One phase problem for two positive harmonic function: below the codimension 1 threshold

2022/05/07 by Volberg, Alexander · 1 citation
#31B25 #35J25 #35J70 #42B37 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #F.2.2 #FOS: Mathematics

paper · doi:10.48550/arxiv.2205.03687

Abstract

What can be said about the domain \Om in \bRⁿ for which its Green's function G(z) satisfies G(z)\asymp \dist (z, \pd\Om)^δ? What can we say about \Om if the Boundary Harnack Principle holds in the form u/v=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.

Cited by

• On the dimension drop for harmonic measure on uniformly non-flat Ahlfors-David regular boundaries

Related

• A note on higher regularity boundary Harnack inequality
• Boundary Harnack principle for Δ+ Δ^α/2
• Harmonic balls and two-phase Schwarz function
• Boundary Harnack estimates in slit domains and applications to thin free boundary problems
• A note on the critical set of harmonic functions near the boundary