Publication date:
20 August 2017
Source:Advances in Mathematics, Volume 316
Author(s): Xavier Ros-Oton, Joaquim Serra
We study the regularity of the free boundary in the fully nonlinear thin obstacle problem. Our main result establishes that the free boundary is $C^1$ near any regular point. This extends to the fully nonlinear setting the celebrated result of Athanasopoulos–Caffarelli–Salsa [1]. The proofs we present here are completely independent from those in [1], and do not rely on any monotonicity formula. Furthermore, an interesting and novel feature of our proofs is that we establish the regularity of the free boundary without classifying blow-ups, a priori they could be non-homogeneous and/or non-unique. We do not classify blow-ups but only prove that they are 1D on $\{x_n=0\}$.