Calc. Var. (2017) 56:129
Calculus of Variations
Optimal regularity for the thin obstacle problem
· Wenhui Shi
Received: 14 November 2016 / Accepted: 30 July 2017 / Published online: 23 August 2017
© Springer-Verlag GmbH Germany 2017
Abstract 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 (Invent Math 204(1):1–82, 2016. doi:10.
1007/s00222-015-0608-6) and the epiperimetric inequality from Focardi and Spadaro (Adv
Differ Equ 21(1–2):153–200, 2016), Garofalo, Petrosyan and Smit Vega Garcia (J Math
Pures Appl 105(6):745–787, 2016. doi:10.1016/j.matpur.2015.11.013), we prove the optimal
regularity of solutions in the presence of C
φ. Moreover we investigate the regularity of the regular free boundary and show that it has
the structure of a C
manifold for some γ ∈ (0, 1).
Mathematics Subject Classiﬁcation 35R35
Communicated by O.Savin.
A.R. acknowledges a Junior Research Fellowship at Christ Church. W.S. was partially supported by the
Hausdorff Center for Mathematics and the Centre for Mathematics of the University of Coimbra -
UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the
European Regional Development Fund through the Partnership Agreement PT2020.
Mathematical Institute of the University of Oxford, Andrew Wiles Building, Radcliffe Observatory
Quarter, Woodstock Road, Oxford OX2 6GG, UK
Department of Mathematics, University of Coimbra, Apartado 3008, 3001-501 Coimbra, Portugal