C^{0,alpha}$ Coefficients
Optimal Regularity for the Thin Obstacle Problem with $C^{0,alpha}$ Coefficients
复制标题
具有 $C^{0,alpha}$ 系数的薄障碍问题的最优正则性
DOI：
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发表时间：
2016
期刊：
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通讯作者：
Wenhui Shi
中科院分区：
文献类型：
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作者：
Angkana Ruland;Wenhui Shi
在本文中，我们研究了在系数、障碍和基础流形上的低正则性假设下的(内部)薄障碍问题的解。结合Andersson{An16}的线性化方法和Cite{FS16}，Cite{GPSVG15}的边长不等式，我们证明了当存在$C^{0，α}$系数$a^{ij}$和$C^{1，α}$障碍$Phi$时，解的最优正则性态.此外，我们还研究了正则自由边界的正则性，证明了它具有$C^{1，Gamma}$流形的结构，其中$Gamma在(0，1)$中。
In this article we study solutions to the (interior) thin obstacle problem under low regularity assumptions on the coefficients, the obstacle and the underlying manifold. Combining the linearization method of Andersson cite{An16} and the epiperimetric inequality from cite{FS16}, cite{GPSVG15}, we prove the optimal $C^{1,min{alpha,1/2}}$ regularity of solutions in the presence of $C^{0,alpha}$ coefficients $a^{ij}$ and $C^{1,alpha}$ obstacles $phi$. Moreover we investigate the regularity of the regular free boundary and show that it has the structure of a $C^{1,gamma}$ manifold for some $gamma in (0,1)$.