Regularity below the continuous threshold in a two-phase parabolic free boundary problem - Nyström, Kaj (Mathematica Scandinavica, 2006)
In this paper we study free boundary regularity in a parabolic two-phase problem below the continuous threshold. We consider unbounded domains $Ω\subset\mathsf{R}^{n+1}$ assuming that $\partialΩ$ separates $\mathsf{R}^{n+1}$ into two connected components $Ω^1=Ω$ and $Ω^2=\mathsf{R}^{n+1}\setminus\overlineΩ$. We furthermore assume that both $Ω^1$ and $Ω^2$ are parabolic NTA-domains, that $\partialΩ$ is Ahlfors regular and for $i\in\{1,2\}$ we define $ω^i(\hat{X}^i,\hat{t}^i,·)$ to be the caloric measure at $(\hat{X}^i,\hat{t}^i)\in Ω^i$ defined with respect to $Ω^i$. In the paper we make the additional assumption that $ω^i(\hat{X}^i,\hat{t}^i,·)$, for $i\in\{1,2\}$, is absolutely continuous with respect to an appropriate surface measure $σ$ on $\partialΩ$ and that the Poisson kernels $k^i(\hat{X}^i,\hat{t}^i,·)=dω^i(\hat{X}^i,\hat{t}^i,·)/dσ$ are such that $\log k^i(\hat{X}^i,\hat{t}^i,·)\in VMO(dσ)$. Our main result (Theorem 1) states that, under these assumptions, $C_r(X,t)\cap\partialΩ$ is Reifenberg flat with vanishing constant whenever $(X,t)\in\partialΩ$ 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, $Ω^1$ and $Ω^2$ are parabolic NTA-domains and $\partialΩ$ is Ahlfors regular then if $Ω$ is close to being a chord arc domain with vanishing constant we can in fact conclude that $Ω$ is a chord arc domain with vanishing constant.