Boundary problems in the theory of univalent functions

Regularity and free boundary regularity for the $p$-Laplace operator in Reifenberg flat and Ahlfors regular domains

“…While the results in [17], [18], [19], [20] concern harmonic functions and harmonic measure, i.e., the case p = 2, the results proved in this paper are valid for 1 < p < ∞ and our results are completely new when p = 2. Consequently we also establish versions, valid in all dimensions, for the p-Laplace equation, 1 < p < ∞, of the classical results of Lavrentiev [22] and Pommerenke [35] mentioned above.…”

“…A classical result concerning the harmonic measure, due to Lavrentiev [22], states that if Ω ⊂ R 2 is a chord arc domain, then ω is mutually absolutely continuous with respect to σ, i.e., dω = kdσ, where k is the associated Poisson kernel. Moreover, Lavrentiev [22] proved that log k is in the space of functions of bounded mean oscillation, defined with respect to σ, on ∂Ω. Later Pommerenke [35] proved that Ω is a vanishing chord arc if and only if log k is in the space of functions of vanishing mean oscillation, defined with respect to σ, on ∂Ω.…”

Brownian motion and sets of harmonic measure zero

“…However, for non-rectifiable boundaries it is not true in general that harmonic measure is equivalet to 1-dimensional Hausdorff measure on 3Z7, even if U is simply connected. Lavrentiev [12] was the first to give an example of a Jordan domain U with a subset E oί dU of zero length and \ a (E) > 0. A simpler example can be found in McMillan and Piranian [IS].…”

On singularity of harmonic measure in space

“…Suppose D is a bounded domain in R m , m>2, R m \D satisfies the corkscrew condition at each point on dD; and E is a set on dD lying also on a BMOχ surface, which is more general than a hyperpiane; then we can prove that if E has m -1 dimensional Hausdorff measure zero then it must have harmonic measure zero with respect to D. Lavrentiev (1936) found a simply-connected domain D in R 2 and a set E on dD which has zero linear measure and positive harmonic measure with respect to D [5]. McMillan and Piranian subsequently simplified the example [6].…”