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Poisson kernel characterization of Reifenberg flat chord arc domains. (English) Zbl 1027.31005

The authors prove a conjecture previously stated by themselves in [Ann. Math. (2) 150, 369-454 (1999; Zbl 0946.31001)] regarding the free boundary regularity problem for the Poisson kernel below the continuous threshold. It is shown that the weak regularity of the Poisson kernel of a domain fully determines the geometry of its boundary. Precisely, the authors prove that if \(\Omega\) is a \(\delta\)-Reifenberg flat chord arc domain for small enough \(\delta>0\) and the logarithm of its Poisson kernel has vanishing mean oscillation then the unit normal vector to \(\partial\Omega\) also has vanishing mean oscillation. The machinery used is based on blow up arguments which combine geometric and analytic information about the free boundary regularity problem for the Poisson kernel via potential theory.

MSC:

31C05 Harmonic, subharmonic, superharmonic functions on other spaces
35R35 Free boundary problems for PDEs
31B25 Boundary behavior of harmonic functions in higher dimensions

Citations:

Zbl 0946.31001
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References:

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