The variable coefficient thin obstacle problem: optimal regularity and regularity of the regular free boundary
This article deals with the variable coefficient thin obstacle problem in n+1 dimensions. We address the regular free boundary regularity, the behavior of the solution close to the free boundary and the optimal regularity of the solution in a low regularity set-up. We first discuss the case of zero...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article (Journal) |
| Language: | English |
| Published: |
2017
|
| In: |
Annales de l'Institut Henri Poincaré. C, Nonlinear analysis
Year: 2016, Volume: 34, Issue: 4, Pages: 845-897 |
| ISSN: | 1873-1430 |
| DOI: | 10.1016/j.anihpc.2016.08.001 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1016/j.anihpc.2016.08.001 Verlag, lizenzpflichtig, Volltext: https://www.sciencedirect.com/science/article/pii/S0294144916300452 
|
| Author Notes: | Herbert Koch, Angkana Rüland, Wenhui Shi |
| Summary: | This article deals with the variable coefficient thin obstacle problem in n+1 dimensions. We address the regular free boundary regularity, the behavior of the solution close to the free boundary and the optimal regularity of the solution in a low regularity set-up. We first discuss the case of zero obstacle and W1,p metrics with p∈(n+1,∞]. In this framework, we prove the C1,α regularity of the regular free boundary and derive the leading order asymptotic expansion of solutions at regular free boundary points. We further show the optimal C1,min{1−n+1p,12} regularity of solutions. New ingredients include the use of the Reifenberg flatness of the regular free boundary, the construction of an (almost) optimal barrier function and the introduction of an appropriate splitting of the solution. Important insights depend on the consideration of various intrinsic geometric structures. Based on variations of the arguments in [18] and the present article, we then also discuss the case of non-zero and interior thin obstacles. We obtain the optimal regularity of the solutions and the regularity of the regular free boundary for W1,p metrics and W2,p obstacles with p∈(2(n+1),∞]. |
|---|---|
| Item Description: | Available online 11 August 2016 Gesehen am 27.05.2021 |
| Physical Description: | Online Resource |
| ISSN: | 1873-1430 |
| DOI: | 10.1016/j.anihpc.2016.08.001 |