Harmonic polynomials and tangent measures of harmonic measure
Matthew Badger
TL;DR
This paper investigates the structure of tangent measures of harmonic measure on NTA domains, proving that polynomial tangent measures are homogeneous and deriving geometric insights for a related free boundary problem.
Contribution
It establishes that polynomial tangent measures are necessarily homogeneous and connects this to geometric properties in two-phase free boundary problems.
Findings
Polynomial tangent measures are homogeneous.
Geometric information for two-phase free boundary solutions is derived.
Results apply to harmonic measures on NTA domains.
Abstract
We show that on an NTA domain if each tangent measure to harmonic measure at a point is a polynomial harmonic measure then the associated polynomials are homogeneous. Geometric information for solutions of a two-phase free boundary problem studied by Kenig and Toro is derived.
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