Optimal regularity for the thin obstacle problem with $$C^{0,\alpha }$$ C 0 , α coefficients

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 $$C^{1,\min \{\alpha ,1/2\}}$$ regularity of solutions in the presence of $$C^{0,\alpha }$$ coefficients $$a^{ij}$$ and $$C^{1,\alpha }$$ obstacles $$\phi $$ . Moreover we investigate the regularity of the regular free boundary and show that it has the structure of a $$C^{1,\gamma }$$ manifold for some $$\gamma \in (0,1)$$ .