On the regularity of the Non-dynamic Parabolic Fractional Obstacle Problem

TL;DR

This paper establishes higher and optimal regularity results for solutions and free boundaries in non-dynamic fractional parabolic obstacle problems, including Hölder continuity of the time derivative at certain free boundary points.

Contribution

It provides new regularity results for solutions and free boundaries in fractional parabolic obstacle problems, including space derivatives and time regularity.

Findings

Higher regularity of solutions and derivatives

Hölder continuity of the time derivative at free boundary points

Regularity of the free boundary for all fractional orders

Abstract

In the class of the so called non-dynamic Fractional Obstacle Problems of parabolic type, it is shown how to obtain higher regularity as well as optimal regularity of the space derivatives of the solution. Furthermore, at free boundary points of positive parabolic density, it is proven that the time derivative of the solution is H\"{o}lder continuous. Finally, at regular free boundary points, space-time regularity of the corresponding free boundary is obtained for any fraction $s\in(0,1)$.