In non-variational two-phase free boundary problems for harmonic measure, we examine how the relationship between the interior and exterior harmonic measures of a domain Ω ⊂ Rn influences the geometry of its boundary. This type of free boundary problem was initially studied by Kenig and Toro in 2006, and was further examined in a series of separate and joint investigations by several authors. The focus of the present paper is on the singular set in the free boundary, where the boundary looks infinitesimally like zero sets of homogeneous harmonic polynomials of degree at least 2. We prove that if the Radon–Nikodym derivative of the exterior harmonic measure with respect to the interior harmonic measure has a Hölder continuous logarithm, then the free boundary admits unique geometric blowups at every singular point and the singular set can be covered by countably many C1,β submanifolds of dimension at most n − 3. This result is partly obtained by adapting tools such as Garofalo and Petrosyan’s Weiss type monotonicity formula and an epiperimetric inequality for harmonic functions from the variational to the non-variational setting.
Bibliographical noteFunding Information:
M. Badger was partially supported by NSF DMS grants 1500382 and 1650546. M. Engelstein was partially supported by an NSF postdoctoral fellowship, NSF DMS 1703306, and by NSF DMS grant 1500771. T. Toro was partially supported by NSF DMS grants 1361823 and 1664867 and by the Craig McKibben & Sarah Merner Professorship in Mathematics. This material is based upon work supported by NSF DMS grant 1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.
- Epiperimetric inequalities
- Harmonic measure
- Harmonic polynomials
- Higher order rectifiability
- Singular set
- Two-phase free boundary problems
- Uniqueness of blowups
- Weiss-type monotonicity formula