The two-phase problem for harmonic measure in VMO and the chord-arc condition

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Abstract: Let

O m e g a^{+} s u b s e t m a t h b b R^{n + 1}

be a bounded

d e l t a

-Reifenberg flat domain, with

d e l t a > 0

small enough, possibly with locally infinite surface measure. Assume also that

O m e g a^{-} = m a t h b b R^{n + 1} s e t m i n u s o v e r l i n e O m e g a^{+}

is an NTA domain as well and denote by

o m e g a^{+}

and

o m e g a^{-}

the respective harmonic measures of

O m e g a^{+}

and

O m e g a^{-}

with poles

p^{p} m i n O m e g a^{p} m

. In this paper we show that the condition that

l o g d f r a c d o m e g a^{-} d o m e g a^{+} i n V M O (o m e g a^{+})

is equivalent to

O m e g a^{+}

being a chord-arc domain with inner normal belonging to

V M O (H^{n} |_{p a r t i a l O m e g a^{+}})

.

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