Regularity of the free boundary in a nonlocal one-dimensional parabolic free boundary value problem

TL;DR

This paper investigates the regularity properties of the free boundary in a one-dimensional nonlocal parabolic free boundary problem, establishing conditions for its smoothness and infinite differentiability.

Contribution

It provides new criteria for the $C^m$-regularity and infinite differentiability of the free boundary in a nonlocal parabolic setting.

Findings

Established $C^m$-regularity of the free boundary.

Derived necessary and sufficient conditions for infinite differentiability.

Provided insights into the smoothness criteria for nonlocal free boundary problems.

Abstract

We consider one-dimensional parabolic free boundary value problem with a nonlocal (integro-differential) condition on the free boundary. Results on $C^m$-regularity of the free boundary are obtained. In particular, a necessary and sufficient condition for infinite differentiability of the free boundary is given.