REGULARITY AND FREE BOUNDARY REGULARITY FOR THE p-LAPLACE OPERATOR IN REIFENBERG FLAT AND AHLFORS REGULAR DOMAINS

Let ω(·) = ω(·, x) denote the harmonic measure associated to the Laplace operator and defined with respect to Ω and x ∈ Ω. A classical result concerning the harmonic measure, due to Lavrentiev [22], states that if Ω ⊂ R 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. Thus Pommerenke’s theorem gives a characterization of the set of all planar vanishing chord arc domains in terms of the behaviour of the Poisson kernel. Concerning higher-dimensional analogues of the results of Lavrentiev and Pommerenke, such results have recently been obtained by Kenig and Toro in a sequence of papers; see [17], [18], [19], [20]. In these papers the authors establish a number of results concerning the regularity and free boundary regularity, below the continuous threshold, for the Laplace equation in Reifenberg flat and Ahlfors regular domains. In particular, as an analogue of Pommerenke’s result, the authors obtain a characterization, in terms of the behaviour of the Poisson kernel, of what they refer to as ‘chord arc domains with vanishing constant’. For generalizations to divergence form elliptic operators with C-coefficients, we refer to [34].