A symmetry theorem revisited

Abstract

Abstract. We show that if harmonic measure and Hausdorff measure are equal on the boundary of certain domains in Euclidean n-space, then these domains are necessarily balls.

A characterization of the unit disk and the harmonic measure doubling condition

“…A related characterization of the disk in terms of harmonic measure was proved in [9]. For another interesting result, which classifies when a domain is a disk or a ball in terms of harmonic measure properties, we refer the reader to [8].…”

Stability of Lewis and Vogel’s result

“…The proof of the Main Lemma is a slight variation of the proof that appears in [LV1]. We sketch the proof and try to indicate as we go along what the ideas behind the calculations are.…”

“…We sketch the proof and try to indicate as we go along what the ideas behind the calculations are. For further details we refer the reader to [LV1] and [LV2].…”

“…Lewis and Vogel proved (see [LV1], [LV2]) that a bounded domain whose harmonic measure (with respect to a fixed point) is a constant multiple of the surface measure to the boundary (i.e. a domain whose Poisson kernel is constant) is a ball, provided the surface measure has at most Euclidean growth.…”