The variable coefficient thin obstacle problem: Carleman inequalities
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 compact...
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| Main Authors: | , , |
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| Format: | Article (Journal) Chapter/Article |
| Language: | English |
| Published: |
19 Jan 2015
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| In: |
Arxiv
Year: 2015, Pages: 1-38 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: http://arxiv.org/abs/1501.04496
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| Author Notes: | Herbert Koch, Angkana Rüland, and Wenhui Shi |
MARC
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| 520 | |a 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. the coefficients are only $W^{1,p}$ regular for some $p>n+1$. These results provide the basis for our further analysis of the free boundary, the optimal ($C^{1,1/2}$-) 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 in the framework of $W^{1,p}$, $p>2(n+1)$, regular coefficients. | ||
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