Pointwise regularity of the free boundary for the parabolic obstacle problem

TL;DR

This paper investigates the regularity and structure of the free boundary in the parabolic obstacle problem, establishing monotonicity formulas and detailed geometric descriptions of regular and singular points.

Contribution

It introduces two new monotonicity formulas and proves second order Taylor expansions at singular points, advancing understanding of free boundary regularity in parabolic obstacle problems.

Findings

Regular points form a locally $C^1$-surface.

Singular points are contained in unions of $C^1$ manifolds.

Monotonicity formulas applicable to general and singular free boundary points.

Abstract

We study the parabolic obstacle problem $$\lap u-u_t=f\chi_{\{u>0\}}, \quad u\geq 0,\quad f\in L^p \quad \mbox{with}\quad f(0)=1$$ and obtain two monotonicity formulae, one that applies for general free boundary points and one for singular free boundary points. These are used to prove a second order Taylor expansion at singular points (under a pointwise Dini condition), with an estimate of the error (under a pointwise double Dini condition). Moreover, under the assumption that $f$ is Dini continuous, we prove that the set of regular points is locally a (parabolic) $C^1$-surface and that the set of singular points is locally contained in a union of (parabolic) $C^1$ manifolds.