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...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article (Journal) |
| Language: | English |
| Published: |
11 July 2016
|
| In: |
Advances in mathematics
Year: 2016, Volume: 301, Pages: 820-866 |
| ISSN: | 1090-2082 |
| DOI: | 10.1016/j.aim.2016.06.023 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1016/j.aim.2016.06.023 Verlag, lizenzpflichtig, Volltext: https://www.sciencedirect.com/science/article/pii/S0001870816308350 
|
| Author Notes: | Herbert Koch, Angkana Rüland, Wenhui Shi |
| Summary: | 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 B1+⊂R+n+1, the coefficients are only W1,p regular for some 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 W1,p, p>2(n+1), regular coefficients and W2,p, p>2(n+1), regular non-zero obstacles. |
|---|---|
| Item Description: | Gesehen am 27.05.2021 |
| Physical Description: | Online Resource |
| ISSN: | 1090-2082 |
| DOI: | 10.1016/j.aim.2016.06.023 |