Regularity below the continuous threshold in a two-phase parabolic free boundary problem

Abstract

In this paper we study free boundary regularity in a parabolic two-phase problem below the continuous threshold. We consider unbounded domains $\Omega\subset\mathsf{R}^{n+1}$ assuming that $\partial\Omega$ separates $\mathsf{R}^{n+1}$ into two connected components $\Omega^1=\Omega$ and $\Omega^2=\mathsf{R}^{n+1}\setminus\overline\Omega$. We furthermore assume that both $\Omega^1$ and $\Omega^2$ are parabolic NTA-domains, that $\partial\Omega$ is Ahlfors regular and for $i\in\{1,2\}$ we define $\omega^i(\hat{X}^i,\hat{t}^i,\cdot)$ to be the caloric measure at $(\hat{X}^i,\hat{t}^i)\in \Omega^i$ defined with respect to $\Omega^i$. In the paper we make the additional assumption that $\omega^i(\hat{X}^i,\hat{t}^i,\cdot)$, for $i\in\{1,2\}$, is absolutely continuous with respect to an appropriate surface measure $\sigma$ on $\partial\Omega$ and that the Poisson kernels $k^i(\hat{X}^i,\hat{t}^i,\cdot)=d\omega^i(\hat{X}^i,\hat{t}^i,\cdot)/d\sigma$ are such that $\log k^i(\hat{X}^i,\hat{t}^i,\cdot)\in \mathrm{VMO}(d\sigma)$. Our main result (Theorem 1) states that, under these assumptions, $C_r(X,t)\cap\partial\Omega$ is Reifenberg flat with vanishing constant whenever $(X,t)\in\partial\Omega$ and $\min\{\hat{t}^1,\hat{t}^2\}>t+4r^2$. This result has an important consequence (Theorem 3) stating that if the two-phase condition on the Poisson kernels is fulfilled, $\Omega^1$ and $\Omega^2$ are parabolic NTA-domains and $\partial\Omega$ is Ahlfors regular then if $\Omega$ is close to being a chord arc domain with vanishing constant we can in fact conclude that $\Omega$ is a chord arc domain with vanishing constant.