The variable coefficient thin obstacle problem: Carleman inequalities

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• 作者：Herbert Koch^a ; ^{koch@math.uni-bonn.de" class="auth_mail" title="E-mail the corresponding author} ; Angkana Rü ; land^b ; ^{ruland@maths.ox.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; Wenhui Shi^c ; ^{wenhui.shi@hcm.uni-bonn.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35R35
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：1 October 2016
• 年：2016
• 卷：301
• 期：Complete
• 页码：820-866
• 全文大小：735 K

文摘

In this article we present a new strategy of addressing the (variable coefficient) thin obstacle problem. Our approach is based on a (variable coefficient) Carleman estimate. This yields semi-continuity of the vanishing order, lower and uniform upper growth bounds of solutions and sufficient compactness properties in order to carry out a blow-up procedure. Moreover, the Carleman estimate implies the existence of homogeneous blow-up limits along certain sequences and ultimately leads to an almost optimal regularity statement. As it is a very robust tool, it allows us to consider the problem in the setting of Sobolev metrics, i.e. working on the upper half ball $B_1^{+}\subset {\mathbb{R}}_{+}^{n+1}$, the coefficients are only W^1,p$W^{1,p}$ regular for some p>n+1$p>n+1$.

These results provide the basis for our further analysis of the free boundary, the optimal regularity of solutions and a first order asymptotic expansion of solutions at the regular free boundary which is carried out in a follow-up article, [21], in the framework of W^1,p$W^{1,p}$, p>2(n+1)$p>2(n+1)$, regular coefficients and W^2,p$W^{2,p}$, p>2(n+1)$p>2(n+1)$, regular non-zero obstacles.