2019/04/01 by Prats, Martí, Tolsa, Xavier
#28A75 #28A78 #31B15 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1904.00751
Let Ω+⊂\mathbb Rn+1 be an NTA domain and let Ω-= \mathbb Rn+1∖ Ω+ be an NTA domain as well. Denote by ω+ and ω- their respective harmonic measures. Assume that Ω+ is a δ-Reifenberg flat domain for some δ>0 small enough. In this paper we show that log(dω-)/(dω+)∈ VMO(ω+) if and only if Ω+ is vanishing Reifenberg flat, Ω+ and Ω- have joint big pieces of chord-arc subdomains, and the inner unit normal of Ω+ has vanishing oscillation with respect to the approximate normal. This result can be considered as a two-phase counterpart of a more well known related one-phase problem for harmonic measure solved by Kenig and Toro.