We provide a potential theoretic characterization of vanishing chord-arc domains under mild assumptions. In particular we show that, if a domain has Ahlfors regular boundary, the oscillation of the logarithm of the interior and exterior Poisson kernels yields a great deal of geometric information about the domain. We use techniques from classical calculus of variations, potential theory, and quantitative geometric measure theory to accomplish this. One feature of this work, compared to [KT06] and [BH16], is that a priori we only require that the domains in question are connected.
Bibliographical noteFunding Information:
Some of this work was done while the first, second, fourth, and fifth authors were in residence at the MSRI Harmonic Analysis program, supported by the National Science Foundation (grant noD˙ MS-1440140). During the preparation of this manuscript the first author was partially supported by the NSF INSPIRE Award (no. DMS-1344235) and the second author was supported by several grants (NSF DMS-2000288, NSF MSPRF DMS-1703306, and David Jerison’s grant DMS 1500771). The third author was also partially supported by the NSF (grant nos. DMS-1664867 and DMS-1500098). The fourth author received partial support as the Craig McKibben & Sarah Merner Professor in Mathematics (NSF grant no. DMS-1664867) and by a Simons Foundation Fellowship (no. 614610). Finally, the fifth author was partially supported by the Institute for Advanced Study and also by the NSF (grant nos. DMS-1664867 and DMS-1902756).
Indiana University Mathematics Journal ©
- chord-arc domains
- harmonic measure
- Reifenberg flatness
- Sets of locally finite perimeter
- two-phase free boundary problems