$H^{1+\alpha}$ estimates for the fully nonlinear parabolic thin obstacle problem

TL;DR

This paper establishes local $H^{1+eta}$ regularity for solutions to the fully nonlinear parabolic thin obstacle problem, extending previous elliptic results to the parabolic setting without symmetry assumptions.

Contribution

It extends the $H^{1+eta}$ regularity results from elliptic to fully nonlinear parabolic thin obstacle problems, without symmetry assumptions and with local estimates.

Findings

Solutions are locally $H^{1+eta}$ on each side of the obstacle.

The results extend previous elliptic work to the parabolic case.

Estimates are local and do not depend on boundary data.

Abstract

We study the regularity of the viscosity solution to the fully nonlinear parabolic thin obstacle problem. In particular, we prove that the solution is local $H^{1+\alpha}$ on each side of the smooth obstacle, for some small $\alpha>0.$ Following the method which was first introduced for the harmonic case by Caffarelli in 1979, we extend the results of Fern\'{a}ndez-Real (2016) who treated the fully nonlinear elliptic case. Our results also extend those of Chatzigeorgiou (2019) in two ways. First, we do not assume solutions nor operators to be symmetric. Second, our estimates are local, in the sense that do not rely on the boundary data.