Pointwise regularity of the free boundary for the parabolic obstacle problem

Abstract

We study the parabolic obstacle problem $$\begin{aligned} \Delta u-u_t=f\chi _{\{u>0\}}, \quad u\ge 0,\quad f\in L^p \quad \text{ with }\quad f(0)=1 \end{aligned}$$ 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.