# Regularity for the fully nonlinear parabolic thin obstacle problem

## TL;DR

This paper establishes $C^{1, eta}$ regularity for viscosity solutions of a fully nonlinear parabolic boundary obstacle problem, extending classical methods to a more general nonlinear setting.

## Contribution

It extends the regularity results for parabolic obstacle problems to fully nonlinear equations, building on and generalizing previous linear and elliptic cases.

## Key findings

- Proves $C^{1, eta}$ regularity for solutions.
- Extends Caffarelli's method to fully nonlinear parabolic equations.
- Bridges gap between linear and nonlinear obstacle problem theories.

## Abstract

We prove $C^{1, \alpha}$ regularity (in the parabolic sense) for the viscosity solution of a boundary obstacle problem with a fully nonlinear parabolic equation in the interior. Following the method which was first introduced for the harmonic case by L. Caffarelli in 1979, we extend the results of I. Athanasopoulos (1982) who studied the linear parabolic case and the results of E. Milakis and L. Silvestre (2008) who treated the fully nonlinear elliptic case.