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 ΩR^n+1 assuming that Ω separates R^n+1 into two connected components Ω^1=Ω and Ω^2=R^n+1Ω. We furthermore assume that both Ω^1 and Ω^2 are parabolic NTA-domains, that Ω is Ahlfors regular and for i{1,2} we define ω^i(X^i,t^i,·) to be the caloric measure at (X^i,t^i)Ω^i defined with respect to Ω^i. In the paper we make the additional assumption that ω^i(X^i,t^i,·), for i{1,2}, is absolutely continuous with respect to an appropriate surface measure σ on Ω and that the Poisson kernels k^i(X^i,t^i,·)=dω^i(X^i,t^i,·)/dσ are such that k^i(X^i,t^i,·)VMO(dσ). Our main result (Theorem 1) states that, under these assumptions, C_r(X,t)Ω is Reifenberg flat with vanishing constant whenever (X,t)Ω and {t^1,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 Ω 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.