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One phase problem for two positive harmonic function: below the codimension $1$ threshold
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abstract
What can be said about the domain $\Om$ in $\bR^n$ for which its Green's function $G(z)$ satisfies $G(z)\asymp \dist (z, \pd\Om)^\delta$? What can we say about $\Om$ if the Boundary Harnack Principle holds in the form $u/v=\text{real analytic}$ on the part $E$ of its boundary? Here $u, v$ are positive harmonic functions on $\Om$ vanishing on $E$. Is this part of the boundary also nice? We discuss these questions below and give answers in very special cases.
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Cited by 2 Pith papers
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Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries
On s-Ahlfors regular sets with uniform L² non-flatness β₂ ≥ δ₀, harmonic measure has dimension strictly less than s for all s ∈ (n − c δ₀², n].
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On the dimension drop for harmonic measure on uniformly non-flat Ahlfors-David regular boundaries
For uniformly non-flat Ahlfors-David regular boundaries of dimension between n−1−δ0 and n−1 in R^n (n≥3), harmonic measure is concentrated on a set of dimension strictly less than the boundary dimension.
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